Bifurcations and Amplitude Death from Distributed Delays in Coupled Landau-Stuart Oscillators and a Chaotic Parametrically Forced van der Pol-Rayleigh System
aa r X i v : . [ n li n . C D ] F e b February 14, 2020 1:42 main
International Journal of Bifurcation and Chaosc (cid:13)
World Scientific Publishing Company
Bifurcations and Amplitude Death from Distributed Delays inCoupled Landau-Stuart Oscillators and a Chaotic ParametricallyForced van der Pol-Rayleigh System
S. ROY CHOUDHURY
Department of Mathematics, University of Central Florida, AddressOrlando, Florida 32816, [email protected]
Ryan Roopnarain
Department of Mathematics, University of Central Florida, AddressOrlando, Florida 32816, [email protected]
Received (to be inserted by publisher)
Distributed delays modeled by ’weak generic kernels’ are introduced in the well-known cou-pled Landau-Stuart system, as well as a chaotic van der Pol-Rayleigh system with parametricforcing. The systems are closed via the ’linear chain trick’. Linear stability analysis of the sys-tems and conditions for Hopf bifurcation which initiates oscillations are investigated, includingderiving the normal form at bifurcation, and deducing the stability of the resulting limit cycleattractor. The value of the delay parameter a = a Hopf at Hopf bifurcation picks out the onsetof Amplitude Death(AD) in all three systems, with oscillations at larger values (correspondingto weaker delay). In the Landau-Stuart system, the Hopf-generated limit cycles for a > a
Hopf turn out to be remarkably stable under very large variations of all other system parameters be-yond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold,flip, transcritical or Neimark-Sacker bifurcations. This is to be expected as the correspondingundelayed systems are robust oscillators over very wide ranges of their respective parameters.Numerical simulations reveal strong distortion and rotation of the limit cycles in phase spaceas the parameters are pushed far into the post-Hopf regime, and also reveal other features,such as how the oscillation amplitudes and time periods of the physical variables on the limitcycle attractor change as the delay and other parameters are varied. For the chaotic system,very strong delays may still lead to the cessation of oscillations and the onset of AD (even forrelatively large values of the system forcing which tends to oppose this phenomenon). Varyingof the other important system parameter, the parametric excitation, leads to a rich sequence ofdynamical behaviors, with the bifurcations leading from one regime (or type of attractor) intothe next being carefully tracked.
Keywords : amplitude death; distributed delays; bifurcation analysis; chaotic attractor
1. Introduction
As is well-known, nonlinear dynamical systems, especially coupled ones, are of wide interest in manyareas of science and technology. When such systems which, in isolation are capable of a great variety of ebruary 14, 2020 1:42 main behaviors, are coupled, a host of novel phenomena are seen. These depend on the specific features, bothof the individual systems, as well as the type of coupling.One important area of application of such systems is what might imprecisely be referred to as ’stabi-lization’, i.e., the creation of simpler system attractors via the coupling. The best known among these issuppression of oscillations, most often termed as Amplitude Death (AD)[Saxena et al. , 2012] , even whenthe uncoupled systems themselves do not exhibit such stationary behavior. Coupling-induced AD is aninstance of a more general phenomenon that may include actual cessation of oscillations, or the conver-sion of chaotic dynamics to periodic or quasiperiodic dynamics. In the case of oscillation suppression bycoupling, two separate phenomena are now recognized. The first is suppression of oscillation to a sin-gle or homogeneous steady state (nowadays referred to as AD), versus the second or Oscillation Death(OD)[Koseska et al. , 2013] where the oscillators asymptotically populate different fixed points or ’inho-mogeneous steady states’, some of which may not have been stable, or perhaps not even present, for theuncoupled oscillators.Both AD and OD are known to occur in various settings. These are reviewed in [Saxena et al. ,2012] and [Koseska et al. , 2013], and include mismatched oscillators[Crowley & Field, 1981], [Bar-Eli,1984], [Bar-Eli, 2011], and [Koseska et al. , 2010], delayed interactions [Reddy et al. , 1998],[Reddy et al. ,1999],[Reddy et al. , 2000],[Reddy et al. , 2000], and [Senthilkumar & Kurths, 2010] (including distributeddelays[Atay, 2003] and cumulative signals[Saxena et al. , 2010] and [Saxena et al. , 2011]), conjugatecoupling[Kim, 2005], [Kim et al. , 2005], [Karnatak et al. , 2010], [Karnatak et al. , 2009], and [Zhang et al. ,2011], dynamic coupling [Konishi, 2003], nonlinear coupling[Prasad et al. , 2010] and [Prasad et al. , 2003],linear augmentation [Sharma et al. , 2011] and [Resmi et al. , 2010],velocity coupling [Saxena et al. , 2012],and other schemes.In this paper, we consider the effect of distributed delays on a variety of coupled systems carefully.While discrete delays have been considered in some detail, distributed delay effects are less-investigated,although they are known to provide stronger AD or OD effects. In order to facilitate analytical investiga-tion to the extent possible, we use the so-called ’chain trick’ together with the ’weak generic kernel’ formof distributed delay[Krise & Choudhury, 2003], [Cushing, 1977], and [MacDonald, 1978]. We consider theeffect of incorporating such delays in two different models viz. two different Van der Pol type oscillators,and a chaotic oscillator[Warminski, 2003].The remainder of this paper is organized as follows. Section 2 briefly reviews the linear stabilityanalysis of the two oscillator systems above in the absence of delay, while Section 3 repeats that analysiswith the inclusion of ’weak generic kernel’ delays in some of the nonlinear interaction terms, thus givinga first set of modifications of the dynamics. The normal form at Hopf bifurcation is derived in Section4. Section 5 then considers detailed numerical results contrasting the behavior of the undelayed systemsto the modifications created by the weak generic delays. Finally, Section 6 summarizes the results andconclusions.
2. Linear Stability
In this section we briefly recapitulate the linear stability of the undelayed systems we will be considering.
