On periodic perturbations of asymmetric Duffing-Van-der-Pol equation
AAugust 6, 2018 20:45 Morozov-Kostromina
ON PERIODIC PERTURBATIONS OF ASYMMETRIC DUFFING–VAN-DER-POLEQUATION
Albert D. Morozov and Olga S. Kostromina
Department of Mechanics and Mathematics, Lobachevsky State University of Nizhny Novgorod,Nizhny Novgorod, 603950, RussiaCorrespondence should be addressed to Albert D. Morozov, [email protected]
Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to anintegrable equation with a homoclinic “figure-eight” of a saddle are considered. The behavior ofsolutions outside the neighborhood of “figure-eight” is studied analytically. The problem of limitcycles for an autonomous equation is solved and resonance zones for a nonautonomous equationare analyzed. The behavior of the separatrices of a fixed saddle point of the Poincar´e map inthe small neighborhood of the unperturbed “figure-eight” is ascertained. The results obtainedare illustrated by numerical computations.
Keywords : limit cycles, resonances, homoclinic structures
1. Introduction
The theory of time-periodic systems close to two-dimensional nonlinear Hamiltonian systems has beengreatly advanced by now (see, e.g., [Guckenheimer & Holmes, 1983], [Morozov & Shil’nikov, 1983], [Wiggins,1990], [Morozov, 1998]). However, many problems remain unsolved, and new examples should be addressed.In this paper, we will consider one such example – an asymmetric variant of the classical Duffing–Van-der-Pol equation: ¨ x + αx + βx = ε [( γ + γ x + γ x ) ˙ x + γ sin γ t ] , (1)where α, β, γ ÷ γ are parameters, and ε is a small positive parameter. It is always possible to set α = ± , β = ± α = β = − α = 1 , β = ± γ x ˙ x does not play a significant role, it is essential only in the case α = − , β = 1. Equation (1) for α = 1 , β = ± α = − , β = 1. The phase plane of an unperturbed equation has two saddle separatrix loops O (0 , γ , γ , γ one may be excluded to yield the followingequation ¨ x − x + x = ε [( p + p x − x ) ˙ x + p sin p t ] , (2)where p ÷ p are parameters . An analysis of Eq. (2) implies the solution of the following tasks: 1) for theautonomous equation ( p = 0) – partition the plane of the parameters ( p , p ) into regions with differenttopological structures and specify the structures; 2) for the nonautonomous equation ( p (cid:54) = 0) – determinepossible structures of resonance zones outside the neighborhood of “figure-eight” and the conditions of The equation with parametric perturbation was considered in [Litvak-Hinenzon & Rom-Kedar, 1997].1 a r X i v : . [ m a t h . D S ] D ec ugust 6, 2018 20:45 Morozov-Kostromina Albert D. Morozov and Olga S. Kostromina existence of structurally stable and unstable homoclinic Poincar´e structures in the neighborhood of “figure-eight”. The existence of a homoclinic structure specifies complicated behavior of solutions, in other words,it leads to chaos. Bifurcations in the neighborhood of “figure-eight” at a nonzero saddle value of theunperturbed autonomous system were recently considered in [Gonchenko et al. , 2013].The Duffing–Van-der-Pol equation is widely used in the theory of oscillations (see, e.g., [Morozov,1973], [Guckenheimer & Holmes, 1983], [Morozov, 1998]). Along with numerous applied problems in whichthere arises Eq. (2), we can mention a purely mathematical problem of vector field bifurcations on a planethat are invariant to the turn of angle π [Arnold, 1978]. In this problem, in Eq. (1) we have γ = γ = 0and the coefficients of the linear terms αx + εγ ˙ x , unlike our case, are the parameters of deformation. Notealso the work [Bautin, 1975] (as well as [Bautin & Leontovich, 1976]) where an autonomous system withcubic nonlinearity without small parameter describing an electric circuit with tunnel diode was considered.Possible local bifurcations were determined and phase portraits were constructed to an accuracy of an evennumber of limit cycles.The presence of the term p x ˙ x in Eq. (2) greatly complicates the problem: there may exist in theautonomous equation ( p = 0) two limit cycles enclosing any of the equilibrium states O ± ( ± , O ± ( ± ,
0) that is absent in the unperturbed equation[Morozov & Fedorov, 1976], [Kostromina & Morozov, 2012].
