On regular and chaotic dynamics of a non- PT -symmetric Hamiltonian system of a coupled Duffing oscillator with balanced loss and gain
aa r X i v : . [ n li n . C D ] S e p On regular and chaotic dynamics of a non-
P T -symmetricHamiltonian system of a coupled Duffing oscillator withbalanced loss and gain
Pijush K. Ghosh ∗ and Puspendu Roy † Department of Physics, Siksha-Bhavana,Visva-Bharati University,Santiniketan, PIN 731 235, India.
Abstract
A non- PT -symmetric Hamiltonian system of a Duffing oscillator coupled to an anti-damped oscillator with a variable angular frequency is shown to admit periodic solutions.The result implies that PT -symmetry of a Hamiltonian system with balanced loss and gain isnot necessary in order to admit periodic solutions. The Hamiltonian describes a multistabledynamical system —three out of five equilibrium points are stable. The dynamics of themodel is investigated in detail by using perturbative as well as numerical methods andshown to admit periodic solutions in some regions in the space of parameters. The phasetransition from periodic to unbounded solution is to be understood without any referenceto PT -symmetry. The numerical analysis reveals chaotic behaviour in the system beyonda critical value of the parameter that couples the Duffing oscillator to the anti-dampedharmonic oscillator, thereby providing the first example of Hamiltonian chaos in a systemwith balanced loss and gain. The method of multiple time-scales is used for investigatingthe system perturbatively. The dynamics of the amplitude in the leading order of theperturbation is governed by an effective dimer model with balanced loss and gain that isnon- PT -symmetric Hamiltonian system. The dimer model is solved exactly by using theStokes variables and shown to admit periodic solutions in some regions of the parameterspace. Keywords:
System with balanced loss and gain, Duffing oscillator, Chaos, Dimer Model ∗ email: [email protected] † email :[email protected] ontents PT -Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 Equilibrium Points and the Dirichlet Theorem . . . . . . . . . . . . . . . 82.3.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 PT -symmetric Dimer with balanced loss and gain 186 Conclusions & Discussions 227 Acknowledgements 238 Appendix-I: Perturbative solution for Γ ≪ , α ≪ The PT -symmetric classical Hamiltonian of coupled harmonic oscillators with balanced lossand gain admits periodic solution in the unbroken PT phase of the system[1]. The boundedsolution becomes unbounded as a result of phase-transition from an unbroken to a broken PT phase. The change in the nature of solutions accompanied by the PT phase-transition hasimportant physical consequences[2]. Further, the corresponding quantum system is well definedin appropriate Stokes wedges and admits bound states in the same unbroken PT phase[1]. Thishas lead to introduction of many PT -symmetric Hamiltonian systems with balanced loss andgain [3, 4, 5, 6, 7, 8, 9, 10]. The examples include many-particle systems[3, 7, 8, 9], systemswith nonlinear interaction[4, 5, 6, 7, 8, 9], systems with space-dependent loss-gain terms[8],systems with Lorentz interaction[10] etc.. The important results which are common to all thesemodels is that PT -symmetric Hamiltonian systems with balanced loss and gain may admitbounded and periodic solutions in some regions of the space of parameters. The bounded andunbounded solutions exist in the unbroken and broken PT -phases of the system, respectively.The corresponding quantum system admits bound states in well defined Stokes wedges. A few PT symmetric, but non-Hamiltonian classical systems with balanced loss and gain are also knownto share the above properties[11, 12].The investigations on classical Hamiltonian systems with balanced loss and gain are mostlyrestricted to PT symmetric models. One plausible reason is that PT -symmetric non-hermitianquantum systems admit entirely real spectra and unitary time-evolution in unbroken PT -phase[13].The same result is valid for a non- PT -symmetric Hamiltonian provided it is pseudo-hermitianwith respect to a positive-definite similarity operator[14] and/or admits an antilinear symmetry.A few examples of non- PT symmetric non-hermitian Hamiltonian admitting entirely real spec-tra and unitary time evolution may be found in the Refs.[14, 15, 16, 17, 18, 19, 20, 21]. It is2nown[7, 8] that systems involving a non-conventional T -symmetry also share the same proper-ties. In classical mechanics, there is no notion of pseudo-hermiticity or anti-linear symmetry andthe time-reversal symmetry is unique. Consequently, the criterion for a classical system withbalanced loss and gain to admit periodic solution is solely based on PT -symmetry.There are no compelling reasons to accept that only PT -symmetric classical Hamiltoniansystem with balanced loss and gain may admit periodic solutions. There should be enoughprovision for accommodating non- PT -symmetric classical Hamiltonian in the investigations onsystems with balanced loss and gain, which upon quantization may lead to a pseudo-hermitiansystem and/or a Hamiltonian with an anti-linear symmetry different from the PT -symmetry.It may be noted here that the Hamiltonian formulation of generic systems with balanced lossand gain does not require any assumption on an underlying discrete or continuous symmetry[7,8, 9, 10]. In particular, a Hamiltonian system in its standard formulation is necessarily non-dissipative, since the flow in the position-velocity state space preserves the volume. Thus, asystem for which individual degrees of freedom are subjected to loss or gain is Hamiltonian onlyif the net loss-gain is zero[10]. This condition for the case of constant loss and gain essentiallyimplies PT symmetry of the term H = ( P x + γy )( P y − γx ) that appears in the Hamiltonian H = H + V ( x, y ), where P x and P y are canonical conjugate momenta corresponding to thecoordinates x and y , respectively and γ is the gain-loss parameter[1, 3, 4, 5, 6, 7, 8, 9, 10].The Hamiltonian H is not even required to be PT symmetric for the case of space-dependentloss-gain terms, i.e. γ ≡ γ ( x, y )[9, 10]. The most important point is that the potential V ( x, y )need not be PT -symmetric as far as the Hamiltonian formulation of systems with balanced lossand gain is considered[7, 8, 9, 10]. Further, there is no general result in a model independentway to suggest that PT -symmetry of V ( x, y ) and hence, of H is necessary in order to haveperiodic solutions. On the contrary, it is known that non-Hamiltonian dimer model withoutany PT symmetry due to imbalanced loss and gain admits stable nonlinear supermodes[22].Similarly, within the mean field description of Bose-Einstein condensate, stationary ground stateis obtained for non- PT -symmetric confining potential[23]. This raises the possibility that non- PT -symmetric Hamiltonian system with balanced loss and gain may admit periodic solutionswith possible applications. It seems that non- PT symmetric Hamiltonian with balanced loss andgain has not been investigated so far for system with finite degrees of freedom.The purpose of this article is to investigate regular as well as chaotic dynamics of a non- PT symmetric Hamiltonian system with balanced loss and gain. In particular, one of the objectivesis to show that transition from periodic to unbounded solutions may be present in non- PT symmetric Hamiltonian systems with balanced loss and gain. The system is non- PT symmetricto start with and there is no question of attributing existence of these solutions to broken orunbroken PT -phases. Such transitions are quite common in generic dynamical systems with orwithout any specific symmetry and/or an underlying Hamiltonian structure. The inclusion ofloss-gain terms with a Hamiltonian description of the system does not make much difference.The standard techniques may determine the region in the parameter space in which periodicsolutions are obtained.The second objective is to investigate chaotic dynamics of a non- PT symmetric Hamiltoniansystem with balanced loss and gain. It may be noted in this regard that chaos in PT symmetricsystems has been studied earlier in different contexts[24, 25, 26]. Classical