The Landau-Stuart Equation
The coupled Landau-Stuart system is given by [Reddy et al. , 1998],[Reddy et al. , 1999],[Reddy et al. ,2000],[Reddy et al. , 2000], and [Senthilkumar & Kurths, 2010] ˙ z ( t ) = (1 + iω − | z ( t ) | ) z ( t ) + ε ( z ( t ) − z ( t ))˙ z ( t ) = (1 + iω − | z ( t ) | ) z ( t ) + ε ( z ( t ) − z ( t )) (1)ebruary 14, 2020 1:42 main where z i ( t ) are complex and ω i > for i = 1 , and ε > . In order to work with the system we firstconvert it in to a real system by defining z k ( t ) = x k ( t ) + iy k ( t ) for each k = 1 , which gives: ˙ x = x − ω y − ( x + y ) x + ε ( x − x )˙ y = y + ω x − ( x + y ) y + ε ( y − y )˙ x = x − ω y − ( x + y ) x + ε ( x − x )˙ y = y + ω x − ( x + y ) y + ε ( y − y ) (2)The only fixed point of this system is the trivial one P : P = ( x , , y , , x , , y , ) = (0 , , , (3)The Jacobian matrix of (2) is given by: − ε − x − y − ω − x y ε ω − x y − ε − x − y εε − ε − x − y − ω − x y ε ω − x y − ε − x − y (4)and evaluating at the fixed point P gives: − ε − ω ε ω − ε εε − ε − ω ε ω − ε (5)The eigenvalues of this matrix then satisfy the characteristic equation (to be considered later) λ + ( − ε ) λ + (6 − ε + 4 ε + ω + ω ) λ + ( − ε − ε − ω + 2 εω − ω + 2 εω ) λ + (1 − ε + 4 ε + ω − εω + ε ω + 2 ε ω ω + ω − εω + ε ω + ω ω ) = 0 (6)which will be considered later. Chaotic System
The chaotic system we consider is a coupled van der Pol-Rayleigh oscillator system with parametricexcitation, and is given by[Warminski, 2003] ¨ x + ( − α + β ˙ x ) ˙ x + δ x + γ x + ( δ − µ cos(2 νt ))( x − y ) = q cos( νt )¨ y + M ( − α + β ˙ y ) ˙ y + M δ y + γ y − M ( δ − µ cos(2 νt ))( y − x ) = 0 (7)In order to work with the system we first convert it in to a first-order system by defining x ( t ) = x ( t ) , x ( t ) = ˙ x ( t ) , y ( t ) = y ( t ) , y ( t ) = ˙ y ( t ) which gives: ˙ x = x ˙ x = ( α − β x ) x − δ x − γ x − ( δ − µ cos(2 νt ))( x − y ) + q cos( νt )˙ y = y ˙ y = M ( α − β y ) y − M δ y − γ y + M ( δ − µ cos(2 νt ))( y − x ) (8)Considering the homogeneous system q = 0 , we find the fixed point: P = ( x , , x , , y , , y , ) = (0 , , , (9)ebruary 14, 2020 1:42 main and if, in addition, we have δ /γ = δ /γ < then there are two additional fixed points: P = ( x , , x , , y , , y , ) = s − δ γ , , s − δ γ , ! (10) P = ( x , , x , , y , , y , ) = − s − δ γ , , − s − δ γ , ! (11)Next we convert the system to an autonomous system by defining T ( t ) = t : ˙ T = 1˙ x = x ˙ x = ( α − β x ) x − δ x − γ x − ( δ − µ cos(2 νT ))( x − y
1) + q cos( νT )˙ y = y ˙ y = M ( α − β y ) y − M δ y − γ y + M ( δ − µ cos(2 νT ))( y − x ) (12)The Jacobian matrix of (12) is given by: c c α − β x δ − µ cos(2 νT ) 00 0 0 0 1 c M ( δ − µ cos(2 νT ) 0 c M ( α − β y ) (13)where c = − µν ( x − y ) sin(2 νT ) (14) c = − δ − δ − γ x + µ cos(2 νT ) (15) c = 2 M µν ( x − y ) sin(2 νT ) (16) c = M ( − δ − γ y − δ + µ cos(2 νT )) (17)and evaluating at the fixed point P gives: − δ − δ + µ cos(2 νT ) α δ − µ cos(2 νT ) 00 0 0 0 10 M ( δ − µ cos(2 νT ) 0 M ( − δ − δ + µ cos(2 νT )) M ( α − β y ) (18)The eigenvalues of this matrix then satisfy the characteristic equation which will be considered later λ ( λ + ( − α − α M ) λ + ( δ + δ + α α M − µ cos(2 νT )+ M ( δ + δ − µ cos(2 νT ))) λ + ( − α δ M − α δ M + α M µ cos(2 νT ) − α M ( δ + δ − µ cos(2 νT ))) λ + − M ( δ − µ cos(2 νT )) + δ M ( δ + δ − µ cos(2 νT )) + δ M ( δ + δ − µ cos(2 νT )) − M µ cos(2 νT )( δ + δ − µ cos(2 νT ))) = 0 (19)Next, evaluating the Jacobian at either of the fixed points P or P gives the matrix: δ − δ + µ cos(2 νT ) α δ − µ cos(2 νT ) 00 0 0 0 10 M ( δ − µ cos(2 νT ) 0 M (2 δ − δ + µ cos(2 νT )) M α (20)ebruary 14, 2020 1:42 main The eigenvalues of this matrix then satisfy the characteristic equation (to be considered later): λ ( λ + ( − α − α M ) λ + ( − δ + δ + α α M − µ cos(2 νT ) − M ( − δ + 2 δ + µ cos(2 νT ))) λ + (2 α δ M − α δ M + α M µ cos(2 νT ) + α M ( − δ + 2 δ + µ cos(2 νT ))) λ − M ( δ − µ cos(2 νT )) + 2 δ M ( − δ + 2 δ + µ cos(2 νT )) − δ M ( − δ + 2 δ + µ cos(2 νT )) + M µ cos(2 νT )( − δ + 2 δ + µ cos(2 νT ))) = 0 (21)
3. Linear Stability and Hopf Bifurcation Analysis of the Delayed Systems
In this section we introduce the delayed systems and perform the linear stability and Hopf bifurcationanalysis on them.