Fig. 1. Phase portrait of unperturbed equation.
Solution of problems 2) and 3) rests upon the solution of problem 1). Therefore, we will start with thefirst problem.
2. Investigation of an autonomous equation2.1.
Poincar´e–Pontryagin generating functions
The first integral of the unperturbed equation is H ( x, y ) ≡ y / − x / x / h . The values h ∈ ( − . , G ± inside “figure-eight”, and h > G outside “figure-eight”(Fig. 1). The value h = 0 corresponds to two symmetric saddle loops (“figure-eight”).The main problem in studying Eq. (2) for p = 0 is limit cycles. Its solution results in finding realzeros of the Poincar´e–Pontryagin generating functions B ( h ) [Morozov, 1998].ugust 6, 2018 20:45 Morozov-Kostromina On periodic perturbations of asymmetric Duffing–Van-der-Pol equation According to [Kostromina & Morozov, 2012], we have B = B ± ( ρ ( h )) = 430 π (2 − ρ ) / { p − ρ − − ρ ) K ( k )++ (cid:2) p (2 − ρ ) − ρ − ρ + 1) (cid:3) E ( k ) ± p √ πρ (cid:112) − ρ } ≡≡ π (2 − ρ ) / B ± ( ρ ) (3)for the domains G ± and B = B ( ρ ( h )) = 830 π (2 ρ − / { [5 p (2 ρ − − ρ ) − ρ − − ρ )] K ( k )++ (cid:2) p (2 ρ − − ρ − ρ + 1) (cid:3) E ( k ) (cid:9) ≡ π (2 ρ − / B ( ρ ) (4)for the domain G . Here, K ( k ) , E ( k ) are complete elliptic integrals and ρ = k . Note that ρ is a moreconvenient variable than h . In the formula (3) we have ρ ( h ) = 2 √ h/ (1 + √ h ) ( ρ ∈ (0 , ρ ( h ) = (1 + √ h ) / √ h ( ρ ∈ (1 / , ρ = 1 corresponds to “figure-eight”. Note thatthe function B ( ρ ) does not depend on the parameter p . Limit cycles
Investigation of the functions B ± ( ρ ) , B ( ρ ) gives the following results [Kostromina & Morozov, 2012]. Theorem 1.
For sufficiently small ε , the number of limit cycles in each domain G ± and G of Eq. (2)does not exceed two. Theorem 2.
For sufficiently small ε , the number of limit cycles of Eq. (2) does not exceed three. The authors of [Kostromina & Morozov, 2012] partitioned the parameter plane into 22 domains D m , m = 1 , . . . ,
22, gave the basic phase portraits at different values of parameters from those domains andfound all bifurcations.By virtue of the invariance of Eq. (2) to the change ( p , x, y ) → ( − p , − x, − y ) the partitioning of theplane of the parameters ( p , p ) is symmetric to the p axis. Therefore, only the upper half plane p ≥ D m , m = 1 , . . . ,
13 is shown in Fig. 2. A magnified fragment of Fig. 2 is presentedschematically in Fig. 3. By virtue of the symmetry, the phase portraits for the domain p < π of the corresponding phase portrait from the domain p > i, j, k ) that means the existence of i limit cycles inside the right loop, j inside the left loop, and k outside “figure-eight”. It was proved that D is of the type (0 , , D − (0 , , , D − (0 , , D − (0 , , D − (0 , , D − (1 , , D − (1 , , D − (1 , , D − (0 , , D − (0 , , D − (1 , , D − (2 , , D − (1 , , L ± : p ± p − O ± ( ± ,
0) in domains G ± , respectively. A + ( − , ) – the point on the straight line L +1 from which the double cycle line originates. The doublecycle line is plotted using the system B +10 ( ρ, p , p ) = 0 , [ dB +10 ( ρ, p , p ) /dρ ] = 0, ρ ∈ (0 , A + corresponds to ρ = 0. This point is readily found by the power series expansion of the function B +10 ( ρ ) inthe neighborhood of ρ = 0 in the case of a structurally unstable focus and zero first Lyapunov exponent(with the second Lyapunov exponent being nonzero). The extreme point A s + (0 , .