chaos has been studiedin complex phase space for the kicked rotor and the double pendulum in Ref. [24], while theemphasis is on quantum kicked rotor and top in Ref. [25]. Chaotic behaviour in PT -symmetricmodels in optomechanics and magnomechanics has also been investigated[26]. No generic featurerelating chaotic regime with that of broken or unbroken PT symmetry of the system is apparentfrom these investigations in a model independent way. Within this background, the requirementof PT symmetry appears to be too restrictive to explore chaotic behaviour in a larger class of3ystems with balanced loss and gain which may have possible technological applications. Thus,the emphasis is on non- PT -symmetric Hamiltonian system with balanced loss and gain.The Duffing oscillator is a prototype example in the study of nonlinear dynamics[27, 28, 29].It describes a damped harmonic oscillator with an additional cubic nonlinear term. The systemadmits different types of solutions in different regions of the space of parameters, including chaoticbehaviour if the system is subjected to an external forcing. A Hamiltonian formulation for theDuffing oscillator is not known for non-vanishing damping term. However, following the methodsdescribed in Refs. [7, 9, 10], a Hamiltonian may be obtained for a system where the Duffingoscillator is coupled to an anti-damped harmonic oscillator in a nontrivial way. The coupling tothe anti-damped oscillator effectively acts as a forcing term, thereby preparing the ground forinvestigating chaotic behaviour in the system. The dynamics of the Duffing oscillator completelydecouples from the system in a particular limit, while the dynamics of the anti-damped oscillatoris unidirectionally coupled to it. It should be emphasised here that even in this limit the anti-damped oscillator is not a time-reversed version of the Duffing oscillator. The system as a wholeis non- PT -symmetric by construction. This is the Hamiltonian system of balanced loss and gainthat will be studied in detail in this article for regular and chaotic dynamics.The coupled Duffing oscillator with balanced loss and gain is analyzed by perturbative aswell as numerical methods. The method of multiple scales[27, 30] is used to find approximatesolutions in the leading order of perturbation parameter. The solutions are periodic in a regionwhich is also obtained by linear stability analysis for the existence of periodic solution. It shouldbe mentioned here that the linear stability analysis may or may not be valid for a Hamiltoniansystem with nonlinear interaction. However, periodic solutions are also obtained in the sameregion by numerically solving the equations of motions. There are unbounded solutions outsidethis region. This is one of the main results of this article —a non- PT -symmetric Hamiltoniansystem with balanced loss and gain admits periodic solution. The phase transition from periodicto unbounded solutions is specific to the model without any reference to PT -symmetry. TheHamiltonian is a multistable system —three out of the five equilibrium points are stable. Thephase-space of the system has a rich structure. The bifurcation diagram of the system alsoshows chaotic behaviour beyond a critical value of the parameter that couples Duffing oscillatorto the anti-damped oscillator. The role of this coupling term is similar to the forcing term ofthe standard forced Duffing oscillator, albeit in a nontrivial way. The chaotic behaviour of thesystem is confirmed independently by various methods. The chaotic behaviour in the coupledDuffing oscillator system with balanced loss and gain is another important result.The present article also deals with a solvable non- PT -symmetric dimer Hamiltonian withbalanced loss and gain which admits periodic solutions. The model of dimer arises as a byproductof the perturbative analysis of the coupled Duffing oscillator model. In particular, differentchoices of the small parameters for implementing the perturbation scheme lead to different setsof equation for the time-evolution of the amplitude which may be interpreted as models of dimers.It is shown within this context that a model of non- PT -symmetric dimer with balanced loss andgain is exactly solvable and admits periodic solution in some regions of the parameter space.It seems that this is the first example of a non- PT -symmetric Hamiltonian dimer model withbalanced loss and gain that admits periodic solution.The plan of the article is the following. The model is introduced in Sec. 2. along withdiscussions on PT -symmetric limit of the system and linear stability analysis. A perturbativeanalysis of the system by using the method of multiple-scale analysis is presented in Sec. 3.The numerical result for the coupled Duffing-oscillator system is presented in Sec. 4. Non- PT -symmetric Hamiltonian of a dimer with balanced loss and gain is presented in Sec. 5. Finally,discussions on the results are presented in Sec. 6. Perturbative analysis of the model by treatingthe gain-loss coefficient and the coupling constant of the nonlinear interaction as small parameters4re presented in Appendix-I in Sec. 8. The system described by the equations of motion,¨ x + 2 γ ˙ x + ω x + β y + gx = 0 , ¨ y − γ ˙ y + ω y + β x + 3 gx y = 0 , (1)is a model of coupled oscillators with nonlinear interaction and subjected to gain and loss. Theparameters γ, ω and g correspond to the loss-gain strength, angular frequency of the harmonictrap and the nonlinear coupling strength, respectively. The linear coupling between the x and y degrees of freedom is denoted by the real constants β and β . The linear coupling is asymmetricfor β = β . The above equation describes a system of balanced loss and gain in the sense thatthe flow in the position-velocity state space preserves the volume, although individual degrees offreedom are subjected to gain or loss. The system admits a Hamiltonian, H = P x P y + γ ( yP y − xP x ) + (cid:0) ω − γ (cid:1) xy + 12 (cid:0) β x + β y (cid:1) + gx y, (2)where P x and P y are canonical momenta, P x = ˙ y − γy, P y = ˙ x + γx. (3)The equations of motion (1) may be obtained from H . The Hamiltonian H reduces to a systemof coupled harmonic oscillators with balanced loss and gain for g = 0 and β = β which admitsperiodic solutions[1] in the unbroken PT phase of the system. Generalisations of the coupledoscillators model of Ref. [1] have been considered earlier by including cubic nonlinearity andpreserving the PT -symmetry[4, 6] of the system. As will be shown below, the Hamiltonian H for g = 0 is not PT -symmetric and there is no question of a broken or unbroken PT phases of it.However, the system admits periodic solutions in restricted region of the parameter space. Thisis a major difference from previous investigations on models of coupled oscillators with balancedloss and gain, where the periodic solutions are attributed to unbroken PT -phases.The system described by Eqs. (1,2) has an interesting limit β = 0 for which the x degree offreedom completely decouples from the y degree of freedom and describes an unforced Duffingoscillator. The Hamiltonian H for β = 0 corresponds to the Hamiltonian of Duffing oscillator inan ambient space of two dimensions, where the auxiliary system is described in terms of y degreeof freedom and corresponds to a forced anti-damped harmonic oscillator with time-dependentfrequency. The time-dependence of the frequency ω + 3 gx is implicit via its dependence on x . Similarly, the time-dependence of the forcing term β x is determined by the solutions ofthe unforced Duffing oscillator. This paves the way for using well known tools and techniquesassociated with a Hamiltonian system to analyse Duffing oscillator analytically. For example,methods of canonical perturbation theory, canonical quantization, integrability, Hamiltonianchaos etc. can be used for investigating Duffing oscillator. The present article deals with onlythe dynamical behaviour of the system for β = 0.The distinction between ambient and target spaces ceases to exist for β = 0 and H constitutesa new class of system with balanced loss and gain. The system governed by H with generic valuesof the parameters may be interpreted as describing a non-standard forced