Delayed Landau-Stuart Equation
Now we consider here the case where the Landau-Stuart oscillators are coupled with a weak distributedtime delay in the first equation: ˙ z ( t ) = (1 + iω − | z ( t ) | ) z ( t ) + ε (cid:18)Z t −∞ z ( τ ) ae − a ( t − τ ) dτ − z ( t ) (cid:19) ˙ z ( t ) = (1 + iω − | z ( t ) | ) z ( t ) + ε ( z ( t ) − z ( t )) (22)By defining z ( t ) = Z t −∞ z ( τ ) ae − a ( t − τ ) dτ we can reduce the system (22) to the system of differential equations: ˙ z ( t ) = (1 + iω − | z ( t ) | ) z ( t ) + ε ( z ( t ) − z ( t ))˙ z ( t ) = (1 + iω − | z ( t ) | ) z ( t ) + ε ( z ( t ) − z ( t ))˙ z ( t ) = a ( z − z ) (23)As in the undelayed case, in order to work with this system we convert it to a real system by defining z k ( t ) = x k ( t ) + iy k ( t ) for each k = 1 , , , which gives: ˙ x = x − ω y − ( x + y ) x + ε ( x − x )˙ y = y + ω x − ( x + y ) y + ε ( y − y )˙ x = x − ω y − ( x + y ) x + ε ( x − x )˙ y = y + ω x − ( x + y ) y + ε ( y − y )˙ x = a ( x − x )˙ y = a ( y − y ) (24)The only fixed point of this system is the trivial one P : P = ( x , , y , , x , , y , , x , , y , ) = (0 , , , , , (25)The Jacobian matrix of (24) is: − ε − x − y − ω − x y ε ω − x y − ε − x − y εε − ε − x − y − ω − x y ε ω − x y − ε − x − y a − a
00 0 0 a − a (26)ebruary 14, 2020 1:42 main which, evaluated at the fixed point P , gives: − ε − ω ε ω − ε εε − ε − ω ε ω − ε a − a
00 0 0 a − a (27)The eigenvalues of this matrix satisfy the characteristic equation λ + b λ + b λ + b λ + b λ + b λ + b = 0 (28)where b = 2( − a + 2 ε ) b = 6 + a + 8 a ( − ε ) − ε + 6 ε + ω + ω b = 2(2 a ( − ε ) + ( − ε )(2 − ε + 2 ε + ω + ω )+ a (6 − ε + 5 ε + ω + ω )) b = (1 − ε + ε + ω )(1 − ε + ε + ω ) + 4 a ( − ε )(2 − ε + ε + ω + ω )+ a (6 − ε + 4 ε + ω + ω ) b = 2 a ( − ε + (1 + ω )(1 + ω ) − ε (2 + ω + ω ) + ε (5 + ω + ω ω + ω ) − a (2 + 4 ε + ω + ω − ε (6 + ω + ω ))) b = a ((1 + ω )(1 + ω ) − ε (2 + ω + ω ) + ε (4 + ω + 2 ω ω + ω )) (29)For P to be a stable fixed point within the linearized analysis, all the eigenvalues must have negativereal parts. From the Routh-Hurwitz criteria, the necessary and sufficient conditions for (24) to haveRe ( λ , , , , , < are: b > (30) b > (31) b b − b > (32) b ( b b + b ) − b − b b > (33) b ( b b b − b b + 2 b b − b b ) − b b − b + b b b + b ( − b + b b ) > (34) − b b b + b b b − b + b b − b b + b ( − b b + b b b + 2 b b b ) − b ( b b + b b ( − b b + b b ) + b ( − b b + 3 b b )) > (35)When the final condition (35) becomes an equality, the characteristic polynomial has one pair ofpurely imaginary complex conjugate roots. Here, we consider a to be the bifurcation parameter and de-note the left hand side of (35) by f ( a ) which is a ninth degree polynomial in a whose coefficients, whichare too large to include, depend on ω , ω , and ε . In order to solve the above conditions for parametersets possibly leading to a Hopf bifurcation, we must first fix a value for ε . Then, with our fixed value of ε , we reduce the conditions (30) to (34) along with the condition f ( a ) = 0 using computer algebra, to ob-tain conditions on the remaining parameters that may possibly lead to a Hopf bifurcation in the delayedsystem.For example, fixing ε = 2 , one of the several sets of conditions for a Hopf bifurcation we obtain is that < ω ≤ √ ebruary 14, 2020 1:42 main − ω − ω + s ω − ω (1 + ω ) < ω < ω + 2 √ and that a is the second root of the polynomial: ( −
12 + ω + ω + 22 ω ω + 2 ω ω + ω − ω ω + ω ω + 2 ω ω − ω ω + ω + ω ω )+ ( − − ω + 2 ω + 32 ω ω − ω ω − ω + 12 ω ω − ω ω + 2 ω ) x + ( −
216 + 6 ω + ω − ω ω − ω ω + 6 ω + 2 ω ω − ω ω + ω ) x + ( −
96 + 8 ω − ω ω + 8 ω ) x + ( −
12 + ω − ω ω + ω ) x In particular we can fix ω = 15 to obtain the condition . < ω < . . Then fixing ω , say to ω = 15 , we obtain that a be the second root of the polynomial − − x + 468 x + 8 x + x ) ,or a ≈ . . So we have that the parameter set ( ε, ω , ω , a ) = (2 , , , . possibly results in aHopf bifurcation in the delayed system. Delayed Chaotic System
Now we consider here the case where the Landau-Stuart oscillators are coupled with a weak distributedtime delay in the first equation: ¨ x + ( − α + β ˙ x ) ˙ x + δ x + γ x + ( δ − µ cos(2 νt ))( x − z ) = q cos( νt )¨ y + M ( − α + β ˙ y ) ˙ y + M δ y + γ y − M ( δ − µ cos(2 νt ))( y − x ) = 0 (36)where z ( t ) = Z t −∞ y ( τ ) ae − a ( t − τ ) dτ and we can reduce the system (36) to the system of differential equations: ¨ x + ( − α + β ˙ x ) ˙ x + δ x + γ x + ( δ − µ cos(2 νt ))( x − z ) = q cos( νt )¨ y + M ( − α + β ˙ y ) ˙ y + M δ y + γ y − M ( δ − µ cos(2 νt ))( y − x ) = 0˙ z − a ( y − z ) = 0 (37)As in the undelayed case, we first convert it in to a first-order system by defining x ( t ) = x ( t ) , x ( t ) =˙ x ( t ) , y ( t ) = y ( t ) , y ( t ) = ˙ y ( t ) which gives: ˙ x = x ˙ x = ( α − β x ) x − δ x − γ x − ( δ − µ cos(2 νt ))( x − z ) + q cos( νt )˙ y = y ˙ y = M ( α − β y ) y − M δ y − γ y + M ( δ − µ cos(2 νt ))( y − x )˙ z = a ( y − z ) (38)The fixed points of the delayed system are: P = P = ( x , , x , , y , , y , , z ) = (0 , , , , (39)and if, in addition, we have δ /γ = δ /γ < then there are two additional fixed points: P = ( x , , x , , y , , y , , z ) = s − δ γ , , s − δ γ , , s − δ γ ! (40) P = ( x , , x , , y , , y , , z ) = − s − δ γ , , − s − δ γ , , − s − δ γ ! (41) when the roots are ordered in increasing real part, with real roots listed before complex roots and complex conjugate pairs listednext to each other ebruary 14, 2020 1:42 main However, in what follows the parameter regimes we will consider will include the case δ /γ = δ /γ ,and so these two additional fixed points will not exist in our case. Next we convert the system to anautonomous system by defining T ( t ) = t : ˙ T = 1˙ x = x ˙ x = ( α − β x ) x − δ x − γ x − ( δ − µ cos(2 νt ))( x − z ) + q cos( νt )˙ y = y ˙ y = M ( α − β y ) y − M δ y − γ y + M ( δ − µ cos(2 νt ))( y − x )˙ z = a ( y − z ) (42)The Jacobian matrix of (42) is: c c α − β x δ − µ cos(2 νT )0 0 0 0 1 0 c M ( δ − µ cos(2 νT ) 0 c M ( α − β y ) 00 0 0 a − a (43)where c = − µν ( x − z ) sin(2 νT ) (44) c = − δ − δ − γ x + µ cos(2 νT ) (45) c = 2 M µν ( x − y ) sin(2 νT ) (46) c = M ( − δ − γ y − δ + µ cos(2 νT )) (47)Evaluating at the fixed point P of the original nonautonomous system gives: − δ − δ + µ cos(2 νT ) α δ − µ cos(2 νT )0 0 0 0 1 00 M ( δ − µ cos(2 νT ) 0 M ( − δ − δ + µ cos(2 νT )) M α
