arXiv:2006.12957v1 [math.DS] 23 Jun 2020 iua,saa ieeta qain ihtime-dependen with equations differential scalar ticular, limitin the of solutions the as conditions behavior some asymptotic under same that the Note systems. limiting sponding ouin eecniee sbfraigojcsadtebi the and objects contained bifurcating solutions. are as systems considered compl were non-autonomous to solutions for equations theory limiting general in of bifurcations equatio of autonomous chang transfer asymptotically a the for as discussed Bifurcations were attractor. tion pullback st the [ pullback in of a studied structure of were change equations Similar the states. with associated are bifurcations correspond non-au the of of examples solutions Several the general. than in differently true completely not is this ever, [ [ condi in in investigated and were considered plane solutions the of autonomous on periodicity limiting almost systems corresponding autonomous the asymptotically of of solutions [ the in and considered asymptoticall first is were system systems perturbed the that such turbations oee yteter fsaiiyadbfrain [ bifurcations and stability t of systems, theory autonomous the by For covered equations. differential of theory TBLT N IUCTO HNMN NASYMPTOTICALLY IN PHENOMENA BIFURCATION AND STABILITY -aladdress E-mail iuctosi o-uooossseshv eetybe d been recently have systems non-autonomous in Bifurcations h nuneo etrain ntesaiiyo solutions of stability the on perturbations of influence The nttt fMteais f eea eerhCnr,Rus Centre, Research Federal Ufa Mathematics, of Institute Abstract. ahmtc ujc Classification: Subject Mathematics function Lyapunov tion, Keywords: th in on states discussed. bifurcations of repelling is dependence perturbations or The attracting system. new s autonomous Lyapunov cally of of equatio emergence change the b the a long-term of and with the terms associated and non-autonomous bifurcations stability and tigates the e nonlinear the case on consider at this vanish depend is In and type time system. in center infinity turbed of at decay equilibrium perturbations an the with system tonian 6 ,weeteams eidcsltoswr prxmtdby approximated were solutions periodic almost the where ], : [email protected] h nuneo iedpnetprubtoso nautonom an on perturbations time-dependent of influence The o-uooossses etrain,aypois sta asymptotics, perturbations, systems, non-autonomous 1,Censesysre,Ua usa 450008 Russia, Ufa, street, Chernyshevsky 112, AITNA SYSTEMS HAMILTONIAN 4 ,weeterltosbtentesltoso h complete the of solutions the between relations the where ], SA .SULTANOV A. OSKAR . 1. 10 ,weetebfrainwsudrto sacag nthe in change a as understood was bifurcation the where ], Introduction 1 1 42,3D0 42,37J65 34D20, 34D10, 34C23, – 5 3 .Amr iecaso ytm ntepaewas plane the on systems of class wide more A ]. .Ti ae sdvtdt o-uooosper- non-autonomous to devoted is paper This ]. offiinswr osdrdi [ in considered were coefficients t t qain eedsrbd h elements The described. were equations ete uooos smttclyautonomous Asymptotically autonomous. y ucto a nesoda rnhn of branching a as understood was furcation n iiigsseswr tde n[ in studied were systems limiting ing si [ in ns eslto fsc rbe seffectively is problem a such of solution he ntesrcueo h oano attrac- of domain the of structure the in e ooossseswoesltosbehave solutions whose systems tonomous blt n h perneo e stable new of appearance the and ability h ouin facmlt ytmhave system complete a of solutions the , n[ in ytm(e,freape [ example, for (see, system g ytmwr icse.Aseilclass special A discussed. were system sacasclpolmi h qualitative the in problem classical a is in htgaatetesaiiyand stability the guarantee that tions 12 11 susdi eea aes npar- In papers. several in iscussed inAaeyo Sciences, of Academy sian ,weesm atclrbounded particular some where ], aiiyo h equilibrium the of tability ,weesm odtosensuring conditions some where ], h tutr fdecaying of structure the ulbimo h unper- the of quilibrium etre asymptoti- perturbed e d ti sue that assumed is It ed. hvoro trajectories of ehaviour s h ae inves- paper The ns. ouin ftecorre- the of solutions iiy bifurca- bility, u Hamil- ous 7 9 ) How- ]). ,where ], system 8 ]. 2 STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS
The present paper considers a class of asymptotically Hamiltonian systems with the equilibrium and investigates the effects of decaying time-dependent perturbations on the stability and bifurcations of solutions. To the best of our knowledge, such problems have not been thoroughly studied. The paper is organized as follows. In section 2, the mathematical formulation of the problem is given and the class of non-autonomous perturbations is described. The proposed method of stability and bifurcation analysis is based on a change of variables associated with a Lyapunov function for a complete asymptotically autonomous system. The construction of this transformation is described in section 3. Section 4 is devoted to bifurcations associated with a change of the stability of the equi- librium. Bifurcations associated with limit cycles are discussed in section 5. The results of sections 4 and 5 are applied in section 6 for a description of bifurcations in the complete system under various restrictions on the perturbations. The paper concludes with a brief discussion of the results obtained.