96) of the double cycleline corresponds to ρ = 1 (see Fig. 3). When the saddle value σ c = εp vanishes to zero at p = 0 thedouble cycle merges with the separatrix . In the lower half plane we have the points A − ( − , − ) and A s − (0 , − . ugust 6, 2018 20:45 Morozov-Kostromina Albert D. Morozov and Olga S. Kostromina
Fig. 2. Partition of the plane of parameters ( p , p ) into domains with different phase portrait topology.Fig. 3. Magnified fragment of Fig. 2. L ± : 5 p ± √ π p − O (0 ,
0) in domains G ± . These straight lines are given by the Melnikov formula forautonomous systems. L – double cycle line in domain G corresponding to the bifurcation value p ≈ . L – line of the “big” loop of the separatrix of saddle O (0 , p , p ) plane are presented in Fig. 4. The dots show the equilibrium states ((0 ,
0) saddle pointand ( ± ,
0) focus), the arrows indicate directions of motion on the separatrices. Note also that the limitugust 6, 2018 20:45 Morozov-Kostromina
On periodic perturbations of asymmetric Duffing–Van-der-Pol equation D (a); D (b); D (c); D (d); D (e); D (f); D (g); D (h); D (i); D (j); D (k); D (l); D (m). cycle near “figure-eight” is unstable. The other phase portraits for symmetric domains may be obtainedby rotation of angle π . The simplest phase portraits are obtained for the dissipation domain D .ugust 6, 2018 20:45 Morozov-Kostromina Albert D. Morozov and Olga S. Kostromina
3. Analysis of resonance zones topology
In the domains filled with closed phase curves of the unperturbed equation ( ε = 0) and separated fromthe unperturbed separatrices, we will pass in Eq. (2) to the “ I action– θ angle ” variables by the followingformulas I ( h ) = 12 π (cid:73) y ( x, h ) dx,θ = ∂S ( x, I ) ∂I , S = (cid:90) xx y ( x, h ( I )) dx, (5)where S ( x, I ) is the generating function of this canonical transformation. The resulting system will bewritten in the form ˙ I = ε [( p + p x − x ) y + p sin ϕ ] x (cid:48) θ ≡ εF ( I, θ, ϕ ) , ˙ θ = ω ( I ) + ε [( p + p x − x ) y + p sin ϕ ] x (cid:48) I ≡ ω ( I ) + εF ( I, θ, ϕ ) , ˙ ϕ = p , (6)where ω is the frequency of self-excited oscillations. Consider the resonance case when ω ( I pq ) = ( q/p ) p , (7)where p, q are coprime integer numbers. The level I = I pq (closed phase curve H ( x, y ) = h pq of theunperturbed system) will be referred to as the resonance level. The neighborhood U √ ε = { ( I, θ ) : I pq − C √ ε < I < I pq + C √ ε, ≤ θ < π, C = const > } will be called the resonance zone.By the substitution θ = ψ + ( q/p ) ϕ, I = I pq + µη, µ = √ ε, (8)in (6), by averaging the obtained system over the fast variable ϕ and neglecting the terms O ( µ ), we obtainthe system [Morozov, 1998] ˙ u = µA ( v, I pq ) + µ P ( v, I pq ) u, ˙ v = µbu + µ ( b u + Q ( v, I pq )) , (9)where u = η + O ( µ ) , v = ψ + O ( µ ), b = dω ( I pq ) /dI, b = d ω ( I pq ) / dI , A ( v, I pq ) = 12 πp (cid:90) πp F ( I pq , v + qϕ/p, ϕ ) dϕ, (10) P ( v, I pq ) = 12 πp (cid:90) πp [ ∂F ( I pq , v + qϕ/p, ϕ ) /∂I ] dϕ, (11) Q ( v, I pq ) = 12 πp (cid:90) πp F ( I pq , v + qϕ/p, ϕ ) dϕ. (12)The substitution u → u − µQ ( v, I pq ) /b and the transition to “slow time” τ = µt reduces Eqs. (9) toa pendulum equation [Morozov, 1998] d vdτ − bA ( v, I pq ) = µσ ( v, I pq ) dvdτ , (13)where σ ( v, I pq ) = 12 πp (cid:90) πp ( p + p x − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x ( I pq , v + qϕ/p ) y = y ( I pq , v + qϕ/p ) dϕ. (14)Apparently, σ = const .The topology of individual resonance zones may be found from Eq. (13) to an accuracy of terms oforder µ .ugust 6, 2018 20:45 Morozov-Kostromina On periodic perturbations of asymmetric Duffing–Van-der-Pol equation In calculations of the function A ( v, I pq ) and quantities b and σ we distinguish the following cases. Case 1 : ( x, y ) ∈ G ± = { ( x, y ) : y / − x / x / h, h ∈ ( − . , } ; Case 2 : ( x, y ) ∈ G = { ( x, y ) : y / − x / x / h, h > } .The unperturbed solution in (6) is different in each case [Kostromina & Morozov, 2012].We represent the function A ( v, I pq ) in the form A j ( v, I pq ) = (cid:101) A j ( v, I pq ) + B j ( I pq ) and designate b = b j , σ = σ j , j = 1 ,