00 0 0 a − a (48)The eigenvalues of this matrix satisfy the characteristic equation λ ( λ + b λ + b λ + b λ + b λ + b ) = 0 (49)where b = a − α − α Mb = − aα − aα M + α α M + δ + δ M + δ + δ M − M µ cos(2 νT ) − µ cos(2 νT ) b = aα α M + aδ + aδ M + aδ + aδ M − aM µ cos(2 νT ) − aµ cos(2 νT ) − α δ M − α δ M + α M µ cos(2 νT ) − α δ M − α δ M + α M µ cos(2 νT ) b = − aα δ M − aα δ M + aα M µ cos(2 νT ) − aα δ M − aα δ M + aα M µ cos(2 νT ) + δ δ M + δ δ M − δ M µ cos(2 νT ) + δ M + δ δ M − δ M µ cos(2 νT ) − δ M µ cos(2 νT ) + M µ cos (2 νT ) b = aδ δ M + aδ δ M − aδ M µ cos(2 νT ) + aδ M + aδ δ M − aM ( δ − µ cos(2 νT )) − aδ M µ cos(2 νT ) − aδ M µ cos(2 νT ) + aM µ cos (2 νT ) (50)ebruary 14, 2020 1:42 main For P to be a stable fixed point within the linearized analysis, all the eigenvalues must have negative realparts. Since λ = 0 is a root of the characteristic polynomial, we can consider the remaining eigenvaluesby looking at the polynomial λ + b λ + b λ + b λ + b λ + b , and from the Routh-Hurwitz criterion,the necessary and sufficient conditions for the roots of this polynomial to have Re ( λ , , , , < are: b > (51) b > (52) b b − b > (53) b ( b b + b ) − b − b b > (54) b ( b b b − b b + 2 b b ) − b b − b + b b b − b b > (55)When the final condition (55) becomes an equality, the characteristic polynomial has one pair ofpurely imaginary complex conjugate roots. Here we consider the delay parameter a to be the bifurca-tion parameter. Denote the left hand side of (55) by f ( a ) , which is a fourth degree polynomial in a , andwhose coefficients, which are too large to include, depend on the remaining parameters. In order to solvethe above conditions for parameter regimes which contains a Hopf bifurcation, we fix values for all pa-rameters except µ and a . Then, with our fixed parameter values, we reduce the conditions (51) to (54)along with the condition f ( a ) = 0 using computer algebra. The objective is to either obtain conditions onthe µ and a that guarantee a Hopf bifurcation setting with a conjugate pair of imaginary roots, or see thata Hopf bifurcation is not possible for the chosen parameter values.In particular we will consider the following parameter set: α = 0 . , β = 0 . , γ = 3 . , α = 0 . , β = 0 . , γ = 3 . ,M = 0 . , δ = − . , δ = − . , ν = 2 . , δ = 0 . (56)Reducing our Routh-Hurwitz Conditions and Hopf Condition for these parameters with computeralgebra shows that for no values of µ, a, and T are all of the conditions satisfied. Thus the system does nothave a Hopf bifurcation for the above parameter values . However, note that a systematic parameter searchin Section 5 reveals a rich array of Hopf and other bifurcations, and various dynamical behaviors in oursystem.
4. Multiple Scales for the Delayed Landau-Stuart Equation
In this section, we will use the method of multiple scales to construct analytical approximations for theperiodic orbits arising through the Hopf bifurcation of the fixed point of the delayed Landau Stuart sys-tem 24 discussed above. The parameter a will be used as the bifurcation parameter. The limit cycle isdetermined by expanding about the fixed point using progressively slower time scales. The expansionstake the form x = x + X n =1 δ n x n ( T , T , T ) + ..., (57) y = y + X n =1 δ n y n ( T , T , T ) + ..., (58) z = z + X n =1 δ n z n ( T , T , T ) + ..., (59) x = x + X n =1 δ n x n ( T , T , T ) + ..., (60)ebruary 14, 2020 1:42 main y = y + X n =1 δ n y n ( T , T , T ) + ..., (61) z = z + X n =1 δ n z n ( T , T , T ) + ..., (62)where T n = δ n t and δ is a small positive non-dimensional parameter that is introduced as a bookkeepingdevice and will be set to unity in the final analysis. Utilizing the chain rule, the time derivative becomes ddt = D + δD + δ D + δ D ..., (63)where D n = ∂/∂T n . Using the standard expansion for Hopf bifurcations, the delay parameter a is orderedas a = a + X n =1 δ n a n ( T , T , T ) + ..., , (64)where a is given by satisfying the Routh-Hurwitz conditions (30) to (34) and (35) with equality. Thisallows the influence from the nonlinear terms and the control parameter to occur at the same order.Using (57)-(64) in (24) and equating like powers of δ yields equations at O ( δ i ) , i = 1 , , of the form : L ( x i , y i , z i , x i , y i , z i ) = S i, (65) L ( x i , y i , z i , x i , y i , z i ) = S i, (66) L ( x i , y i , z i , x i , y i , z i ) = S i, (67) L ( x i , y i , z i , x i , y i , z i ) = S i, (68) L ( x i , y i , z i , x i , y i , z i ) = S i, (69) L ( x i , y i , z i , x i , y i , z i ) = S i, (70)where the L i , i = 1 , , , , , are the differential operators L ( x i , y i , z i , x i , y i , z i ) = D x i + ( ε − x i − εx i + ω y i (71) L ( x i , y i , z i , x i , y i , z i ) = D y i + ( ε − y i − εy i − ω x i (72) L ( x i , y i , z i , x i , y i , z i ) = D x i + ( ε − x i − εx i + ω y i (73) L ( x i , y i , z i , x i , y i , z i ) = D y i + ( ε − y i − εy i − ω x i (74) L ( x i , y i , z i , x i , y i , z i ) = D x i + a ( x i − x i ) (75) L ( x i , y i , z i , x i , y i , z i ) = D y i + a ( y i − y i ) (76)The source terms S i,j for i = 1 , , and j = 1 , , , , , i.e. at O ( δ ) , O ( δ ) , and O ( δ ) are given asfollows. The first order sources S ,j = 0 for j = 1 , , , , , . The second order sources are: S = − D x S = − D y S = − D x S = − D y S = − D x + a ( x − x ) S = − D y + a ( y − y ) (77)ebruary 14, 2020 1:42 main and the third order sources are: S = − D x − D x − x y − x S = − D y − D y − x y − y S = − D x − D x − x y − x S = − D y − D y − x y − y S = − D x − D x + a ( x − x ) + a ( x − x ) S , = − D y − D y + a ( y − y ) + a ( y − y ) (78)Next, equation (70) may be solved for y i in terms of y i . Using this in (68), we can solve for y i in terms of y i and x i . Then, we replace y i in (66) and add ω /ε multiplied by (67) to (66) which thenenables us to solve for x i in terms of y i . Next, replacing x i and y i in equation (65), we can solve for x i in terms of y i and x i . Then in (69) we can replace x i and add to it ω ω /ε multiplied by (67), which thenallows us to solve for x i in terms of y i . Finally, using these relations in equation (67) gives the compositeequation L c w i = Γ i (79)where L c = D + β D + β D + β D + β D + β D + β (80)and β = − a + 4 εβ = 6 + a + 8 a ( ε − − ε + 6 ε + ω + ω β = 2(2 a ( ε −