2. Problem statement Consider the system of two differential equations: dx dy (1) = ∂ H(x,y,t), = ∂ H(x,y,t)+ F (x,y,t), t> 0. dt y dt − x It is assumed that the functions H(x,y,t) and F (x,y,t) are infinitely differentiable and for every compact D R2 H(x,y,t) H (x,y) and F (x,y,t) 0 as t for all (x,y) D. The limiting ∈ → 0 → → ∞ ∈ autonomous system with the Hamiltonian H0(x,y) is assumed to have the isolated fixed point (0, 0) of center type. Without loss of generality, it is assumed that 2 2 x + y 3 2 2 (2) H0(x,y)= + (r ), r = x + y 0. 2 O p → It is also assumed that the level lines (x,y) R2 : H (x,y)= E define a family of closed curves on { ∈ 0 } the phase space (x,y) parameterized by the parameter E for all E (0, E0], E0 = const. Non-autonomous perturbations of the limiting system are described∈ by the functions with power-law asymptotics: ∞ ∞ − k − k (3) H(x,y,t) H (x,y)= t q H (x,y), F (x,y,t)= t q F (x,y), t , q Z . − 0 k k →∞ ∈ + Xk=1 Xk=1 It is assumed that the perturbations preserve the fixed point (0, 0): ∂ H(0, 0,t) 0, ∂ H(0, 0,t) 0, F (0, 0,t) 0. x ≡ y ≡ ≡ The structure of the perturbations can be more complicated, for example, asymptotic series (3) can dif- fer from power series, or the coefficients of asymptotics can explicitly depend on t. Such perturbations, however, are not considered in the present paper. Note that the Painlev´eequations [13], autoresonance models [14,15] and synchronization models [16,17] are reduced to non-autonomous systems of the form (1). Our goal is to describe possible asymptotic regimes in the perturbed system and to reveal the role of decaying perturbations in the corresponding bifurcations. Here, the bifurcations are associated with a change of Lyapunov stability of the equilibrium and the emergence of new attracting or repelling states. Let us note that the decaying perturbations do affect the stability of the system. A simple example is given by the following equation: d2x dx + x = γt−κ , γ,κ R, γ = 0, κ> 0. dt2 dt ∈ 6 2 2 This equation in the variables x,y =x ˙ has the form (1) with H(x,y,t) H0(x,y) (x + y )/2 and F (x,y,t) γt−κy. It can easily be checked that the unperturbed autonomous≡ equation≡ (γ = 0) has the following≡ general solution: x(t; a, ϕ) = a cos(ϕ + t). The long-term asymptotics for a two- parameter family of solutions to the perturbed equation (γ = 0) is constructed with using WKB 6 STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS 3 approximations [18]: x(t; a, ϕ)= a cos(ϕ + t)+ (t1−κ) , κ> 1; O x(t; a, ϕ)= atγ/2 cos(ϕ + t)+ (t−1) , κ = 1; O γt1−κ x(t; a, ϕ)= a exp cos ϕ + t + (t1−2κ)+ (log t) + (t−κ) , κ< 1, 1 κ n O O O o − where a, ϕ R are arbitrary parameters. It follows that the stability of the trivial solution x(t) 0 or the fixed∈ point (0, 0) depends on the parameters γ and κ. In particular, if κ> 1, the fixed point≡ is marginally stable. In this case, the solutions of the non-autonomous equation have the same behaviour as the solutions of the limiting equation. The fixed point becomes attracting if γ < 0 (polynomially stable when κ = 1 and exponentially stable when 0 <κ< 1), and loses stability if γ > 0. In the general case, the long-term asymptotics for solutions are obtained not so easily, and the stability of the equilibrium depends on nonlinear terms of equations. The examples of nonlinear equations are contained in section 6.