2, where B , B are the Poincar´e–Pontryagin generating functions (3) and (4),respectively.Following [Morozov, 1998], we refer to the resonance level I = I pq as splittable if the equation A j ( v ; I pq ) = 0 has simple roots. The nonsplittable resonance level I = I pq for which | A j ( v ; I pq ) | > I = I pq is called partially passable, if B j ( I pq ) (cid:54) = 0 andimpassable, if B j ( I pq ) = 0.Note that the behavior of solutions of the initial equation (2) in the neighborhood of passable, partiallypassable and impassable resonance levels is defined by the theorems from [Morozov, 1998]. Case 1
Using the unperturbed solutions at the resonance level and formulas (10), (14), we find for q = 1 d vdτ − b ( p A cos pv + B ) = µσ dvdτ , (15)where b = π − ρ ) / [2(1 − ρ ) K ( ρ ) − (2 − ρ ) E ( ρ )] ρ (1 − ρ ) K ( ρ ) , (16) σ = p − − ρ ) K ( ρ ) E ( ρ ) , (17) A = −√ p a p a p , a = exp (cid:18) − π K ( √ − ρ ) K ( ρ ) (cid:19) . (18)For q > d vdτ − b B = µσ dvdτ . (19)Thus, for q = 1 the topology of resonance zones is described by Eq. (15). The phase portraits of thisequation are well known [Morozov, 1998] (Fig. 5). At B ( I pq ) = 0, we have by definition an impassableresonance (Fig. 5(c)). In this case, the resonance level I = I pq coincides with the level I = I in theneighborhood of which the autonomous equation has a limit cycle. A partially passable resonance ispresented in Fig. 5(b), and a passable resonance in Fig. 5(a). According to (19), at q > B ( I pq ) (cid:54) = 0we have a passable resonance.Consider briefly the bifurcations of the transition from the impassable to the partially passableresonance. Let us set in Eq. (15) B = µγ , where γ defines the deviation of the resonance level I = I pq from I = I . We denote by γ ± the bifurcation values of γ at which Eq. (15) has, respectively, the upperor lower loop enclosing a phase cylinder. As γ deviates from the bifurcation value, the loop gives birthto a limit cycle enclosing a phase cylinder, and the resonance level becomes partially passable. The limitcycle corresponds to the two-dimensional torus in the initial equation (for the Poincar´e map it is a closedinvariant curve shown in Fig. 6). More details about these bifurcations can be found in [Morozov, 1998].The frequency ω ( I ) of the self-excited oscillations in domains G ± meets the condition ω ( I ) ∈ (0 , √ p > p / √
2. Therefore, onlythe resonance levels H ( x, y ) = y / − x / x / h p , for which p > p / √
2, are split. Note that theresonance levels with larger values of p are closer to the unperturbed separatrix.ugust 6, 2018 20:45 Morozov-Kostromina Albert D. Morozov and Olga S. Kostromina - p /p p /p vu - p /p p /p vv - p /p p /p v (a) (b) (c)Fig. 5. Phase portraits for Eq. (15). Case 2
Analogously to Case 1, we find the equation d vdτ − b ( p A cos pv + B ) = µσ dvdτ , (20)that defines the topology of the resonance zones at odd p and q = 1. Otherwise, the resonance zonestopology is defined by the equation d vdτ − b B = µσ dvdτ . (21)In (20) and (21) we have b = π ρ − / [(1 − ρ ) K ( ρ ) + (2 ρ − E ( ρ )] ρ (1 − ρ ) K ( ρ ) , (22) σ = p − ρ − K ( ρ ) ( E ( ρ ) + ( ρ − K ( ρ )) , (23) A = − √ p a p/ a p . (24)For even p and/or q >