1) + ( ε − − ε + 2 ε + ω + ω )+ a (6 − ε + 5 ε + ω + ω )) β = (1 − ε + ε + ω )(1 − ε + ε + ω ) + 4 a ( ε − − ε + ε + ω + ω )+ a (6 − ε + 4 ε + ω + ω ) β = 2 a ( − ε + (1 + ω )(1 + ω ) − ε (2 + ω + ω ) + ε (5 + ω + ω ω + ω ) − a (2 + 4 ε + ω + ω − ε (6 + ω + ω ))) β = a ((1 + ω )(1 + ω ) − ε (2 + ω + ω ) + ε (4 + ω + 2 ω ω + ω )) The composite source Γ i is equal to r S i + r S i + r S i + r S i + r S i + r S i + r D S i + r D S i + r D S i + r D S i + r D S i + r D S i + r D S i + r D S i + r D S i + r D S i + r D S i + r D S i + r D S i + r D S i + r D S i − a D S i + (4 − a − ε ) D S i − D S i whereebruary 14, 2020 1:42 main r = − a ( ε − ε ( ω + ω ) r = a ε ( − ε + ω ω ) r = − a ( − εω + (1 + ω ) ω + ε ( ω + ω )) r = a (1 + 2 ε + ω − ε (3 + ω )) r = − a ( ε − ε ( ω + ω ) r = a (2 ε − (1 + ω )(1 + ω ) + 2 ε (2 + ω + ω ) − ε (5 + ω + ω ω + ω )) r = − a ε ( ε − a )( ω + ω ) r = a ε ( − − a ( ε −
1) + 2 ε − ε + ω ω ) r = − a (1 + 2 a ( ε − − ε + ε + ω ) ω r = − a (( ε − − ε + ε + ω ) + a (3 − ε + 2 ε + ω )) r = − a ε ( ω + ω ) r = − (1 − ε + ε + ω )(1 − ε + ε + ω ) − a ( ε − − ε + ε + ω + ω ) r = − a ε ( ω + ω ) r = − a ε ( − a + 2 ε ) r = − a ( − a + 2 ε ) ω r = − a (3 + 3 a ( ε − − ε + 3 ε + ω ) r = − ε − − ε + 2 ε + ω + ω ) − a (6 − ε + 5 ε + ω + ω ) r = − a εr = − a ω r = − a ( − a + 3 ε ) r = − − a ( ε −
1) + 12 ε − ε − ω − ω We use (79) later to identify and suppress secular terms in the solutions of (65)-(70)Let us now turn to finding the solutions of (65)-(70), solving order by order in the usual way.For i = 1 or O ( ǫ ) we know S ,k = 0 for k = 1 , ..., . Hence we pick up a solution for the first order fieldsusing the eigenvalues (from the previous section) at Hopf bifurcation, which we denote λ = iω and it’scomplex conjugate λ , i.e. y = α [ T , T , T ] e − iωt + β [ T , T , T ] e iωt (81)where β = ¯ α is the complex conjugate of α since λ = ¯ λ and y is real. As is evident, the α and β modescorrespond to the center manifold where λ , are purely imaginary and where the Hopf bifurcation occurs.Since we wish to construct and analyze the stability of the periodic orbits which lie in the center manifold,we suppress the other eigenvalues with non-zero real parts.Using (81) in (65)-(70) for i = 1 and the process used to derive the composite equation we have: y = e − iωT a (cid:0) ( a − iω ) α [ T , T , T
3] + e iωT ( a + iω ) β [ T , T , T (cid:1) (82)ebruary 14, 2020 1:42 main x = e − iωT a ( ω + ω ) ( a ( ε + ω ω ) − ( ε − ω ω ) (cid:18) iω (cid:0) a + 6 a ( ε −
1) + 3 ε − ε + ω + ω + 3 (cid:1) (cid:0) α ( T , T , T ) − e iωT β ( T , T , T ) (cid:1) − ω (cid:0) a ( ε −
1) + a (cid:0) ε − ε + 2 ω + ω ω + 2 ω + 6 (cid:1) +( ε − (cid:0) ε − ε + ω − ω ω + ω + 1 (cid:1)(cid:1) (cid:0) α ( T , T , T ) + e iωT β ( T , T , T ) (cid:1) − iω (cid:0) a (cid:0) ε − ε + ω + ω ω + ω + 3 (cid:1) + a ( ε − (cid:0) ε − ε + 2 (cid:0) ω + ω + 1 (cid:1)(cid:1) + ω ω (cid:0) − ε + 2 ε + ω ω − (cid:1)(cid:1) (cid:0) α ( T , T , T ) − e iωT β ( T , T , T ) (cid:1) − a (cid:0) a (cid:0) ε − ε (cid:0) ω + ω ω + ω + 3 (cid:1) + ω + ω ω + ω + 1 (cid:1) − ω ω (2 ε + ω ω − (cid:0) α ( T , T , T ) + e iωT β ( T , T , T ) (cid:1) + ω (2 a + 3 ε − (cid:0) α ( T , T , T ) + e iωT β ( T , T , T ) (cid:1) − iω α ( T , T , T )+ iω e iωT β ( T , T , T ) (cid:19) (83) x = e − iωT a ε ( ω + ω ) (cid:18) (cid:0) a (cid:0) ε ( − − iω ) − ω + 2 iω + ω + 1 (cid:1) − iω (cid:0) ε + ε ( − − iω ) − ω + 2 iω + ω + 1 (cid:1)(cid:1) α ( T , T , T )+ e iωT (cid:0) a (cid:0) iε ( ω + i ) − ω − iω + ω + 1 (cid:1) + iω (cid:0) ε + 2 iε ( ω + i ) − ω − iω + ω + 1 (cid:1)(cid:1) β ( T , T , T ) (cid:19) (84)where we have omitted y and x as the expressions for them are too long to include. Now that the firstorder solutions are known, the second-order sources S , S , S , S , S , S may be evaluated using(77). Computing the second-order composite source Γ , we find that the entire source is secular and thatthe Setting the coefficients of the secular e ± iωt terms in these sources to zero yields D α = ∂α∂T = 0 , D β = ∂β∂T = 0 (85)Next, using the second-order sources, and (85) , the second-order particular solution is taken in theusual form to balance the zeroth and second harmonic terms at this order, i.e., y = y , + y , e iωt (86)Then since the entire second order source was secular, upon removing the secular terms with (85) wefind the second order source is now zero. Thus using (86) in (79) for i = 2 we find the coefficients in thesecond-order particular solution are y , = y , = 0 , thus y = 0 . Then using y in (65)-(70) for i = 2 ,together with the second-order sources, yields that the other second-order fields are also zero, y = y = x = x = x = 0 Using these, together with the first-order results, we may evaluate the coefficients of the secular termsin the composite source Γ , from (78) and (79). Suppressing these secular, first-harmonic, terms to obtainuniform expansions yields the final equation for the evolution of the coefficients in the linear solutions onthe slow second-order time scales ∂β∂T = C β + C αβ (87)ebruary 14, 2020 1:42 main where the very large expressions for the coefficients C i are omitted for the sake of brevity.This equation (87) is the normal form, or simplified system in the center-manifold, in the vicinityof the Hopf bifurcation point. We shall now proceed to compare the predictions for the post-bifurcationdynamics from this normal form with actual numerical simulations.