3. Change of variables The proposed method of study of asymptotic regimes in system (1) is based on the construction of appropriate Lyapunov functions. Recently, it was noted in [19–21] that such functions are effective in the asymptotic analysis of solutions to nonlinear non-autonomous systems. See also [22] for application of the second Lyapunov method to asymptotic analysis of equations with a small parameter. Here, a Lyapunov function is used as a new dependent variable. In this section, the construction of such function and the change of variables are presented in a form suitable for further bifurcation analysis of system (1). First, consider the limiting system dx dy (4) = ∂ H (x,y), = ∂ H (x,y). dt y 0 dt − x 0 To each level line (x,y) R2 : H (x,y) = E , E (0, E ] there correspond a periodic solution { ∈ 0 } ∈ 0 x0(t, E), y0(t, E) of system (4) with a period T (E) = 2π/ω(E), where ω(E) = 0 for all E [0, E0] and ω(E)=1+ (E) as E 0. The value E = 0 corresponds to the fixed point6 (0, 0). ∈ O → Define auxiliary 2π-periodic functions X(ϕ, E) = x0(ϕ/ω(E), E) and Y (ϕ, E) = y0(ϕ/ω(E), E), satisfying the system: ∂X ∂Y ω(E) = ∂ H (X,Y ), ω(E) = ∂ H (X,Y ). ∂ϕ Y 0 ∂ϕ − X 0 These functions are used for rewriting system (1) in the action-angle variables (E, ϕ): (5) x(t)= X(ϕ(t), E(t)), y(t)= Y (ϕ(t), E(t)). From the identity H (X(ϕ, E),Y (ϕ, E)) E it follows that 0 ≡ ∂ X ∂ X 1 ϕ E = = 0. ∂ Y ∂ Y ϕ E ω(E) 6
The last inequality guarantees the reversibility of transformation (5) for all E (0, E0) and ϕ R. It can easily be checked that in new variables (E, ϕ) system (1) takes the form: ∈ ∈ dE dϕ (6) = f(E,ϕ,t), = ω(E)+ g(E,ϕ,t), dt dt where
f(E,ϕ,t) ω(E) ∂ϕH X(ϕ, E),Y (ϕ, E),t F X(ϕ, E),Y (ϕ, E),t ∂ϕX(ϕ, E) , ≡ − − g(E,ϕ,t) ω(E) ∂EH X(ϕ, E),Y (ϕ, E),t 1 F X(ϕ, E),Y (ϕ, E),t ∂EX(ϕ, E) ≡ − − 4 STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS are 2π-periodic functions with respect to ϕ. Since (0, 0) is the equilibrium of system (1), we have f(0,ϕ,t) 0. From (3) it follows that ≡ ∞ ∞ − k − k f(E,ϕ,t)= t q f (E, ϕ), g(E,ϕ,t)= t q g (E, ϕ), t , k k →∞ Xk=1 Xk=1 where
fk(E, ϕ) ω(E) ∂ϕHk X(ϕ, E),Y (ϕ, E) Fk X(ϕ, E),Y (ϕ, E) ∂ϕX(ϕ, E) , (7) ≡− − gk(E, ϕ) ω(E) ∂EHk X(ϕ, E),Y (ϕ, E) Fk X(ϕ, E),Y (ϕ, E) ∂EX(ϕ, E) . ≡ − To simplify the first equation in (6), we consider the transformation of the variable E in the form:
N − k (8) VN (E,ϕ,t)= E + t q vk(E, ϕ), Xk=1 where the coefficients vk(E, ϕ) are chosen in such a way that the right-hand side of the equation for the new variable v(t) V (E(t), ϕ(t),t) does not depend on ϕ at least in the first terms of the asymptotics: ≡ N N dv − k − N+1 (9) = t q Λ (v)+ R (v,ϕ,t),R (v,ϕ,t)= (t q ), t . dt k N+1 N+1 O →∞ Xk=1 Under the transformation (E, ϕ) (v, ϕ) the form of the second equation in (6) changes slightly: 7→ ∞ dϕ − k (10) = ω(v)+ G (v,ϕ,t), G (v,ϕ,t)= t q g (v, ϕ), t . dt N N N,k →∞ Xk=1
Here, each function gN,k(v, ϕ) is 2π-periodic with respect to ϕ and is expressed through v1, . . . , vk and g1,...,gk. For example,