1, the resonance is passable if B ( I pq ) (cid:54) = 0.The Poincar´e map for Eq. (2) at different parameter values was constructed using the WInSet software[Morozov & Dragunov, 2003] . It was found that at small values of ε numerical results are in a goodagreement with the theoretical study. Figure 6 illustrates the structure of the neighborhood of the splittablelevels I = I , I = I . The impassable resonance zones are shown in Figs. 6(a) and 6(c), and the partiallypassable zones in Figs. 6(b) and 6(d). The dots in Fig. 6 correspond to points of period-2, as well as tothe fixed points of the Poincar´e map in domain G +1 (Figs. 6(a) and 6(b)) and periodic points of period-3in domain G (Figs. 6(c) and 6(d)). Besides, a closed invariant curve of the Poincar´e map in domain G +1 isshown in Fig. 6(b) and in domain G in Fig. 6(d). The stable separatrices are plotted by the blue curves,the unstable separatrices by the red ones.Note that the fixed and periodic points in resonance zones correspond to resonance periodic solutionsof period 2 πp/p in the initial equation, and the closed invariant curves to quasiperiodic (double-frequency)solutions (two-dimensional tori). The first version of the software was described in [Morozov et al. , 1999]. ugust 6, 2018 20:45 Morozov-Kostromina
On periodic perturbations of asymmetric Duffing–Van-der-Pol equation ε = 0 . p = 1 , p = − . , p = 0 . , p = 2 . p = 1 , p = − . , p = 0 . , p = 2 . p = 1 , p = 0 . , p = 1 , p = 3 .
36 (c); p = 1 , p = 0 . , p = 1 , p = 3 (d).
4. On global behavior of solutions outside the neighborhood of “figure-eight”
The resonance levels corresponding to the limit cycles in the autonomous equation ( p = 0 in Eq. (2)) aresplittable as B j ( I pq ) = 0, j = 1 , G ± , G is no more than two. Let us remove from these domains the neighborhoods of such levelsand designate the remaining domains without the neighborhood of “figure-eight” by V . Using the resultsobtained in [Morozov, 1998] and Eqs. (15), (18), (20) and (24), we obtain the following theorem. Theorem 3.
There are only finitely many splittable resonance levels in V . It follows from this theorem that for relatively small ε > D in the bifurcationdiagram) and ρ = ρ (1) , ρ = ρ (2) be simple roots of the Poincar´e–Pontryagin function B +1 ( ρ ) in domain G +1 .Let us fix the parameter p = 1 .
22 and find the values of the parameters p , p at which the cycle ρ = ρ (1) coincides with the resonance level I = I , and the cycle ρ = ρ (2) with the level I = I . Fromthe resonance condition (7) we have p = 2 ω ( ρ (1) , p ) = 3 ω ( ρ (2) , p ). From this relation and the equationsdefining the limit cycles B +1 ( ρ (1) , p ) = 0, B +1 ( ρ (2) , p ) = 0, we find p ≈ − . ρ (1) ≈ . ρ (2) ≈ . Albert D. Morozov and Olga S. Kostromina y x y x (a) (b)Fig. 7. Behavior of the trajectories of the Poincar´e map for Eq. (2) in the case of two impassable resonances with p = 2 and p = 3 in domain G +1 (a), and a fragment of this domain (b). Here, ε = 0 . p = − . p = 1 . p = 1, p = 2 . Then, at p ≈ .
782 we will have impassable resonance zones. There exists in the initial equation a stableperiodic solution of period 6 π/p (stable periodic points of period-3 in the red zone in Fig. 7(a)) andan unstable periodic solution of period 4 π/p (unstable periodic points of period-2 in the white zone inFig. 7(a)). All the other resonance levels between the above ones will be passable. Between the resonancezones, trajectories fill the cell under consideration (see Fig. 7(a); actually, these trajectories are slowlyspiraling and tend to stable periodic points of period-3 as t → ∞ ). A higher-order partially passableresonance can be seen near the unperturbed separatrix loop in Fig. 7(a). A magnified fragment with atrajectory inside the resonance zone with p = 2 is presented in Fig. 7(b), where one can see passableresonances. Passable resonances were also observed between the resonance zones with p = 2 and p = 3.The behavior of the invariant curves of the Poincar´e map in the neighborhood of the impassableresonance zone with p = 2 is shown in more detail in Fig. 6(a).