5. Numerical Results and Discussion5.1.
Landau-Stuart Equation
We may immediately make two additional points here regarding the Hopf bifurcation. In most systems[Krise & Choudhury, 2003], the Hopf bifurcation may occur either below or above the critical value ofthe system’s chosen bifurcation parameter, and one needs to test which in fact occurs. Since we havechosen the delay a as bifurcation parameter, and larger delays or lower a values have a stabilizing effect,we know that for our delayed Landau-Stuart system, the post-Hopf regime is for a values larger thanthe a Hopf value found using the second root of the polynomial in the last equation of Section 3.1. For a < a
Hopf , the strong delay stabilizes the oscillations and yields a stable fixed point. This is thus theregime of Amplitude Death(AD) for the system caused by the delay. The a = a Hopf point is thus theexact value of the delay parameter where AD sets in, and this may be precisely pinpointed here via thesemi-analytic treatment in Section 3.1.Note also that, in principle, the Hopf bifurcation might be either supercritical with stable oscillationsseen above a = a Hopf or at weaker delays, or subcritical where the Hopf-created periodic orbit is unstableand coexists with the stable fixed point in the a < a
Hopf or Amplitude Death regime. In the latter case,there would be no nearby system attractor for a > a
Hopf , and the dynamics in that regime would featureany of the three following scenarios: a. jumping to a distant periodic attractor if one exists, b. flying offto infinity in finite time (an attractor at infinity), or c. an aperiodic attractor on which the system orbitsevolve.However, we may plausibly rule out the occurrence of this latter, subcritical Hopf scenario. This isbecause the undelayed Landau-Stuart system is a robust oscillator showing stable periodic behavior, that,under the effect of delay, persists in the a > a
Hopf regime of a post-supercritical Hopf bifurcation, whilebeing reduced to Amplitude Death by stronger delays for a < a
Hopf . This does in fact turn out to becorrect, as will be verified below via both the normal form and numerical simulations.By approximating the flow of the system in a computer model, we can easily analyze the behavior ofthe system for various sets of parameters. Here we will consider the case in section 3.1 where ε = 2 , ω =15 , and ω = 15 and values of a around the Hopf bifurcation value a Hopf ≈ . . Fig. 1. Periodic oscillations in y for a = 10 . ebruary 14, 2020 1:42 main ( x , y , y ) phase space for the parameters of Figure 1 and the approach from the initial conditions.Fig. 3. The smaller delayed limit cycle in red and undelayed limit cycle in blue plotted in ( x , y , y ) phase space for theparameters of Figure 1. Figures 1 through 3 show the limit cycle for a = 10 above the Hopf bifurcation value a Hopf . Aspredicted from the normal form, and our plausibility argument above, we have stable periodic behaviorabove the bifurcation point as shown in Figure 1 for y ( t ) . Figure 2 shows the limit cycle in ( x , y , y ) phase space and the approach from the initial conditions. Figure 3 shows both the delayed (in red) andundelayed (in blue) limit cycles in ( x , y , y ) phase space from which we can see the stabilizing effect ofebruary 14, 2020 1:42 main the delay causing the limit cycle to shrink towards the fixed point at the origin, as well as rotate in phasespace.Figure 4 shows the limit cycle for a = 5 . just above the bifurcation point a Hopf in red and theundelayed system in blue in ( x , y , y ) phase space. Here we can see that, as we further decrease theparameter a towards the bifurcation value or increase the delay, the limit cycle continues to shrink towardsthe fixed point at the origin. Fig. 4. The delayed limit cycle in red and undelayed limit cycle in blue plotted in ( x , y , y ) phase space for a = 5 . . Next, Figures 5 and 6 show the delayed solution for an even larger delay a = 5 . which is now belowthe bifurcation value a Hopf . Here, we see the system exhibit Amplitude Death as the solutions spiraltowards the now stabilized origin. - × - × - ty1 Fig. 5. Amplitude death in y for a = 5 . . ebruary 14, 2020 1:42 main ( x , y , y ) phase spacefor a = 5 . . Finally Figure 7 shows the delayed time series for y when a = 2 . Figure 8 shows both the delayedsolution in red and the undelayed solution in blue, as well as their approach from the initial conditions,where the delayed system again exhibits Amplitude Death. We also observe that the smaller the value of a , or the greater the delay, the faster the approach to the origin. Fig. 7. Amplitude death in y for a = 2 . ebruary 14, 2020 1:42 main ( x , y , y ) phase spacefor a = 2 . In this delayed system, as mentioned above, the limit cycles in the a > a
Hopf regime are very robust,as one might expect since the undelayed Landau-Stuart system is well-known to demonstrate stable peri-odic behavior over wide ranges of the system parameters. However, it is quite possible that these robustlimit cycles might be quickly disrupted by secondary symmetry breaking, cyclic-fold, flip, transcritical, orNeimark-Sacker bifurcations when some other system parameter is changed. To investigate this, for cho-sen values of a well above a Hopf , we varied the other system parameters deep into this post-Hopf regime,i.e. far from the starting values ε = 2 , ω = 15 , and ω = 15 used above. The post-supercritical Hopflimit cycle proves extremely robust under variation of all three of these parameters. No further complexdynamics arises in this delayed system from additional bifurcations of the Hopf-created limit cycles, notsurprisingly since the undelayed Landau-Stuart system is a stable oscillator over a wide range of theseparameters. Chaotic System