g (v, ϕ) = g (v, ϕ) ω′(v)v (v, ϕ), N,1 1 − 1 ′ ′′ 2 gN,2(v, ϕ) = g2(v, ϕ) ω (v) v2(v, ϕ) ∂vv1(v, ϕ)v1(v, ϕ) ∂vg1(v, ϕ)v1(v, ϕ)+ ω (v)v1 (v, ϕ). − − − Note that such a transformation is usually applied in the averaging of systems with a small parameter and is associated with a fast variable elimination [23]. Here, ϕ can serve as an analogue of a fast variable. However, the presence of a small parameter is not assumed in the system, and the terms “fast” and “slow” variables are not appropriate. Let us move on to the calculation of the coefficients vk(E, ϕ). The total derivative of the function VN (E,ϕ,t) with respect to t along the trajectories of system (6) has the following form
dVN := ∂tVN (E,ϕ,t)+ f(E,ϕ,t)∂E VN (E,ϕ,t)+ ω(E)+ g(E,ϕ,t) ∂ϕVN (E,ϕ,t) dt (6) ∞ − k k q = t q ω(E)∂ v (E, ϕ)+ f (E, ϕ) − v (E, ϕ) (11) ϕ k k − q k−q Xk=1 ∞ − k + t q fj(E, ϕ)∂E vi(E, ϕ)+ gj(E, ϕ)∂ϕvi(E, ϕ) , Xk=2 i+Xj=k where it is assumed that vj(E, ϕ) 0 for j 0 and j > N. Substituting (8) into the right-hand side of (9) and the comparison of the result≡ with≤ (11) lead to the following chain of differential equations:
(12) ω(E)∂ v = Λ (E) f (E, ϕ)+ Z (E, ϕ), k = 1, 2,...,N, ϕ k k − k k STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS 5
where each function Zk(E, ϕ) is expressed through v1, . . . , vk−1. In particular, Z 0, 1 ≡ Z v ∂ Λ f ∂ v + g ∂ v , 2 ≡ 1 E 1 − 1 E 1 1 ϕ 1 1 Z v ∂ Λ + v ∂ Λ + v2∂2 Λ f ∂ v + g ∂ v , 3 ≡ 2 E 1 1 E 2 2 1 E 1 − j E i j ϕ i i+Xj=3 k q Z C vα1 vα2 . . . vαi ∂mΛ f ∂ v + g ∂ v + − v , k ≡ i,α,m 1 2 i E j − j E i j ϕ i q k−q j+α1+2αX2+···+iαi=k i+Xj=k α1+···+αi=m≥1 where Ci,α,m = const. Define (13) Λ (E)= f (E, ϕ) Z (E, ϕ) , k h k i − h k i where 2π def 1 f (E, ϕ) = f (E, φ) dφ. h k i 2π Z k 0 Hence, for every k 1 the right-hand side of (12) is 2π-periodic function with respect to ϕ with zero average. By integrating≥ (12) with respect to ϕ, we obtain
vk(E, ϕ)=Hk(X(ϕ, E),Y (ϕ, E)) ϕ 1 + Λ (E) ω(E)F (X(φ, E),Y (φ, E))∂ X(φ, E)+ Z (E, φ) dφ ω(E) Z k − k φ k 0 for k = 1,...,N. It can easily be checked that each vk(E, ϕ) is a smooth 2π-periodic function with respect to ϕ such that v (0, ϕ) 0. From (13) it follows that Λ (v) = (v) as v 0. The function k ≡ k O → RN+1(E,ϕ,t) has the following form: ∞ k−1 − k k q R (E,ϕ,t) t q f − v + f ∂ v + g ∂ v . N+1 ≡ k − q k−q k−j E j k−j ϕ j k=XN+1 Xj=1
It is clear that RN+1(E,ϕ,t) is 2π-periodic functions with respect to ϕ such that RN+1(0,ϕ,t) 0 −(N+1)/q ≡ and RN+1(v,ϕ,t)= (t ) as t for all v [0, d0], ϕ R with d0 = (1 σ)E0. From (8) it followsO that for all σ (0→∞, 1) there exists∈ t > 0 such∈ that − ∈ 0 (14) (1 σ)E V (E,ϕ,t) (1 + σ)E, (1 σ) ∂ V (E,ϕ,t) (1 + σ) − ≤ N ≤ − ≤ E N ≤ for all E [0, E0], ϕ R and t t0. Hence, the transformation (E, ϕ) (v, ϕ) is reversible. Thus, we∈ have ∈ ≥ 7→ Lemma 1. There exists a reversible change of variables (x,y) (v, ϕ) which reduces system (1) into the form (9), (10). 7→