5. Analysis of the behavior of solutions in the small neighborhood of“figure-eight”
The unperturbed equation ¨ x − x + x = 0 has a right loop Γ r = Γ rs (cid:83) Γ ru of saddle separatrix O (0 ,
0) andthe left loop Γ l = Γ ls (cid:83) Γ lu (Fig. 1).It is known that under the action of perturbations, the separatrices of the fixed saddle point of thePoincar´e map may intersect forming homoclinic structures of two types: 1) Γ rs (cid:84) Γ ru (cid:54) = (cid:11) and/or Γ ls (cid:84) Γ lu (cid:54) = (cid:11) ; 2) Γ ru (cid:84) Γ ls (cid:54) = (cid:11) or Γ rs (cid:84) Γ lu (cid:54) = (cid:11) , when p (cid:54) = 0.Existence of a homoclinic structure results in complicated behavior of solutions in its neighborhoodor, in other words, in a nontrivial hyperbolic set [Shil’nikov, 1967]. The problem of the existence of type1) homoclinic structure is solved using the Melnikov formula [Mel’nikov, 1963] ∆( t ) = ε ∆ ( t ) + O ( ε ),where ∆( t ) is the distance between the related branches of the separatrix into which the unperturbedseparatrix splits. The substitution of x = ξ + εx ( t ) + O ( ε ), where x ( t ) = − p p sin ( p t ) , (25)in (2) yields the following equation¨ ξ − ξ + ξ = ε (cid:20) ( p + p ξ − ξ ) ˙ ξ + 3 p p ξ sin ( p t ) (cid:21) . (26)ugust 6, 2018 20:45 Morozov-Kostromina On periodic perturbations of asymmetric Duffing–Van-der-Pol equation Applying the Melnikov formula to this equation, we find∆ ( t ) = 2 (cid:18) p ± π √ p − (cid:19) + 3 πp πp / p cos ( p t ) . (27)If ∆ ( t ) is an alternating function, which holds under the condition | p | > p ∗ = 43 (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) p ± π √ p − (cid:19) cosh( πp / πp (cid:12)(cid:12)(cid:12)(cid:12) , (28)then there occurs transversal intersection of the stable and unstable manifolds of the fixed point.If ∆ ( t ) is a constant-sign function, then the corresponding separatrix manifolds of the saddle fixedpoint do not intersect. However, if the value of | p − p ∗ | is small enough, then, as follows from [Gavrilov &Shil’nikov, 1972], [Morozov, 1976], a nontrivial hyperbolic set exists in the neighborhood of “figure-eight”.Under the condition | p | = 43 (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) p ± π √ p − (cid:19) cosh( πp / πp (cid:12)(cid:12)(cid:12)(cid:12) (29)the corresponding separatrices of the fixed point (0 ,
0) are tangent to each other (to an accuracy of termsof order ε ).Making use of the Melnikov formula, it is easy to represent all possible cases of relative position ofthe separatrices as a result of splitting of the left or right separatrix loop. For example, for p = 0, thecondition 23 p + π √ p −
815 = 0specifies the existence of the right separatrix loop. With allowance for external force, the outgoing andincoming separatrices intersect transversally, forming a homoclinic Poincar´e structure. In this case, for theleft separatrix loop we have ∆ ( t ) = − π √ p + 3 πp πp / p cos ( p t ) . (30) x -1.2 -0.6 0.6 1.2 y -0.60.6 x -1.2 -0.6 0.6 1.2 y -0.60.6 x -1.2 -0.6 0.6 1.2 y -0.60.6 (a) (b) (c)Fig. 8. Behavior of separatrices of the fixed point (0 ,
0) for Eq. (26) on the ( ξ = x, ˙ ξ = y ) plane at ε = 0 . p = 0 . p = 0 . p = 4 and p = 1 .
13 (a), p = 1 . p = 2 .
83 (c).
The separatrices of the fixed point (0 ,
0) of the Poincar´e map on the ( ξ = x, ˙ ξ = y ) plane are shownin Fig. 8 for ε = 0 . p = 0 . p = 0 . p = 4 and p = 1 .
13 (a), p = 1 . p = 2 .