Duffing oscillator withthe identification of β y in the first equation of (1) as the ‘forcing term’. The forcing term incase of standard Duffing oscillator can be chosen. However, for the case of non-standard Duffing5scillator it is determined in a nontrivial way from the second equation of (1) which is also coupledto the first equation. Eq. (1) can be also interpreted as two coupled undamped, unforced Duffingoscillators with velocity as well as space mediated coupling terms. In particular, Eq. (1) can berewritten as,¨ u + Ω + u + gu + (cid:26) γ ˙ v + β − β v − g v (cid:0) v − u (cid:1)(cid:27) = 0 , ¨ v + Ω − v + gv + (cid:26) γ ˙ u − β − β u − g u (cid:0) u − v (cid:1)(cid:27) = 0 , Ω ± ≡ ω ± β + β u, v ) defined by the relations, u = x + y √ , v = x − y √ , P u = P x + P y √ , P v = P x − P y √ . (5)In general, the coefficients Ω ± of the harmonic terms are different and becomes identical, Ω + =Ω − = ω for β = − β . Moreover, either Ω + or Ω − can be chosen to be zero for ω = − β + β or ω = β + β , respectively. The loss and gain terms are hidden in the ( u, v ) co-ordinate system andgive rise to velocity mediated coupling between the two Duffing oscillators. The space mediatedcoupling between the oscillators comprise of linear as well as nonlinear terms. The linear termvanishes for β = β and in the limit of vanishing strength of the nonlinear coupling betweenthe Duffing oscillators, i.e. g →
0, the system describes a linear system that has been studiedearlier[1]. The Hamiltonian in the ( u, v ) co-ordinate system has the following form: H u = 12 ( P u − γv ) −
12 ( P v + γu ) + Ω + u − Ω − v + g (cid:0) u − v (cid:1) + uv (cid:2) β − β + g ( u − v ) (cid:3) . (6)It is expected that some of the behaviours of the standard forced Duffing oscillator will persistfor the model under investigation. The ( u, v ) co-ordinates are used solely for the purpose ofinterpreting the model as coupled Duffing oscillators. The rest of the discussions in this articlewill be based on ( x, y ) co-ordinates. The following transformations are employed, t → ω − t, x → | β | − x, y → | β | − y, β = 0 , β = 0 , (7)in order to fix the independent scales in the system. This allows a reduction in total numberof independent parameters which is convenient for analyzing the system. The model can bedescribed in terms of three independent parameters Γ , β and α defined as,Γ = γω , β = p | β || β | ω , α = g | β | ω , (8)and the equations of motion have the following expressions:¨ x + 2Γ ˙ x + x + sgn( β ) βy + αx = 0 , ¨ y −
2Γ ˙ y + y + sgn( β ) βx + 3 αx y = 0 . (9)The sign-function sgn( x ) is defined for x = 0 as sgn( x ) = x | x | , x ∈ R . The limit to the linearsystem g → α →
0. The linear coupling between the two oscillator modeswith the strength β comprises of two distinct cases:6 Linear Symmetric Coupling(LSC) : The signs of the linear coupling terms in Eq. (9)are same for this case and it occurs either for (a) β , β > β , β <
0. It maybe noted that the linear coupling terms ( sgn( β ) βy, sgn( β ) βx ) appearing in Eq. (9)reduce to ( βy, βx ) and ( − βy, − βx ) for the case (a) and the case (b), respectively. Thesetwo cases correspond to positive and negative linear coupling strengths, since β >
0. It isapparent that Eq. (9) for the case (a) is related to the same equation with case (b) via thetransformation β → − β . Thus, it is suffice to consider the positive LSC only from whichthe results for the negative LSC may be obtained by taking β → − β . The effect of theasymmetric linear coupling between the oscillators in Eq. (1) is encoded in the nonlinearcoupling α through its dependence on | β | . • Linear Anti-symmetric Coupling(LAC) : A relative sign difference between the linearcoupling terms in Eq. (9) is termed as LAC which may be obtained either for β > , β < β < , β >
0. It can be shown that the linear model, i.e. α = 0, for the anti-symmetriccoupling does not admit any periodic solutions. Numerical analysis for the nonlinear modelin a limited region of the parameter space indicates that it may not admit periodic and/orstable solutions. An exhaustive numerical analysis is required to ascertain this which isbeyond the scope of this article and LAC will not be pursued further for perturbative andnumerical analysis.The scale transformation (7) implies, P x → ω √ β ˜ P x , P y → ω √ β ˜ P y , H → β − ˜ H, (10)where ˜ P x = ˙ y − Γ y, ˜ P y = ˙ x + Γ x and˜ H = ˜ P x ˜ P y + Γ (cid:16) y ˜ P y − x ˜ P x (cid:17) + (cid:0) − Γ (cid:1) xy + β (cid:2) sgn( β ) x + sgn( β ) y (cid:3) + αx y. (11)The Hamiltonian ˜ H and the equations of motion in (9) will be considered for further analysis andthe results in terms of the original variables may be obtained by inverse scale transformations.Defining generalized momenta Π x = ˜ P x + Γ y, Π y = ˜ P y − Γ x , the Hamiltonian can be rewrittenas, ˜ H = Π x Π y + V ( x, y ) , V ( x, y ) = xy + β (cid:2) sgn( β ) x + sgn( β ) y (cid:3) + αx y. (12)The Hamiltonian ˜ H or equivalently the energy E = ˙ x ˙ y + V ( x, y ) is a constant of motion, butneither semi-positive definite nor bounded from below. The energy may be bounded from belowfor specific orbits in the phase-space to be determined from the equations of motion. PT -Symmetry The Hamiltonian may be interpreted as a two dimensional system with a single particle or asystem of two particles in one dimension. The Hamiltonian H or ˜ H is not PT -symmetric foreither of the cases. For example, with the interpretation of ˜ H describing a system of two particlesin one dimension, the parity( P ) and T symmetry are defined as, T : t → − t, ˜ P x → − ˜ P x , ˜ P y → − ˜ P y P : x → − x, y → − y, ˜ P x → − ˜ P x , ˜ P y → − ˜ P y . (13)7he term linear in Γ is not invariant under P T symmetry. Similarly, the system is not invariantunder PT symmetry even if the parity ( P ) transformation in two dimensions is considered in itsmost general form: P : (cid:18) xy (cid:19) → (cid:18) x ′ y ′ (cid:19) = (cid:18) x cos θ + y sin θx sin θ − y cos θ (cid:19) , (cid:18) ˜ P x ˜ P y (cid:19) → (cid:18) ˜ P ′ x ˜ P ′ y (cid:19) = (cid:18) ˜ P x cos θ + ˜ P y sin θ ˜ P x sin θ − ˜ P y cos θ (cid:19) (14)where θ ∈ (0 , π ). The first term in ˜ H is invariant under T , while it is invariant under P onlyfor two distinct values of θ , namely, θ = π , π . It may be noted that θ = π corresponds to P : ( x, y ) → ( y, x ) , ( ˜ P x , ˜ P y ) → ( ˜ P y , ˜ P x ), while P : ( x, y ) → ( − y, − x ) , ( ˜ P x , ˜ P y ) → ( − ˜ P y , − ˜ P x )for θ = π . The second, third and the fourth terms of ˜ H in Eq. (11) are invariant under PT symmetry for these two values of θ and LSC. However, the nonlinear coupling term breaks PT symmetry for α = 0. It may be noted that H in Eq. (2) is not invariant under PT symmetryfor β = β even for vanishing nonlinear coupling, i.e. g = 0. The scale transformation plays animportant role for showing implicit PT invariance of H with g = 0 and LSC. The fourth termof ˜ H breaks PT symmetry for the LAC and is related to the result that H is not PT symmetricfor β = β . A non-vanishing nonlinear interaction necessarily breaks PT invariance for H and˜ H irrespective of LSC or LAC. The Hamilton’s equations of motion,˙ x = ˜ P y − Γ x, ˙ y = ˜ P x + Γ y, ˙˜ P x = − βx + Γ ˜ P x + (Γ − y − αx y, ˙˜ P y = − βy − Γ ˜ P y + (Γ − x − αx , (15)are equivalent to the coupled second order equations (9) with positive LSC. Results for negativeLSC may be obtained by taking β → − β . The equilibrium points and their stability may beanalyzed by employing standard techniques. In particular, the equilibrium points are determinedby the solutions of the algebraic equations obtained by putting the right hand side of Eq. (15)equal to zero. According to the Dirichlet theorem[31], an equilibrium point is stable providedthe second variation of the Hamiltonian is definite at that point. This is neither a necessarycondition nor the converse is true. If the application of the Dirichlet theorem leads to inconclusiveresults, a linear stability analysis may be performed, which is an approximate method. Themethod involves the study of time-evolution of small fluctuations around an equilibrium pointby keeping only linear terms. The quadratic and higher order fluctuations are neglected