4. Bifurcations of the equilibrium In this section, possible bifurcations of the fixed point (0, 0) of (1) as well as the trivial solution of equation (9) are discussed. From the properties of the function RN+1(v,ϕ,t) it follows that the leading terms of asymptotics for solutions of (9) does not depend on ϕ. The long-term behaviour of solutions v(t) is determined by the functions Λ (v) N . Besides, from (10) it follows that ϕ(t) { k }k=1 →∞ as t , while v(t) [0, d0]. Let→∞n 1 be the least∈ natural number such that Λ (v) 0. Then equation (9) takes the form: ≥ n 6≡ N dv − k = t q Λ (v)+ R (v,ϕ,t), t t . dt k N+1 ≥ 0 kX=n From (2) and (5), it follows that E = r2/2+ (r3) as r 0. Combining this with (14), we see that O → VN (E,ϕ,t) is positive definite function in the vicinity of the fixed point (0, 0). Thus, VN (E,ϕ,t) in the 6 STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS variables (x,y) can be used as a Lyapunov function candidate for system (1). If the total derivative of VN (E,ϕ,t) with respect to t along the trajectories of system (6) is sign definite for E close to zero and ϕ R, then this function can be effectively used for the stability analysis of the equilibrium (0, 0). It ∈ can easily be seen that the right-hand side of (9) coincides with the total derivative of Vn(E,ϕ,t). Lemma 2. Let n 1 be an integer such that Λ (v) 0 for k Proof. Consider VN (E,ϕ,t) with N = n as a Lyapunov function candidate for system (1). From (15) it follows that the function v(t)= Vn(E(t), ϕ(t),t) satisfies the equations: dv − n − 1 = t q v(λ + (v)+ (t q )) dt n O O as t and v 0 for all ϕ R. Hence, for all σ (0, 1) there exist 0 < d1 d0 and t1 t0 such that→ ∞ → ∈ ∈ ≤ ≥ dv − n t q (1 σ)λmv 0 if λn > 0, (16) dt ≥ − ≥ dv − n t q (1 σ) λ v 0 if λ < 0, dt ≤− − | m| ≤ n for all v [0, d1], ϕ R and t t1. Integrating the first estimate in (16) with respect to t yields the instability∈ of the trivial∈ solution≥v(t) 0 of equation (9) for all ϕ R. Indeed, there exists ǫ (0, d /4) ≡ ∈ ∈ 1 such that for all δ (0,ǫ) the solution v(t) with initial data v(t1) = δ exceeds the value ǫ as t>t∗, where ∈ 1 2ǫ (1−σ)λn n t∗ = t1 if = 1; δ q n n 1− q 1− q q n 2ǫ n t∗ = t1 + − log if = 1. (1 σ)λnq δ q 6 − Similarly, by integrating the second estimate in (16), we obtain the following inequalities: t −(1−σ)|λn| n 0 v(t) v(t1) if = 1, ≤ ≤ t1 q (1 σ) λ q n 1− n n n 1− q q (17) 0 v(t) v(t1)exp − | | t t1 if = 1, ≤ ≤ − q n − q 6 − as t t1. From the last estimates it follows that the trivial solution of system (6) is stable for all ϕ ≥R. Moreover, the stability is exponential if n stable if λn < 0 and γm,s < 0; • unstable if λ > 0 and γ > 0. • n m,s Proof. As above, consider Vn(E,ϕ,t) as a Lyapunov function candidate. In this case, its total derivative has the form: dVn dv − m − n = t q Λm(v)+ ̺m(v,ϕ,t) + t q Λn(v)+ ̺n(v,ϕ,t) , dt (6) ≡ dt h i h i where v(t) = V (E(t ), ϕ(t),t), ̺ (v,ϕ,t) Mt−1/qvs, ̺ (v,ϕ,t) Mt−1/qv as v 0, t for n | m | ≤ | n | ≤ → → ∞ all ϕ R with M = const > 0. Therefore, for all σ (0, 1) there exist 0 < d1 d0 and t1 t0 such that ∈ ∈ ≤ ≥ n n dv − s − (1 σ) t q γn,s v + t q λn v 0 if λn < 0, γm,s < 0, dt ≤− − | | | | ≤ n n dv − s − (1 σ) t q γn,sv + t q λnv 0 if λn > 0, γm,s > 0, dt ≥ − ≥ for all v [0, d ], ϕ R and t t . From the last estimates it follows that the solution E(t) 0 to ∈ 1 ∈ ≥ 1 ≡ system (6) is stable if λn < 0, γm,s < 0 and unstable if λn > 0, γm,s > 0. Combining (5) and (14), we obtain the corresponding propositions on the stability of the equilibrium (0, 0) of system (1). Note that in some cases the last proposition can be improved. In particular, we have Lemma 4. Under the conditions of Lemma 3, we have in case m m n dVn dv − s − 1 − − 1 = t q v (γm,s + (v)+ (t q )) + t q v(λn + (v)+ (t q )) dt (6) ≡ dt O O O O as t , v 0 for all ϕ R. The right-hand side of the last expression is not sign definite uniformly for all→∞ small →v and big τ.∈ Indeed, if v ǫ and t ǫ−κ, where 0 < ǫ 1, κ = const, then the sign of dv/dt is determined by λ in case κ n n+qν n dU du −1 − s−1 − − +1 (20) = = u νt + t q (λn + γm,su )+ (t q )+ (t q ) dt (6) dt O O as t for all u [0,U ] and ϕ R with U = const > 0. →∞ ∈ 0 ∈ 0 8 STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS Consider the case m n du − s−1 −ν − 1 = ut q λn + γm,su + (t )+ (t q ) dt O O as t . If λ < 0, then for all σ (0, 1) there exist 0 < U U and t t such that →∞ n ∈ 1 ≤ 0 1 ≥ 0 du − n t q (1 σ) λ u 0 dt ≤− − | n| ≤ for all u [0,U ], ϕ R and t t . Integrating the last inequality with respect to t, we get an ∈ 1 ∈ ≥ 1 estimate of the form (17) as t t1. Combining this with (18), we get exponential stability of the solution E(t) 0 of equation (6≥). ≡ If λ > 0 and γ < 0, the leading term of du/dt has a zero at u = U , U = (λ / γ )1/(s−1). n m,s c c n | m,s| Let us show that U(E(t), ϕ(t),t) Uc as t . Consider the change of variable u(t)= Uc + z(t) in equation (19). Then z(t) satisfies→ the equation:→∞ n dz − s−1 (21) = t q (Uc + z) λn + γm,s(Uc + z) + p(z,ϕ,t), dt where p(z,ϕ,t) Mt−n/q[t−ν + t−1/q] as t t for all z z and ϕ R with positive constants M | |≤ ≥ 1 | |≤ 1 ∈ and z1. It can easily be checked that the unperturbed equation with p(z,ϕ,t) 0 has asymptotically stable solution z(t) 0. Let us show that this solution is stable with respect≡ to the perturbation p(z,ϕ,t). Consider ≡ℓ(z) = z2/2 as a Lyapunov function candidate for equation (21). The total derivative of ℓ(z) has the form: dℓ n 1 − q s−1 2 −ν − q = t z γm,s (s 1)Uc z + (z )+ (t )+ (t ) dt (21) − | | − O O O as z 0 and t . Therefore, for all σ > 0 there exist 0 < z z and t t such that → →∞ 2 ≤ 1 2 ≥ 1 dℓ n − q s−1 −ν −1/q t 2(1 σ) γm,s (s 1)Uc ℓ + Mz2[t + t ] dt (21) ≤ − − | | − −ν −1/q for all z z2, ϕ R and t t2. By integrating the last inequality, we get ℓ(z(t)) = (t )+ (t ) | |≤ ∈ ≥ O−ν/2 O −1/2q as t for solutions with initial data z(t2) z2. Hence, U(E(t), ϕ(t),t)= Uc+ (t )+ (t ) and →∞E(t)= (t−ν) as t . Therefore,| the|≤ solution E(t) 0 of (6) is polynomiallyO stable.O ConsiderO the case m du −1 s−1 −ν − 1 = ut λn + ν + γm,su + (t )+ (t q ) dt O O as t . If λ + ν < 0, then for all σ> 0 there exist 0 < U U and t t such that →∞ n 1 ≤ 0 1 ≥ 0 du t−1( λ + ν + σ)u 0 dt ≤ −| n | ≤ −ν for all u [0,U1], ϕ R and t t1. Integrating the last inequality yields 0 v(t) = t u(t) ∈λn+σ ∈ ≥ ≤ ≤ v(t1)(t/t1) as t t1. Therefore, the solution E(t) 0 is polynomially stable. If λn + ν > 0 and ≥ ≡ −ν/2 γm,s < 0, then, as above, there is a family of solutions such that