83 (c).Note that for p = 0 . p = 0, p = 0 the unstable limit cycle in Eq. (2) coincides with “figure-eight”.Then, for small enough p (cid:54) = 0, the inverse Poincar´e map has a quasiattractor.When a perturbed autonomous equation has a “big” separatrix loop, the Melnikov formula does nothold for a nonautonomous equation. For this case, the separatrices of a fixed saddle point of the Poincar´emap for Eq. (26) on the ( ξ = x, ˙ ξ = y ) plane are shown in Fig. 9 for ε = 0 . p = 0 . p = 1 . p = 4ugust 6, 2018 20:45 Morozov-Kostromina Albert D. Morozov and Olga S. Kostromina x -0.9 0.9 y -0.50.5 x -0.9 0.9 y -0.50.5 (a) (b)Fig. 9. Behavior of separatrices of the fixed point (0 ,
0) for Eq. (26) at ε = 0 . p = 0 . p = 1 . p = 4 and (a) p = 1 .
6, (b) p = − . x -0.9 0.9 y -0.60.6 x -0.9 0.9 y -0.60.6 (a) (b) x -0.9 0.9 y -0.60.6 x -0.9 0.9 y -0.60.6 (c) (d)Fig. 10. Behavior of separatrices of the fixed point (0 ,
0) for Eq. (26) at ε = 0 . p = 0 . p = 1 . p = 4 and (a) p = 1 .
6, (b) p = − . ε = 0 . p = 0 . p = 0 . p = 4, (c) p = 0 .
5, (d) p = − . and (a) p = 1 .
6, (b) p = − .
6. There occurs transversal intersection of the corresponding separatrices(Γ ru (cid:84) Γ ls (cid:54) = (cid:11) in Fig. 9(a) and Γ rs (cid:84) Γ lu (cid:54) = (cid:11) in Fig. 9(b)). Also, homoclinic structures with tangency(Fig. 10) are possible at different values of the parameters.Figure 11 illustrates other homoclinic structures with tangency of stable and unstable separatrices ofthe fixed point (0 ,
0) for Eq. (26) at ε = 0 . p = 4 for the following values of the parameters p , p , p :(a) p = 0 . p = 0 . p = 3; (b) p = 0 . p = 0 . p = 4 .
55; (c) p = 0 . p = 0 . p = 2 .
34; (d) p = 0 . p = 0 . p = 2 .
96; (e) p = 1, p = 0 . p = 2 .
32; (f) p = 0 . p = 0, p = 2; (g) p = 0 . p = 0 . p = 3 .
34; (h) p = 0 . p = 0, p = 1 .
98; (i) p = 0 . p = 0 . p = 2 .
82; (j) p = 0 . p = 0 . p = 2 .
6. Bifurcation diagrams
Using the WInSet and Maple 13 software, we constructed three bifurcation diagrams of the Poincar´e mapfor Eq. (26) on the ( p , p ) plane for fixed values of the parameters ε , p , and p . In the bifurcation curves,ugust 6, 2018 20:45 Morozov-Kostromina On periodic perturbations of asymmetric Duffing–Van-der-Pol equation x -1.2 -0.6 0.6 1.2 y -0.50.5 x -1.2 -0.6 0.6 1.2 y -0.50.5 x -1.2 -0.6 0.6 1.2 y -0.50.5 (a) (b) (c) x -1.2 -0.6 0.6 1.2 y -0.50.5 x -1.2 -0.6 0.6 1.2 y -0.60.6 x -1.2 -0.6 0.6 1.2 y -0.50.5 (d) (e) (f) x -1.2 -0.6 0.6 1.2 y -0.60.6 x -1.2 -0.6 0.6 1.2 y -0.60.6 x -1.2 -0.6 0.6 1.2 y -0.60.6 (g) (h) (i) x -1.2 -0.6 0.6 1.2 y -0.60.6 (j)Fig. 11. Other homoclinic structures with tangency of stable and unstable separatrices of the fixed point (0 ,
0) for Eq. (26). the corresponding separatrices of the fixed point (0,0) are tangent to each other. These curves separatedomains with homoclinic structure on the plane of parameters ( p , p ) (the stable and unstable separatricesof the saddle point (0,0) intersect transversally). The three bifurcation diagrams describe all possible casesof the relative position of separatrices of the fixed saddle point (0,0) for the Poincar´e map.The obtained bifurcation diagrams are symmetric to the p axis. Let us set p > ε = 0 . p = 0 . p = 4, we obtain sixbifurcation curves. Equations for the straight lines M , M , M are found from (29). The other bifurcationcurves M , M , M are obtained numerically by means of the WInSet software. Each pair of lines M and M , M and M , M and M have exactly one common point on the p axis. The first point ( p ≈ . p ≈ . Albert D. Morozov and Olga S. Kostromina and p ≈ . M and M , as well as of the lines M , M and M correspond to double homoclinictangency. The obtained bifurcation curves are presented in Fig. 12. Fig. 12. Bifurcation diagram for the Poincar´e map on the ( p , p ) plane at p = 0 . Setting ε = 0 . p = 0 . p = 4, we obtain three bifurcation curves shown in Fig. 13. As theparameter p changes from 0 .