Since our preliminary search for a Hopf bifurcation yielded a negative result for one set of parameters, letus first vary the value of the delay parameter a and study its effect on the system. While the effect of delaycan be predicted to be stabilizing, a much more complex set of dynamical behaviors occurs for this case,including a rich array of evolving system attractors as a , as well as other system parameters, are varied.Hence, the latter part of this sub-section will also systematically consider the bifurcations and dynamicsas the other important parameter µ , which measures the strength of the parametric excitation, is varied.This will systematically reveal a variety of dynamical behaviors. Chaotic Case µ = 0 . Figure 9 show solutions in ( x , x , y ) phase space of the delayed attractor in red and undelayed attractorin blue in the chaotic case of µ = 0 . (having one positive Lyapunov exponent, three negative exponents,and a fifth one along the time coordinate and hence always having value zero). We first consider theebruary 14, 2020 1:42 main system in the absence of forcing ( q = 0 ) as values of the delay parameter a range from a = 0 . to a = 10 .Here we observe 3 types of behavior as we vary a , the first being a cocoon shaped structure surroundingthe undelayed attractor which occurs for a = 0 . to a = 2 , a = 5 . to a = 10 . The second type of behavioris a double loop type structure for the delayed solutions, again surrounding the undelayed attractor, andoccurring in two different ways, the first oriented as for a = 3 and the second oriented as in the case a = 4 (a rotated version of a = 3 ). The final type of behavior is the case a = 3 . where we see a slightly morecomplicated looping structure surrounding the undelayed attractor. Fig. 9. The delayed (red) and undelayed (blue) solutions of the system in the chaotic case ( µ = 0 . ) with no forcing ( q = 0 ) forvarious values of the delay parameter a . ebruary 14, 2020 1:42 main µ = 0 . ) with forcing ( q = 0 . ) forvarious values of the delay parameter a . Next, in Figure 10 we have plots in ( x , x , y ) phase space of the delayed attractor in red and unde-layed attractor in blue in a forced chaotic case with µ = 0 . and q = 0 . as we vary the delay parameter. Asin the unforced case, for several values of a , the delayed solution is like a cocoon around the undelayedattractor. For the cases a = 1 to a = 3 , a = 5 . , , and a = 9 to a = 10 we see the delayed solution isnow a thin horizontal loop around the undelayed attractor. For the cases a = 4 , . , . the delay makesthe shape of the attractor much more complicated with several loops now surrounding the undelayedattractor. In the case a = 5 we see the delay results in a much thicker smaller attractor while in the case a = 8 we see the delayed attractor is very similar to the undelayed case. Both are expected results, withthe stabilizing effect of the smaller a or larger delay shrinking the attractor, while the case with larger a has only weak delay and so does not differ appreciably from the undelayed system.Finally in Figure 11 we have solutions of the of the delayed and undelayed system for µ = 0 . as wevary both the delay parameter a (increasing down the columns) and forcing parameter q (increasing downthe rows). The first thing to observe is that the most varied behavior occurs in the unforced case, and thatas we increase the forcing the effect of the delay decreases. For instance, for q = 4 , the undelayed anddelayed systems have very similar solutions even as we vary the delay strength. Again this is intuitivelysomething one would expect, with the increasing q or forcing having a destabilizing effect that counteractsthe stabilizing effect of increasing delay as a is reduced.ebruary 14, 2020 1:42 main µ = 0 . ) for values of a =2 , , , , and q = 0 , . , , Note that, unlike in the case of the delayed Landau-Stuart system, even for very large delays or smallvalues of a the system does not exhibit complete Amplitude Death or stabilization of the chaotic behaviorto either a stable limit cycle or, even further, to a stable fixed point. As we shall see below, transition fromchaotic regimes to synchronized periodic oscillations on limit cycles (sometimes referred to as OscillationDeath, or perhaps more accurately Chaos Death in this case) is indeed possible if we look more widely inour parameter space. Hyperchaotic Case µ = 2 In this section we look at numerically generated solutions of the system (38) for hyperchaotic cases with µ = 2 (having two positive, two negative, and one zero (along the time coordinate) Lyapunov exponent).ebruary 14, 2020 1:42 main µ = 2 ) with no forcing ( q = 0 )for various values of the delay parameter a . Figure 12 shows plots in ( x , x , y ) phase space of the delayed attractor in red and undelayed attrac-tor in blue in the hyperchaotic case µ = 2 with no forcing ( q = 0 ) as values of the delay parameter a rangefrom a = 0 . to a = 10 . We see that the delayed attractor is initially thin and long, and oriented vertically.As we increase a from a = 0 . to a = 3 . the top and bottom ends of the attractor form a loop. From a = 4 to a = 5 we see the attractor does not have a more amorphous shape, forming a cocoon around theundelayed attractor. For a = 5 . through a = 10 , the delay causes the system’s attractor to take on a muchmore complicated shape that loops around the undelayed attractor, with the exception of a = 7 where thedelayed solution forms a horizontal loop around the delayed attractor instead.Next in Figure 13 we have plots in ( x , x , y ) phase space of the delayed attractor in red and unde-layed attractor in blue in the forced hyperchaotic case, µ = 2 and q = 2 . as we vary the delay parameter.ebruary 14, 2020 1:42 main From this figure we see that at higher values of a or weak delay, the delayed and undelayed solutionsare, as one would expect, almost the same. At small values of a , the stabilizing effect of the stronger delaycauses the attractor to become much smaller than for the undelayed case. Since the destabilizing effect ofthe forcing is quite strong for q = 2 . , note that only strong delay (corresponding to when a is small) hasa significant effect on the system attractor.In Figure 14 we have solutions of the of the delayed and undelayed system as we vary both the delayparameter a (increasing down the columns) and forcing parameter q (increasing down the rows). We seethat for no forcing the introduction of the delay causes very different behavior as the delay strength variesas we saw in Figure 12. However, increasing the forcing parameter we see that the effects of the delay fordifferent values of a become similar. We also see that at the higher forcing value q = 8 the delayed orbitsare simpler than the undelayed orbit. In particular the case q = 8 shows that unlike in Figure 13 it is notalways the case that the delay only has significant effects on the system at smaller values of a . This is againexpected, as the very strong destabilizing effect of this large forcing would be partially counteracted evenby weak delays. Fig. 13. The delayed (red) and undelayed (blue) solutions of the system in the hyperchaotic case ( µ = 2 ) with forcing ( q = 2 . )for various values of the delay parameter a . ebruary 14, 2020 1:42 main µ = 2 ) for values of a =2 , , , , and q = 0 , . , , Varying the Parametric Forcing