dueto its smallness. The resulting linear system of coupled differential equations can be solved tostudy the time-evolution of the fluctuations. A detailed classification of equilibrium points basedon the time-evolution of small fluctuation may be found in any standard reference on nonlineardifferential equation, including the Refs. [27, 32]. A Hamiltonian system admits either center i.e.closed orbit in the phase-space surrounding an equilibrium point or hyperbolic point signallinginstability[32]. The stable equilibrium point of a Hamiltonian system necessarily corresponds toa center. It should be noted that a center is not asymptotically stable. The system admits five equilibrium points P , P ± , P ± in the phase-space ( x, y, ˜ P x , ˜ P y ) of thesystem, which are determined by the solutions of the algebraic equations obtained by putting8he right hand side of Eq. (15) equal to zero. The equilibrium points are, P = (0 , , , , P ± = ( ± δ + , ± η + , ∓ Γ η + , ± Γ δ + ) , P ± = ( ± δ − , ± η − , ∓ Γ η − , ± Γ δ − ) , (16)where δ ± and η ± are defined as follows: δ ± = 1 √ α h − ± p β i , η ± = − δ ± β h ± p β i . (17)The points P ± are related to each other through the relation P ± = − P ∓ . The projections of thepoints P ± on the ‘ x − y ’-plane are related through a rotation by an angle π . This is a manifestationof the fact that Eq. (9) remains invariant under the transformation x → − x, y → − y for fixed α, β, Γ. Under the same transformation ˜ P x → − ˜ P x , ˜ P y → − ˜ P y , and Eq. (15) remains invariantunder ( x, y, ˜ P x , ˜ P y ) → ( − x, − y, − ˜ P x , − ˜ P y ). The relation P ± = − P ∓ may be explained in thesame way.The equilibrium point P exists all over the parameter space, while all other points exist inrestricted regions in the parameter space. In particular, • α > : P , P ± are equilibrium points for β >
1, while P is the only equilibrium point for0 < β ≤
1. Points P ± are non-existent, since δ ± and η ± are purely imaginary. • α < : P , P ± , P ± are equilibrium points for 0 < β <
1, while P , P ± are equilibriumpoints for β ≥ • α = : Only P is the equilibrium point.The critical points of the Hamiltonian ˜ H are also located at P , P ± , P ± , since the equilibriumpoints correspond to the solutions of the equations ˜ H Z ≡ ∂ ˜ H∂Z = 0 , Z ≡ ( x, y, ˜ P x , ˜ P y ). However,the Hessian ˜ H ZZ of H is not definite at these equilibrium points in any region of the parameter-space. For example, the four eigenvalues of ˜ H ZZ at P are determined as,12 h β + Γ ± (cid:8) − β ) + ( β + Γ ) (cid:9) i , h β − Γ ± (cid:8) β ) + ( β − Γ ) (cid:9) i . (18)The spectrum always consists of positive as well negative eigenvalues for fixed Γ and β — all thefour eigenvalues are neither semi-positive definite nor negative-definite simultaneously. Thus, thesecond variation of ˜ H is not definite at P and no conclusion can be drawn about its stabilityby using the Dirichlet theorem[31]. A similar analytical treatment for the points P ± and P ± become cumbersome. However, numerical investigations for some chosen values of the parametersindicate that the eigenvalues of the Hessian of H for none of the points are definite. In absence of any definite results on the stability of equilibrium points by the use of Dirichlettheorem, a linear stability analysis may be performed. Considering small fluctuations aroundan equilibrium point ( x , y , ˜ P x , ˜ P y ) in the phase space as ( x = x + ξ , y = y + ξ , ˜ P x =˜ P x + ξ , ˜ P y = ˜ P y + ξ ) and keeping only the terms linear in ξ i in Eq. (15), the followingequation is obtained,˙ ξ = M ξ, M = − Γ 0 0 10 Γ 1 0 − ( β + 6 αx y ) Γ − − αx Γ 0Γ − − αx − β − Γ , (19)9here ξ = ( ξ , ξ , ξ , ξ ) T and A T denotes transpose of A . The values of x , y , ˜ P x , ˜ P y differ foreach equilibrium point and may be substituted at an appropriate step of the stability analysis.The characteristic equation of the matrix M and its solutions ± iλ j , j = 1 , λ + 2 (cid:0) αx − (cid:1) λ + (1 + 3 αx ) − β − αβx y = 0 ,λ j = (cid:20) − + 3 αx − ( − j +1 q β + 4Γ − (1 + 3 αx ) + 6 αβx y (cid:21) . (20)The stable solutions are obtained in different regions of the parameter space for which λ j ∈ R ∀ j .Each equilibrium point is analyzed separately for its stability. • Point P : The equilibrium point P corresponds to x = y = 0 and is stable in a regionof parameter-space defined by the conditions, − √ < Γ < √ , (cid:0) − Γ (cid:1) < β < . (21)These inequalities are equivalent to the following conditions:12 < β < , − Γ < Γ < Γ , Γ ≡ √ q − p − β . (22)The point P is a stable equilibrium point for any α and restricted values of β and Γ specifiedby the condition (21). The system with α = 0 corresponds to the coupled oscillator modelof Ref. [1] and appears as linear part of the coupled Duffing oscillators models[4, 11]. Thestable equilibrium point P with the same stability condition has been found for all thesemodels. • Point P ± : It may be noted that x = δ and x y = δ + η + for both the points P ± .The eigenvalues λ j are same for both the points P ± and a simplified expression may beobtained as, λ j = − − + S − ( − j +1 √ p − S + 12Γ + 12Γ − S (12Γ − , j = 1 , , (23)where S = p β . Stable solutions for these two points exist in the same region inparameter space: β > , Γ ≤ √ − . (24)The equilibrium points P ± exist for α > , β > α < , < β <
1. Thus, P ± arestable for α > α <
0. The stable equilibrium points P ± are specific tothe nonlinear interaction of the model. • Point P ± : It may be noted that x = δ − and x y = δ − η − for both the points P ± . Stablesolutions for these two points do not exist anywhere in parameter space, since − λ j = 1 + 2Γ + S + ( − j +1 √ p − S + 12Γ + 12Γ + S (12Γ − , j = 1 , , (25)implies that at least one eigenvalue has a non-vanishing imaginary part.10he Hamiltonian describes a multistable system — the points P and P ± are stable for α > P and the condition (24) for P ± . For α <
0, only P is the stablepoint. No multistable system within the context of systems with balanced loss and gain has beenreported earlier.All the stable equilibrium points are ‘center’[27, 32], i.e. all the eigenvalues of M are purelyimaginary. It should be kept in mind that the linear stability analysis may or may not hold forcenter and/or a Hamiltonian system when the effect of the nonlinear interaction is considered[32].The result is only indicative for scanning a large parameter space in order to find bounded solutionby using perturbative and/or numerical methods. It will be seen that there are bounded solutionfor the model under investigation whenever conditions (21) or (24) are satisfied as well as in otherregions of the parameter space for which no information can be gained from the linear stabilityanalysis. Introducing the following matrices, X = (cid:18) xy (cid:19) , P = (cid:18) ββ (cid:19) , ˜ V ( x, y ) = (cid:18) x x y (cid:19) (26)and denoting the Pauli matrices as σ a , a = 1 , , σ taken to be diagonal, Eq. (9) withpositive LSC can be rewritten as,¨ X + 2Γ σ ˙ X + P X + α ˜ V ( x ) = 0 . (27)Results for negative LSC may be obtained by taking β → − β . The system of coupled linearoscillators with balanced loss-gain corresponds to α = 0 and is exactly solvable. The nonlinearinteraction is treated as perturbation for α ≪
1. The standard perturbation theory fails dueto the appearance of secular terms, which are unbounded in time and lead to divergences inthe long-time behaviour of the solutions. One of the possible remedies is to use the method ofmultiple time-scales[30] in which many time-variables are introduced temporarily by multiplyingthe original time t with different powers of α . In particular, the coordinates are expressed inpowers of the small parameter α and multiple time-scales are introduced as follows, T n = α n t, X = ∞ X n =0 α n X ( n ) ( T , T , . . . ) , X ( n ) = (cid:18) x n y n (cid:19) . (28)This introduces slow and fast time scales in the system. For example, T n +1 is always slower than T n , since α ≪