78 to 0 . M and M in Fig. 12 approach each otherand coincide at p = 0 .
8. As a result we obtain a straight line N with a new type of tangency – doublehomoclinic tangency. An equation for N is found from (29). The lines N and N obtained numericallyhave one common point ( p ≈ . p = 0) corresponding to the “big” separatrix loop in the autonomousequation. Fig. 13. Bifurcation diagram for the Poincar´e map on the ( p , p ) plane at p = 0 . Setting ε = 0 . p = 0 . p = 4, we obtain five bifurcation curves plotted in Fig. 14. Equationsfor the straight lines R , R , R are found from (29). The other bifurcation lines R , R are obtainednumerically using the WInSet software. The intersection point of the curves R and R ( p ≈ . p = 0) corresponds to the “big” separatrix loop in the autonomous equation. The intersection points ofugust 6, 2018 20:45 Morozov-Kostromina On periodic perturbations of asymmetric Duffing–Van-der-Pol equation R and R and of R , R and R give double homoclinic tangencies. Fig. 14. Bifurcation diagram for the Poincar´e map on the ( p , p ) plane at p = 0 . Each of the three bifurcation diagrams has domains with homoclinic structure and a nonsmoothboundary. This phenomenon was explained in ample detail in the work [Gonchenko et al. , 2013].
7. Conclusion
The problem of time-periodic perturbations of two-dimensional Hamiltonian systems with a saddle andtwo separatrix loops in the form of “figure-eight” is a challenging problem for the theory of bifurcations.Bifurcations in the neighborhood of “figure-eight” for the case of an unperturbed autonomous system witha nonzero saddle value were recently considered in [Gonchenko et al. , 2013]. This problem for the case ofa zero saddle value has not been fully understood yet. The asymmetric Duffing–Van-der-Pol equation (2)studied in the present work is a good model for solution of this problem.Despite its fundamental role in the theory of differential equations, the theory of bifurcations, and thetheory of oscillations, Eq. (2) has not been studied thus far for the case when p (cid:54) = 0. We have solved theproblem of limit cycles in the autonomous case. For the nonautonomous case, we have found resonance zonestructures and global behavior of solutions in the cells separated from unperturbed separatrices. Differentresonance periodic solutions and two-dimensional invariant tori have also been found. The problem of theexistence of homoclinic structures in the neighborhood of unperturbed separatrices (in the neighborhoodof “figure-eight”) has been solved. All possible cases of relative position of the separatrices of a trivial fixedsaddle point for the Poincar´e map have been revealed. Three bifurcation diagrams for the Poincar´e map onthe ( p , p ) plane separating domains of existence of different homoclinic structures have been constructed.The results obtained for the separatrix tangency illustrate many specific features found in [Gonchenko etal. , 2013] for two-parametric families of maps in the neighborhood of “figure-eight” with nonzero saddlevalue.Note that some of the problems associated with the presence of homoclinic structures remain open. Forexample, one such problem is to study fractal properties of attraction basin boundaries for stable periodicregimes in the considered equation.
8. Acknowledgments
We dedicate this paper to the memory of the outstanding scientist Leonid P. Shil’nikov, the pioneerof homoclinic bifurcation theory. The authors are grateful to M.I. Malkin for helpful discussions andcomments. This work was partially supported by the Russian Science Foundation, grant No 14-41-0044.ugust 6, 2018 20:45 Morozov-Kostromina REFERENCES
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