U(E(t), ϕ(t),t) = Uν + (t )+ −1/2q 1/(s−1) O (t ) as t , where Uν = ((λn + ν)/ γm,s ) . Taking into account the transformation of Ovariables, we get→ polynomial ∞ stability of the solution| | E(t) 0 to system (6). In the case m n−m dw −1 s−1 −β − = t w η + γm,sw + (t )+ (t q ) dt O O as t for all w [0,W ] and ϕ R with W = const > 0. If γ < 0, then, as above, system has →∞ ∈ 0 ∈ 0 m,s a family of solutions such that W (E(t), ϕ(t),t) = W + (t−η/2)+ (t−(n−m)/2q) as t , where c O O → ∞ W = (η/ γ )1/(s−1). Hence, E(t)= (t−η) as t . c | m,s| O →∞ STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS 9 4 4 4 2 2 2 Du Dw u Du Dw u Du s s s , , , n 0 n 0 n 0 Γ Γ Γ -2 -2 -2 -4 De s Dp s -4 Dp s -4 Dp s -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 Λm Λm Λm (a) n < q (b) n = q (c) q Figure 1. Partition of the parameter plane (λn, γm,s) when m Finally, consider the case q m stable if γm,s < 0 and γn,d < 0; • unstable if γ > 0 and γ > 0. • m,s n,d Proof. If s d, the total derivative of the function V (E,ϕ,t) has the asymptotics: ≤ n n dVn dv − s − 1 = t q v (γn,s + (v)+ (t q )) dt (6) ≡ dt O O as t and v 0 for all ϕ R. If γ > 0, the total derivative is locally positive and the solution →∞ → ∈ n,s E(t) 0 of system (6) is unstable. In the opposite case, when γn,s < 0, the total derivative is locally negative≡ and the trivial solution is stable. Let s > d, γ < 0 and γ < 0. Then for all σ> 0 there exist 0 < d d and t t such that m,s n,d 1 ≤ 0 1 ≥ 0 dv n n−m − q d s−d − q t (1 σ)v γm,s v + γn,d t 0 dt ≤− − | | | | ≤ for all v [0, d ], t t and ϕ R. Hence, the solution E(t) 0 to system (6) is stable. Similarly, if ∈ 1 ≥ 1 ∈ ≡ γm,s > 0 and γn,d > 0, the trivial solution is unstable. 5. Bifurcations of limit cycles Let us show that decaying non-autonomous bifurcations may lead to the appearance of limit cycles. Lemma 6. Let n 1 be an integer such that Λk(v) 0 for k dv − n − 1 = t q Λn(v)+ (t q ) dt O as t for all v [0, d0] and ϕ R. The change of the variable v(t) = Vc + z(t) leads to the following→ ∞ equation: ∈ ∈ dz − n (22) = t q Λ (V + z)+ p(z,ϕ,t), dt n c −(n+1)/q where p(0,ϕ,t) 0, p(z,ϕ,t) Mt for all z [ Vc, d0 + Vc], ϕ R and t t0 with positive 6≡ | |≤′ ∈ − ∈ ≥ constant M > 0. From λn := Λn(Vc) < 0 it follows that the trivial solution of (22) with p(z,ϕ,t) 0 is stable. Let us show that this solution is stable with respect to the perturbation p(z,ϕ,t). Indeed,≡ consider ℓ(z)= z2/2 as a Lyapunov function candidate for (22). Its total derivative has the form: dℓ − n (23) = t q Λn(Vc + z)z + p(z,ϕ,t)z. dt (22) 2 First note that there exists δ1 > 0 such that Λn(Vc + z)z λn z /2 as z δ1. Let us fix 0 <ǫ<δ1 and choose ≤ −| | | |≤ ǫ 4M q δ∗ = , t∗ = max ,t0 , 2 nδ∗ λn o | | then dℓ n λ − 1 n λ − q 2 n −1 q − q 2 n t z | | Mδ∗ t∗ t z | | dt ≤− 2 − ≤− 4 for all δ z ǫ, ϕ R and t t . Hence, any solution z(t) with initial data z(τ ) δ , τ t ∗ ≤ | | ≤ ∈ ≥ ∗ | 0 | ≤ ∗ 0 ≥ ∗ cannot leave the domain z <ǫ as t τ0. Returning to the original variables, we obtain the result of the Lemma. {| | } ≥ STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS 11 Let