The above gives a general idea about the effects of the delay and forcing on the system dynamics. Inorder to understand the various possible dynamical regimes, and the transitions between them, morecomprehensively, we shall next consider the effect of systematically increasing the other, and perhapsmost important, system parameter µ which controls the parametric forcing.We consider the case of weak delay with a = 10 , although smaller a values show qualitatively similarbehavior. At small µ . , we see periodic dynamics, as seen in the phase plot of Figure 15, and the powerspectral density of Figure 16 which shows a single narrow peak at ω ≃ . .ebruary 14, 2020 1:42 main µ = 0 . , and a = 10 , q = 0 . . ω P ( ω ) Fig. 16. The power spectral density for µ = 0 . , and a = 10 , q = 0 . . There is a complete cascade of period doublings for µ ∈ (0 . , . , leading to a more complex chaoticattractor with one positive Lyapunov exponent at µ = 0 . , as seen in the phase plot of Figure 17, and thebroad features in the power spectral density of Figure 18.ebruary 14, 2020 1:42 main
26 Fig. 17. The phase space plot for µ = 0 . , and a = 10 , q = 0 . . ω P ( ω ) Fig. 18. The broad chaotic features in the power spectral density for µ = 0 . , and a = 10 , q = 0 . . Note the secondary singlepeak at ω ≃ . . The chaotic behavior persists over the window µ ∈ (0 . , . and then is destroyed in a boundarycrisis for µ ∈ (3 . , . , leading into a new period doubled attractor at µ = 3 . with a dominant singlepeak at ω ≃ . as seen in Figures 19 and 20. This corresponds to a synchronized state of the twooscillators.ebruary 14, 2020 1:42 main µ = 3 . , and a = 10 , q = 0 . . ω P ( ω ) Fig. 20. The single peaked power spectral density for µ = 0 . , and a = 10 , q = 0 . , with ω ≃ . and a very small secondarypeak still persisting at ω ≃ . . This periodic attractor then immediately undergoes a symmetry breaking bifurcation for µ ∈ (3 . , . , as shown in the power spectral density plot of Figure 21 where the symmetry breaking givesrise to the peak at the second harmonic frequency of ω ≃ . ebruary 14, 2020 1:42 main ω P ( ω ) Fig. 21. The single peaked power spectral density for µ = 3 . , and a = 10 , q = 0 . with ω ≃ . , the second harmonic of thefrequency in Figure 20. As µ is increased further, a small secondary peak at ω ≃ . is created as the oscillators losingsynchronization near µ ≃ . . The behavior is thus now two-period quasiperiodic, and this persists till mu = 83 . , as seen in Figures 22 and 23, showing the attractor and the double-peaked power spectrumat that value. Fig. 22. The two-period quasiperiodic attractor for µ = 83 . , and a = 10 , q = 0 . . ebruary 14, 2020 1:42 main ω P ( ω ) Fig. 23. The power spectral density for µ = 83 . , and a = 10 , q = 0 . , with ω ≃ . and a second peak at an incommensuratefrequency ω ≃ . . Following this, there is a cascade of torus doublings for µ ∈ (83 . , . , leading to a morecomplex chaotic attractor at µ = 83 . with one positive Lyapunov exponent, as seen in the phase spaceplot of Figure 24, and the broad features in the power spectral density of Figure 25. Fig. 24. The phase space plot for µ = 83 . , and a = 10 , q = 0 . after a sequence of torus doublings. ebruary 14, 2020 1:42 main ω P ( ω ) Fig. 25. The broad chaotic features in the power spectral density for µ = 83 . , and a = 10 , q = 0 . . Note the secondary singlepeak at ω ≃ . . As µ in raised further, the chaotic attractor is destroyed by a boundary crisis at µ ≃ . as seen inFigures 26 and 27. In the latter, the earlier two peaks in the power spectral density persist, but sidebandsand a new peak at ω ≃ . have been created. Fig. 26. The phase space plot for µ = 83 . , and a = 10 , q = 0 . . ebruary 14, 2020 1:42 main ω P ( ω ) Fig. 27. The power spectral density for µ = 83 . , and a = 10 , q = 0 . . The earlier two peaks in the power spectral densitypersist, but sidebands and a new peak at ω ≃ . have been created. Finally this exterior crisis begins to terminate in a stable quasiperiodic attractor at µ ≃ . as seenin Figure 28 where the earlier two peaks in the power spectral density persist, but a new peak at ω ≃ . has been created. ω P ( ω ) Fig. 28. The power spectral density for µ = 83 . , and a = 10 , q = 0 . . The earlier two peaks in the power spectral densitypersist, but a new peak at ω ≃ . has been created. For slightly higher µ ≃ . , a new second harmonic peak is born at ω ≃ . by symmetry breaking,and the crisis terminates with the cleaner-looking power spectrum at µ ≃ seen in Figure 29.ebruary 14, 2020 1:42 main REFERENCES ω P ( ω ) Fig. 29. The power spectral density for µ = 85 , and a = 10 , q = 0 . . The new peak at ω ≃ . and its second harmonic nowremain. We shall end our bifurcation sequence here for this case, as the general features are clear by now.To conclude our numerical results, let us very briefly consider the case of strong delay with a =0 . , q = 8 , where we use a stronger forcing to partly balance the stabilizing effect of the very large delay.Now the range of periodic behavior with ω ≃ . at low values of µ persists up to µ ≃ . after whicha second frequency ω ≃ . comes in via Hopf bifurcation. Further bifurcations and changes in systemdynamics as µ is raised then mimic those discussed above for the weak delay case, except that they occurat significantly larger values of µ .
6. Results and Conclusions
We have comprehensively analyzed the effects of distributed ’weak generic kernel’ delays on the cou-pled Landau-Stuart system, as well as a chaotic oscillator system with parametric forcing. As expected,increasing the delay by reducing the delay parameter a is stabilizing, with its Hopf bifurcation value (de-pendent, of course, on the other system parameters) being a point of exact Amplitude death for both theLandau-Stuart and the chaotic van der Pol-Rayleigh parametrically forced system. In the Landau-Stuartsystem, the Hopf-generated limit cycles for a > a Hopf are very robust under large variations of all othersystem parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking,cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. This is to be expected as the correspondingundelayed systems are robust oscillators over very wide ranges of their respective parameters.Numerical simulations reveal strong distortion and rotation of the limit cycles in phase space as theparameters are pushed far into the post-Hopf regime, and also enable tracking of other features, suchas how the oscillation amplitudes and time periods of the physical variables on the limit cycle attractorchange as the delay and other parameters are varied. For the chaotic system, very strong delays may stilllead to the onset of AD (even for relatively large values of the system forcing which tends to oppose thisstabilization phenomenon).Varying of the other important system parameter, the parametric excitation, leads to a rich sequence ofevolving dynamical regimes, with the bifurcations leading from one into the next being carefully trackednumerically here.
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Int. J. Bif. Chaos21(01)