1. Using Eq. (28) in Eq. (27) and equating the terms with the same coefficient α n to zero, the following equations up to O ( α ) are obtained as follows: O ( α ) : ∂ X (0) ∂T + 2Γ σ ∂X (0) ∂T + P X (0) = 0 , (29) O ( α ) : ∂ X (1) ∂T + 2Γ σ ∂X (1) ∂T + P X (1) + 2 ∂ X (0) ∂T ∂T + 2Γ σ ∂X (0) ∂T + (cid:18) x x y (cid:19) = 0 . (30)The unperturbed Eq. (29) has the solution, X (0) = A e − iλ T (cid:18) η (cid:19) + B e − iλ T (cid:18) η (cid:19) + c.c., η j = 1 β (cid:0) λ j + 2 i Γ λ j − (cid:1) , (31)11here A ≡ A ( T , T , . . . ) and B ≡ B ( T , T , . . . ) are independent of T , but, depends onslower time scales T , T , . . . and c.c. denotes complex conjugate. The expressions for theeigenvalues λ , λ are given by Eq. (20) with x = 0 , y = 0, i.e. eigenvalues associated withthe stability of the point P . In order to find dependence of A and B on T , Eq. (30) is to besolved by eliminating secular terms.The O ( α ) Eq. (30) is a linear inhomogeneous equation and the complementary solution is ob-tained by replacing ( A , B ) → ( A , B ) in Eq. (31) describing X (0) , where A ≡ A ( T , T , . . . )and B ≡ B ( T , T , . . . ) are independent of T . The particular solution Y (1) is determined fromthe equation, ∂ Y (1) ∂T + 2Γ σ ∂Y (1) ∂T + P Y (1) = B, (32)where − B is the inhomogeneous part of Eq. (30). With the introduction of two complexparameters z j = Γ + iλ j and substituting the solution X (0) , B has the following expression: − B = e − iλ T z ∗ ∂A ∂T + 3 A (cid:16) | A | + 2 | B | (cid:17) − η z ∂A ∂T + 3 A n | A | (2 η + η ∗ ) + 2 | B | ( η + η + η ∗ ) o + e − iλ T z ∗ ∂B ∂T + 3 B (cid:16) | A | + | B | (cid:17) − z η ∂B ∂T + 3 B n | A | ( η + η ∗ + η ) + | B | (2 η + η ∗ ) o + e − iλ T A (cid:18) η (cid:19) + e − iλ T B (cid:18) η (cid:19) + e − i ( λ +2 λ ) T A B (cid:18) η + 2 η (cid:19) + e − i (2 λ + λ ) T A B (cid:18) η + η (cid:19) + e − i (2 λ − λ ) T A B ∗ (cid:18) η + η ∗ (cid:19) + e − i ( λ − λ ) T A ( B ∗ ) (cid:18) η + 2 η ∗ (cid:19) + c.c. (33)Eq. (32) is a linear homogeneous equation and solutions for each term in B can be obtainedseparately. The first two terms of B and their complex conjugates are secular and needs specialtreatment for obtaining solutions. Using Fredholm Alternative Theorem , the conditions forobtaining particular solutions corresponding to these two secular terms are, ∂A ∂T + 3 A (cid:16) Q | A | + 2 Q | B | (cid:17) = 0 ,∂B ∂T + 3 B (cid:16) Q | A | + Q | B | (cid:17) = 0 , (34)where the complex constants Q i ’s are given by, Q = 1 + η (2 η + η ∗ )2( z ∗ − z η ) , Q = 1 + η ( η + η + η ∗ )2( z ∗ − z η ) ,Q = 1 + η ( η + η ∗ + η )2( z ∗ − z η ) , Q = 1 + η (2 η + η ∗ )2( z ∗ − z η ) . (35) A system of linear equations Oξ s = B s admits solutions only if V † B s = 0 for all vectors V satisfying theequation O † V = 0, where a † denotes adjoint. The constant matrix O for the present case is obtained bysubstituting an ansatz for the particular solution Y (1) = ξ s e − iλ s T , s = 1 , e − iλ s T in Eq. (32).
12 general solution of Eq. (34) determines the T dependence of the constants ( A , B ) which isnot known for generic values of the Q i ’s which depend on the Γ and β . However, it can be shownnumerically that the Q i ’s are purely imaginary numbers for values of the Γ and β satisfyingthe condition (21) of linear stability. It is observed numerically that the Q Ri ∼ O (10 − ) ∀ i −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 Γ −505101520 Q R i , Q I i Q I1 Q I2 Q I3 Q I4 Q Ri (a) β = . β −10123 Q R i , Q I i Q I1 Q I2 Q I3 Q I4 Q Ri (b) Γ = . Figure 1: (Color online) Plots of the real( Q Ri ) and imaginary( Q Ii ) parts of the Q i ; (a) Q R,Ii versus Γ for β = .
99 and (b) Q R,Ii versus β for Γ = . Q Ri ∀ i merge to the horizontal axis forboth the cases.with an upper bound on the computational error of the same order, while the Q Ii take non-zerofinite values for the fixed β and − √ < Γ < √ , where the Q j is written in terms of real andimaginary parts as Q j = Q Rj + iQ Ij . Plots of the Q i ’s as a function of the Γ for fixed β = . Q i ’s versus β for Γ = . β satisfying (21) that the same results hold. Eq.(34) can be solved analytically by assuming Q Ri = 0 ∀ i for which | A | and | B | are constantsof motion. In particular, taking the constant values of A and B as their values at t = 0, i.e. A ≡ A (0) , B ≡ B (0), the solutions are obtained as, A ( t ) = | A (0) | e − iαt [ Q I | A (0) | +2 Q I | B (0) ] ,B ( t ) = | B (0) | e − iαt [ Q I | A (0) | + Q I | B (0) ] . (36)The expressions for Q Ii are not reproduced, since they are too long and does not add muchqualitative information to the discussions. The values of the constants | A (0) | and | B (0) | maybe fixed by using initial conditions. The approximate solution is, X = | A (0) | e − it [ λ +3 α ( Q I | A (0) | +2 Q I | B (0) | )] (cid:18) η (cid:19) + | B (0) | e − it [ λ +3 α ( Q I | A (0) | + Q I | B (0) | )] (cid:18) η (cid:19) + c.c. + O ( α ) . (37)The solution is bounded and consistent with linear stability analysis. The presence of multipletime scales in the solution is apparent, since phases and the amplitudes vary with differenttime scales. The solutions are uniform for t ≤ α − . The particular solutions of Eq. (32)can be obtained by substituting A ( t ) and B ( t ) in B . However, a complete solution which isuniform for t ≤ α − requires to find the time dependence of A ( T ) and B ( T ) appearing in thecomplementary solution of X (1) . This involves removing secular terms of differential equationsappearing at O ( α ), which is beyond the scope of this article.13 comment is in order regarding the choice of the perturbation terms. There are otherpossibilities for choosing small parameters to implement a perturbation scheme. A few viableperturbation schemes are, (a) β << , α <<
1, (b) Γ << , α << << , β << • Case (a) : A multiple scale analysis does not give any bounded solution. The zeroth ordersystem consists of damped and anti-damped oscillators without any coupling between thetwo. There are growing as well as decaying modes. This is consistent with the results oflinear stability analysis which predicts periodic solutions only for Γ < β . However, thiscondition is violated if β is treated as small parameter, while keeping Γ arbitrary. • Case (b) : A multiple scale analysis gives bounded solution that is consistent with lin-ear stability analysis. The perturbative analysis is valid for weak nonlinear interactioncharacterized by α << • Case (c) : The perturbation analysis is valid for strong as well as weak nonlinearity char-acterized by α . It may be noted that the discussion of this section as well the case (b)is restricted to weak α only. It also deserves special attention due to its relevance in thecontext of effective dimer model with balanced loss and gain which is treated separately inSec. 5.All three cases correspond to perturbation around the point P . A perturbation around thepoints P ± is not pursued in this article, since it involves use of the Jacobi elliptic functionsand the analysis of the equation governing the dynamics of the amplitude becomes nontrivial.However, numerical solutions around the points P ± are provided in the next section. The linear stability analysis of Eq. (9) predicts periodic solution in regions of the parameterspace defined by Eqs. (21) and (24). The perturbative solution obtained by using multiple timescale analysis is also periodic in the lowest order of the perturbation. In absence of any globalstability analysis, stability is not guaranteed for the complete Hamiltonian including nonlinearinteraction. Further, the system may admit chaotic solution, since the first equation of Eq. (9)may be interpreted as a forced Duffing oscillator with the identification of βy as a forcing termwhose profile is determined in a nontrivial way by the system itself. Thus, the system may admitchaotic solution for certain regions in the parameter space. In this section, regular as well aschaotic dynamics of Eq. (9) are studied numerically.The system is described in terms of three independent parameters Γ , β and α . Bifurcationdiagram may be investigated by varying one of these parameters and keeping the remaining twoparameters as fixed. The bifurcation diagram for varying β is presented in Fig-2 for Γ = 0 . α = . P . The plot ispresented for β ≥
0. However, it should be mentioned that the bifurcation diagram is symmetricwith respect to β = 0, if extended to negative values of β . The onset of chaos is seen for a criticalvalue β c ∼ .
05 and persists for β ≥ β c . The crossover from regular to chaotic dynamics as β is varied through β c may be understood by interpreting the x degree of freedom as describing aforced duffing oscillator with the identification of βy as the forcing term. Unlike the standardforced Duffing oscillator, the forcing is determined in a nontrivial way from the solution of thesystem. The chaotic behaviour of y degree of freedom is induced via its coupling to the x degreeof freedom. Regular and the chaotic dynamics of the system are studied in some detail in thenext two sections. It should be mentioned here that the numerical investigations have beencarried out for very large values of t ( ∼ , − , t = 0. However, for better14 x -4-2024681012 β y -101234567 Figure 2: (Color online) Bifurcation diagrams for β with Γ = 0 .
01 and α = . x (0) = 0 . y (0) = .
02, ˙ x (0) = .
03, ˙ y (0) = . t ( ∼ − The time-series of the dynamical variables in the vicinity of the point P is shown in Fig. (3)for Γ = . , β = . α = ±
1. Periodic solutions in Figs.(3a) and (3b) correspond to α = 1,while Figs. (3c) and (3d) correspond to α = −
1. It may be noted that the time evolution of t −0.6−0.4−0.20.00.20.40.6 x (a) α = 1 , β = . , Γ = . t −0.6−0.4−0.20.00.20.40.6 y (b) α = 1 , β = . , Γ = . t −0.6−0.4−0.20.00.20.40.6 x (c) α = − , β = . , Γ = . t −0.6−0.4−0.20.00.20.40.6 y (d) α = − , β = . , Γ = . Figure 3: (Color online) Regular solutions of Eq. (9) in the vicinity of the point P with theinitial conditions x (0) = . , y (0) = 0 . , ˙ x (0) = .
03 and ˙ y (0) = . , β show similar oscillatorybehaviour for positive as well as negative α . There are minute changes in amplitudes and phasesand that too in the limit of large t . It has been checked numerically that the same featurealso persists for smaller as well as higher values of α . The periodic solutions in the vicinity ofthe points P ± exist only for α >
0, confirming the results of the linear stability analysis. Thesolutions around P +1 are shown in Fig. (4) for α = 1 , β = 1 . , Γ = .
3. The Lyapunov exponentsand the autocorrelation functions for the time series representing the periodic solutions in Fig.(4) have been calculated to confirm that these solutions are indeed regular. The transition fromregular to chaotic behaviour is seen as β is increased beyond ˜ β c ∼ . α = 1 , γ = . x (0) = . , y (0) = − . , ˙ x (0) = .
02 and ˙ y (0) = .
03. It may be noted thatthe initial conditions for the bifurcation diagram in Fig.-2 is different from the initial conditionsused for periodic solutions around the point P in Fig.-4. Thus, the critical value of β c is differentfor the two cases. The equilibrium points P ± are related to each other as P − = − P +1 for fixed t −0.3−0.2−0.10.00.10.20.3 x (a) α = 1 , β = 1 . , Γ = . t −0.3−0.2−0.10.00.10.20.3 y (b) α = 1 , β = 1 . , Γ = . Figure 4: (Color online) Regular solutions of Eq. (9) in the vicinity of the point P +1 with theinitial conditions x (0) = . , y (0) = − . , ˙ x (0) = .
02 and ˙ y (0) = . α and β . Further, Eq. (9) remains invariant under x → − x, y → − y . Thus, the solutions aroundthe equilibrium point P − for the same values of the parameters α = 1 , β = 1 . , γ = .
3, butwith the initial conditions x (0) = − . , y (0) = 0 . , ˙ x (0) = − .
02 and ˙ y (0) = − .
03 may simply beobtained by taking mirror image of the plot in Fig. 4.(a) with respect to x = 0 and in Fig. 4.(b)with respect to y = 0. No separate numerical solution around P − is presented for this reason. The bifurcation diagram shows that the system with Γ = 0 .
01 and α = . β > β c =1 .
05. The sensitivity of the dynamical variables to the initial conditions are studied in differentregions of the parameters by considering two sets of initial conditions: (a) x (0) = . , y (0) =0 . , ˙ x (0) = . , ˙ y (0) = .
04 and (b) x (0) = . , y (0) = 0 . , ˙ x (0) = . , ˙ y (0) = . y (0) which differ by . β = 1 .
5. Thechaotic behaviour in the model has been confirmed by other independent methods also. In thisregard, the auto-correlation function, Lyapunov exponent, Poincar´ e section and power spectraare plotted in Fig. (6). The Lyapunov exponents are ( . , . , − . , − . − . 16 x ( t ) t (a) Γ = 0 . , β = 1 . , α = . -2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 0 20 40 60 80 100 y ( t ) t (b) Γ = 0 . , β = 1 . , α = . -1.5-1-0.5 0 0.5 1 1.5 10 20
30 40
50 60
70 80 90 100 x ( t ) t (c) Γ = 0 . , β = 1 . , α = 5 -1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 0 20 40 60 80 100 y ( t ) t (d) Γ = 0 . , β = 1 . , α = 5 Figure 5: (Color online) Chaotic solutions of Eq. (9) with two sets of initial conditions (a) x (0) = . , y (0) = 0 . , ˙ x (0) = . , ˙ y (0) = .
04 (violet color) and (b) x (0) = . , y (0) = 0 . , ˙ x (0) = . , ˙ y (0) = .
025 (green colour)The main emphasis of this article is on Hamiltonian system with balanced loss and gain.However, the Hamiltonian with Γ = 0, i.e. no gain-loss regime, deserves special attention due toits rich dynamical properties. For Γ = 0, the point P is stable for 0 < β <
1, while the points P ± are stable for β >
1. Further, the quantities Q i ’s are purely imaginary and Eq. (34) isexactly solvable without any assumptions on Q i ’s. It has been checked numerically that periodicsolutions of Eq. (9) exist for Γ = 0 near the equilibrium points P , P ± . The most important resultis that the Hamiltonian ˜ H with Γ = 0 is chaotic. The Poincar´ e section, Lyapunov exponents,autocorrelation functions and power spectra for Γ = 0 , α = . , β = 1 . x (0) = . , y (0) = . , ˙ x (0) = . , ˙ y (0) = .
04 are presented in Fig.-7. The Lyapunov exponentsare (0 . , . , − . , − . .
01 with all other conditions remaining the same. It may be noted that theDuffing oscillator admits chaotic solutions provided both damping and external driving termsare present. However, for the case of coupled Duffing oscillator model of this article, chaoticbehaviour is observed for the system without any loss-gain terms. The coupling to the linearoscillator with x -dependent angular frequency provides driving force to the Duffing oscillator.The Hamiltonian H u in Eq. (6) takes a simple form for γ = 0 , β = β : H u = (cid:18) P u + Ω + u + g u (cid:19) − (cid:18) P v + Ω − v + g v (cid:19) + g uv (cid:0) u − v (cid:1) , Ω ± = ω ± β . (38)describing two nonlinearly coupled undamped unforced Duffing oscillators. This also provides anexample of Hamiltonian chaos within a simple framework which deserves further investigations.17 -4 -3 -2 -1 0 1 2 3 ˙ y (a) Poincar´ e section: ˙ y ( t ) VS. y ( t ) plot t L y apuno v E x ponen t s -0.5-0.4-0.3-0.2-0.100.10.20.30.40.5 λ = λ = λ = -0.0016145 λ = -0.13244 (b) Lyapunov exponents A u t o c o rr e l a t i o n (c) Autocorrelation function of x ( t ) A u t o c o rr e l a t i o n (d) Autocorrelation function of y ( t ) M a g n i t u d e (e) Powerspectra of x ( t ) M a g n i t u d e (f) Powerspectra of y ( t ) Figure 6: (Color online) Poincar´ e section, Lyapunov exponents, autocorrelation function andpower spectra for Γ = 0 . , β = 1 . , α = . x (0) = . , y (0) =0 . , ˙ x (0) = . , ˙ y (0) = . P T -symmetric Dimer with balanced loss and gain
Different types of dimer models play an important role in many areas of physics and in particular,in the context of PT symmetric systems[4, 6, 12, 22, 23]. In this section, an exactly solvablenon- PT -symmetric Hamiltonian describing a dimer model with balanced loss and gain is shownto admit periodic solutions. A standard route to the occurrence of dimer models is via differentapproximation methods, including the multiple time scale analysis. The time-evolution of theamplitude in the leading order of the perturbation is described by dimer models. For the case ofthe coupled Duffing oscillator model, the resulting dimer models for α ≪ γ ≪ , α ≪ ≪ , β ≪ β are treated as small parameters with the identification of Γ ≡ ǫ Γ , β ≡ ǫ β , ǫ ≪
1. The strength of the nonlinear interaction α is kept arbitrary. The time scales andpower series expansion of the space co-ordinates in terms of ǫ are chosen as, T = t, T n = ǫ n t, n = 1 , , . . . , X = ∞ X n =0 ǫ n +1 X (2 n +1) , (39)so that there is no contribution from the nonlinear interaction in the lowest order. In particular,18 -4 -3 -2 -1 0 1 2 3 4 ˙ y -0.3-0.2-0.100.10.20.3 (a) Poincar´ e section: ˙ y ( t ) VS. y ( t ) plot t L y apuno v E x ponen t s -0.5-0.4-0.3-0.2-0.100.10.20.30.40.5 λ = λ = λ = -0.0043114 λ = -0.22685 (b) Lyapunov exponents A u t o c o rr e l a t i o n (c) Autocorrelation function of x ( t ) A u t o c o rr e l a t i o n (d) Autocorrelation function of y ( t ) M a g n i t u d e (e) Powerspectra of x ( t ) M a g n i t u d e (f) Powerspectra of y ( t ) Figure 7: (Color online) Poincar´ e section, Lyapunov exponents, autocorrelation functions andpower spectra for Γ = 0 , α = . , β = 1 . x (0) = . , y (0) = . , ˙ x (0) = . , ˙ y (0) = . O ( ǫ ) : ∂ X (1) ∂T + X (1) = 0 , (40) O ( ǫ ) : ∂ X (3) ∂T + X (3) + 2 ∂ X (1) ∂T ∂ + 2Γ σ ∂X (1) ∂T + β σ X (1) + α (cid:18) x x y (cid:19) = 0 . (41)The O ( ǫ ) equation describes two decoupled isotropic harmonic oscillators and the general solutionhas the form X (1) = A ( T , T , . . . ) e iT + c.c., A ≡ (cid:18) A ( T , . . . ) B ( T , . . . ) (cid:19) , (42)where the amplitude A depends on slower time scales. The time-evolution of A as a function of T is determined by the equation ,2 i ∂A∂T + (2 i Γ σ + β σ ) A + 3 α (cid:18) | A | A | A | B + A B ∗ (cid:19) = 0 , (43) A detailed derivation is skipped in order to avoid repetition. The necessary steps as outlined in Sec. 3 forthe case of α ≪ X (1) in Eq. (41) and removing the secular terms. Eq. (43) de-scribes a coupled dimer model with nonlinear interaction. The amplitudes A i ’s may be identifiedas wave propagating along the i th wave-guide.Eq. (43) may also be derived from the Lagrangian L or the corresponding Hamiltonian H , L = i (cid:16) A † σ ˙ A − ˙ A † σ A (cid:17) − A † σ A − β A † A − α | A | ( A B ∗ + A ∗ B ) , H = 2Γ A † σ A + β A † A + 3 α | A | ( A B ∗ + A ∗ B ) . (44)The canonically conjugate variables are ( A , iB ∗ ). The Hamiltonian system describes a nonlinearSchr¨ o dinger dimer with balanced loss and gain. The parameter Γ measures the strength ofthe loss-gain, while α is the coupling of the nonlinear interaction. The system differs withmost of the previous studies on Hamiltonian dimer model with balanced loss and gain in onerespect, it is not PT symmetric. The parity and time-reversal transformation are defined as, P : A → σ A, T : T → − T , i → − i . The Hamiltonian is PT symmetric for α = 0. Thenonlinear interaction breaks PT symmetry and H is non- PT -symmetric for α = 0.The Hamiltonian dimer is an integrable system. Introducing the Stokes variables, Z a = 2 A † σ a A, R = 2 A † A = q Z + Z + z , a = 1 , , , (45)it immediately follows from Eq. (43) that Z is the second integral of motion, i.e. ˙ Z = 0. Theconstant Z is fixed to its initial value at T = 0, i.e. Z (0) ≡ C . The dimer equations aretransformed into a set of coupled linear differential equation,˙ Z = N Z, Z ≡ Z Z R , N ≡ a b − a − cb − c , (46)where the parameters a, b, c are defined as, a = β + 3 αC , b = 3 αC , c = 2Γ . (47)One of the eigenvalues of the matrix N is zero and the remaining two eigenvalues are purelyimaginary numbers and complex conjugate of each other for real η : η = 0 , η = iη, η = − iη, η ≡ p a − b − c = s(cid:18) β + 32 αβ C (cid:19) − . (48)There are growing as well as decaying modes whenever η becomes imaginary. The parameter η is real for the following condition, β + 32 αβ C − ≥ . (49)It is interesting to note that for a fixed set of parameters the integration constant C may bealways chosen such that η is real. The general solutions has the expression, Z = C − ca − ba + C i − a ( ab − icη )+ b ( b + c )( b + c ) η − ab − icηb + c e − iηǫ t + C i a ( ab + icη ) − b ( b + c )( b + c ) η − ab + icηb + c e iηǫ t , (50)20here C , C , C are integration constants. The solutions for A and A may be obtained as, A = 12 p R + Z e iθ , A = 12 p R − Z e iθ , (51)where the phases are determined from the equations, dθ dT = β Z R + Z + 34 ( R + Z ) , θ = θ + tan − Z Z . (52)The expressions for the amplitudes | A i | and the relative phase θ − θ are derived from thedefining relations for the Stokes variables. On the other hand, time-evolution of θ is determinedfrom Eq. (43) by using A j = | A j | e iθ j , j = 1 ,
2. The four integration constants C i may bechosen appropriately to implement a variety of initial conditions. There may be restrictions onthe parameters for specified initial conditions such that | A | , | A | are semi-positive definite andphases are well defined.The system admits a stationary mode for which A , A are periodic in time with constantamplitudes. In particular, the amplitudes | A | , | A | are independent of time, while the phasesdepend on time. The stationary solution is obtained by choosing the integration constants as C = aη C , C = C = 0 for which Z T = C η ( − c, − b, a ) and the integration constant C maybe fixed through appropriate initial condition. The expressions for A , A corresponding to thisparticular solution of Z are obtained by using Eqs. (51) and (52), A = s C β η e i η C β η ǫ t ,A = s C η (2 β + 3 αC ) e i (cid:20) η C β η − tan − ( η ) (cid:21) ǫ t . (53)These solutions are physically acceptable in regions of the parameter-space determined by Eq.(49)along with the additional conditions:For α ≥ C β > , For α < < C β < β | α | . (54)It may be noted that C can always be chosen satisfying these conditions for any given set ofvalues for α, β , Γ . The power P i = | A i | for the i th wave-guide remains the same throughoutthe time-evolution, without being effected by the loss gain terms. Such a stationary mode, whichexists for PT -symmetric dimer models, is also seen in this non- PT -symmetric Hamiltoniansystem. Moreover, the allowed ranges of Γ can be varied at ease by choosing appropriate valueof the integration constant C for a fixed set of parameters α and β . This is an advantage overthe previous models.Solutions with time-dependent amplitude as well as phase can also be constructed. Forexample, the initial profile Z (0) T = (0 , ,
1) may be implemented by choosing C = α Γ (1 − β ) , C = a η , C = C = − b + c η for which Z ( t ) has the following expression: Z ( t ) = 1 η abη − a cos( ηǫ t ) − η ab − a + 1 η b − c sin( ηǫ t ) . (55)The solution for A , A may be determined by using the Eqs. (51) and (52). The Hamiltonian H is not PT symmetric, yet it admits periodic solutions. This ascertains that systems with21alanced loss and gain may admit periodic solutions without any PT symmetry of the governingHamiltonian. The periodic solutions become unbounded for the values of the parameter for which η is imaginary. The corresponding solutions in terms of hyperbolic functions may be obtainedby taking the limit η → iη in Z ( t ). & Discussions
It has been shown that a non- PT symmetric Hamiltonian system with balanced loss and gain mayadmit stable periodic solutions in some regions of the parameter-space. The result is importantfrom the viewpoint that all previous investigations are mainly based on PT -symmetric systemsin which the existence of stable periodic solution is attributed to the unbroken PT -phase. Therequirement of PT symmetry is too restrictive and there is no compelling reason for a systemwith balanced loss and gain to be PT -symmetric in order to admit stable periodic solutions.The result of this article paves the way for accommodating a large class of non- PT symmetricHamiltonian in the mainstream of investigations on systems with balanced loss and gain. Further,all the advantages associated with a Hamiltonian system may be used to explore such a modelin detail.A coupled Duffing oscillator Hamiltonian system with balanced loss and gain has been con-sidered as an example to present the results. The Duffing oscillator is coupled to an anti-dampedharmonic oscillator such that the coupling term effectively acts as a forcing term, albeit in anon-trivial way. The frequency of the anti-damped oscillator depends on the degree of freedomcorresponding to the Duffing oscillator. There is an interesting limit in which the dynamics ofthe Duffing oscillator completely decouples from the system, while the anti-damped oscillatoris unidirectionally coupled to it. This limit corresponds to a Hamiltonian formulation for thestandard Duffing oscillator. It should be emphasized that even in this limit the anti-dampedoscillator is not a time-reversed version of the standard Duffing oscillator. This opens the possi-bility of investigating the dynamics of the standard Duffing oscillator using techniques associatedwith a Hamiltonian system. Further, the quantum Duffing oscillator may also be introduced andinvestigated within the canonical quantization scheme.It has been shown that the coupled Duffing oscillator model admits stable periodic solutionin some regions of the parameter-space. The Hamiltonian is non- PT -symmetric and there is noquestion of attributing these periodic solutions to an unbroken PT -phase. These solutions areinvestigated by using perturbative as well as numerical methods. It is known that the drivenDuffing oscillator admits chaotic behaviour. The coupled Duffing oscillator model investigatedin this article also admits chaotic behaviour in some regions of the parameter-space where thecoupling to the anti-damped oscillator effectively acts as a driving term. This is an example of aHamiltonian chaos for systems with balanced loss and gain which has not been observed earlier.The method of multiple scale analysis has been used to investigate the system perturbatively.The amplitude depends on a slower time-scale than the phase and the dynamics of the amplitudeis determined by a set of coupled nonlinear equations which describe a dimer system. Theresulting dimer model in the leading order of the perturbation for small coupling β and loss-gain parameter Γ is also Hamiltonian and non- PT symmetric. Further, it is exactly solvableand admits stable periodic solutions in some regions of the parameters space. This provides anexample of a non- PT -symmetric dimer model admitting stable periodic solution. It should bementioned here that the dimer model obtained by considering the nonlinear coupling α as a smallparameter is also non- PT symmetric and no exact solutions can be found for the generic valuesof β and Γ. However, stable periodic solutions are obtained for Γ and β within a range specifiedby the linear stability analysis. It is known that dimer models with balanced loss and gain22re important in the field of optics and provide many counter-intuitive results. The examplesprovided in this article suggest that non- PT -symmetric systems should be included within theambit of the investigations on dimer models with balanced loss and gainIt is worth recalling some of the results pertaining to PT -symmetric quantum systems [13,14, 15, 16, 20, 21] to place the results obtained in this article in proper perspective. The generalunderstanding on non-hermitian quantum system is that it may admit entirely real spectra withunitary time-evolution provided at least one of the following conditions is satisfied: • The Hamiltonian is PT symmetric and unbroken PT -phase exists[13]. It may be noted inthis context that, unlike in the case of classical mechanics, the time-reversal symmetry isnot unique for quantum system. The non-conventional representation of the time-reversaloperator T has been used in the literature[7, 8]. • The Hamiltonian H is pseudo-hermitian with respect to a positive-definite similarity op-erator η , i.e. H † = ηHη − [14], where H † denotes the hermitian adjoint of H . The systemadmits an anti-linear symmetry[14] which may be identified as PT symmetry for some spe-cial cases. This allows to include non- PT -symmetric Hamiltonians with pseudo-hermiticityor with specific anti-linear symmetry in the main stream of investigations on non-hermitiansystems admitting entirely real spectra and unitary time-evolution[20, 21].The situation changes significantly for a classical system for which the time-reversal symmetryis unique and there is no analogue of pesudo-hermiticity or anti-linear symmetry for the classicalHamiltonian. It appears that the criterion based on PT -symmetry alone is not sufficient topredict the existence of periodic solution in a classical balanced loss-gain system. A possibleresolution of the problem may be to fix the criterion based on the corresponding quantum systemso that anti-linear symmetry and/or pseudo-hermiticity of the quantized Hamiltonian is used.However, an implementation of the scheme is tricky and nontrivial, since there may be more thanone quantum system for a given classical Hamiltonian based on the quantization condition. Aunique identification of the quantized Hamiltonian corresponding to a given classical system withbalanced loss and gain that admits periodic solution requires additional conditions to be imposed.The problem to fix an appropriate criterion for the existence of periodic solution in classicalsystem with balanced loss and gain remains unresolved and requires further investigations. This work of PKG is supported by a grant (
SERB Ref. No. MTR/2018/001036 ) fromthe Science & Engineering Research Board(SERB), Department of Science & Technology, Govt.of India under the
MATRICS scheme. The work of PR is supported by CSIR-NET fellow-ship(
CSIR File No.: 09/202(0072)/2017-EMR-I ) of Govt. of India. Γ ≪ , α ≪ Introducing a small parameter ǫ ≪ ǫ Γ , α = ǫα , Eq. (27) can be rewrittenas, ¨ X + P X + ǫ h σ ˙ X + α ˜ V ( x ) i = 0 . (56)The unperturbed part of the system is described by coupled harmonic oscillators satisfying theequation ¨ X + P X = 0. The terms with the coefficient ǫ in Eq. (56) is treated as perturbation,23hich contain the effect of loss-gain and nonlinear coupling. The standard perturbation theoryfails and the method of multiple time-scales will be employed to analyse Eq. (56). The coordi-nates are expressed in powers of the small parameter ǫ and multiple time-scales are introducedas follow, T n = ǫ n t, X = ∞ X n =0 ǫ n X ( n ) ( T , T , . . . ) . (57)Using Eq. (57) in Eq. (56) and equating the terms with same coefficient ǫ n to zero, the followingequations up to O ( ǫ ) are obtained as follows: O ( ǫ ) : ∂ X (0) ∂T + P X (0) = 0 , (58) O ( ǫ ) : ∂ X (1) ∂T + P X (1) + 2 ∂ X (0) ∂T ∂T + 2Γ σ ∂X (0) ∂T + α (cid:18) x x y (cid:19) = 0 . (59)These equations are to be solved consistently to get the perturbative results.The unperturbed Eq. (58) has the solution, X (0) = A e − iχ T (cid:18) β (cid:19) + B e − iχ T (cid:18) − β (cid:19) + c.c., χ = p β, χ = p − β. (60)The T dependence of A and B are determined by the equations, ∂A ∂T = − iα χ | A | A , ∂B ∂T = − iα χ | B | B , (61)which have been obtained by eliminating secular terms of Eq. (59). These two equations definea Hamiltonian system, H = 3 α " | A | χ + | B | χ , (62)with the canonical conjugate pairs as ( A , iA ∗ ) and ( B , iB ∗ ). It immediately follows that both | A | and | B | are constants of motion and the constant values are chosen to be their value at t = 0. The approximate solution of X is obtained as, X = | A (0) | e − it (cid:20) χ + α | A | χ (cid:21) (cid:18) β (cid:19) + | B (0) | e − it (cid:20) χ + α | B | χ (cid:21) (cid:18) − β (cid:19) + c.c. + O ( ǫ ) , (63)which is periodic and has uniform expansion for t ≤ ǫ − . It may be noted that α = α (cid:16) ΓΓ (cid:17) andthe solution inherits the effect of both the loss-gain and nonlinear interaction. References [1] C. M. Bender, M. Gianfreda, S. K. Ozdemir, B. Peng, and L. Yang, Twofold transition inPT-symmetric coupled oscillators, Phys. Rev. A , 062111 (2013).[2] B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori,C. M. Bender, and L. Yang, Paritytime-symmetric whispering-gallery microcavities, NaturePhysics,
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