us show that H (x(t),y(t)) V as t if n q. From (23) it follows that dℓ/dt 0 → c → ∞ ≤ ≤ t−n/q( λ ℓ + Mδ t−1/q) as t t . By integrating the last inequality in the case n = q, we get −| n| 1 ≥ 0 t −|λn| 1 t −|λn| |λn|− −1 0 ℓ(z(t)) ℓ(z(t0)) + Mδ1t s q ds ≤ ≤ t0 Z t0 with z(t ) δ . Similar estimates hold in the case n Corollary 1. Let 1 n q be an integer such that in Λk(v) 0 for k We assume that vi(E, ϕ) 0 for i < h. According to the scheme described in section 3, the functions v (E, ϕ) are determined from≡ system (12). Therefore, for h k < l, we have k ≤ Z (E, ϕ) 0, Λ (E) f (E, ϕ) 0, v (E, ϕ) H (X(ϕ, E),Y (ϕ, E)). k ≡ k ≡ h k i≡ k ≡ k For l k n 1, condition (25) is used to guarantee the equality Λk(E) 0. Indeed, from (11) it follows that≤ ≤ − ≡ Z (E, ϕ) = f (E, ϕ)∂ v (E, ϕ)+ g (E, ϕ)∂ v (E, ϕ) l − i E j i ϕ j i+Xj=l l−h = f (E, ϕ)∂ v (E, ϕ)+ g (E, ϕ)∂ v (E, ϕ) − l−j E j l−j ϕ j Xj=h l−h = ω(E) ∂ H ∂ H ∂ H ∂ H 0. ϕ l−j E j − E l−j ϕ j ≡ Xj=h Hence, Λ (E) f (E, ϕ) ω(E) F (X(ϕ, E),Y (ϕ, E))∂ X(ϕ, E) l ≡ h l i≡ h l ϕ i ∂ H (X(ϕ, E),Y (ϕ, E))F (X(ϕ, E),Y (ϕ, E)) 0 ≡ h y 0 l i≡ and v (E, ϕ) H (X(ϕ, E),Y (ϕ, E)) +v ˆ (E, ϕ), where l ≡ l l vˆ (E, ϕ)= F (X(ϕ, E),Y (ϕ, E))∂ X(ϕ, E) dϕ. l Z l ϕ For l + 1 k n 1, we have ≤ ≤ − Z (E, ϕ) = f (E, ϕ)∂ v (E, ϕ)+ g (E, ϕ)∂ v (E, ϕ) . k − i E j i ϕ j i+Xj=k Note that fj and gj with j l are not involved in these sums. Such functions have the multipliers ∂ v and ∂ v , correspondingly.≥ If j l, then k j n j 1 n l 1 h 1 and ϕ k−j E k−j ≥ − ≤ − − ≤ − − ≤ − vk−j(E, ϕ) 0. Similarly, the functions ∂ϕvj and ∂ϕvj with j l are not contained in these sums. Therefore, ≡ ≥ Z (E, ϕ) = f (E, ϕ)∂ v (E, ϕ)+ g (E, ϕ)∂ v (E, ϕ) k − i E j i ϕ j i+Xj=k h≤i,j vˆ (E, ϕ)= F (X(ϕ, E),Y (ϕ, E))∂ X(ϕ, E) dϕ. k Z k ϕ We see that v (E, ϕ)= (E), vˆ (E, ϕ)= (E3/2) as E 0 for all ϕ R. k O k O → ∈ STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS 13 For k = n, the function Zn(E, ϕ) has the following form: Z (E, ϕ) = f (E, ϕ)∂ v (E, ϕ)+ g (E, ϕ)∂ v (E, ϕ) n − i E j i ϕ j i+Xj=n h≤i,j≤l = f (E, ϕ)∂ v (E, ϕ)+ g (E, ϕ)∂ v (E, ϕ) − i E n−i i ϕ n−i i∈{Xl,n−l} f (E, ϕ)∂ v (E, ϕ)+ g (E, ϕ)∂ v (E, ϕ) − i E j i ϕ j i+Xj=n h≤i,j≤l−1 Z1(E, ϕ)+ Z2(E, ϕ). ≡ n n If l + h>n, then Z1(E, ϕ) 0. If l + h = n, n ≡ 1 (28) Zn(E, ϕ)= ω(E) ∂ϕHh(∂E vˆl + Fl∂EX) ∂EHh(∂ϕvˆl + Fl∂ϕX) . − It can easily be checked that (29) Z2(E, ϕ)= ω(E) ∂ H ∂ H + ∂ H ∂ H 0. n − ϕ j E i E j ϕ i ≡ i+Xj=n h≤i,j≤l−1 Consequently, Λ (E)= ω(E) F (X(ϕ, E),Y (ϕ, E))∂ X(ϕ, E) Z (E, ϕ) n h n ϕ − n i and vn(E, ϕ)= Hn(E, ϕ) +v ˆn(E, ϕ), where 1 vˆn(E, ϕ)= Λn(E)+ Zn(E, ϕ) ω(E)F (X(ϕ, E),Y (ϕ, E))∂ϕX(ϕ, E) dϕ. ω(E) Z − Since H (X(ϕ, E),Y (ϕ, E)) E, then from (2) and (26) it follows that 0 ≡ X(ϕ, E)= √2E cos ϕ + (E), Y (ϕ, E)= √2E sin ϕ + (E), − O O F (X(ϕ, E),Y (ϕ, E)) = λ √2E sin ϕ + (E),Z (E, ϕ)= (E3/2) n n O n O as E 0 for all ϕ R. Thus → ∈ 1 Λn(E) = ω(E) (√2E sin ϕ + (E))(λn√2E sin ϕ + (E)) Zn(E, ϕ) Dω(E) O O − E = λ E + (E2), E 0. n O → This implies that system (1) satisfies the conditions of Lemma 2 and the stability of the equilibrium is determined by the sign of λn. Example 1. The system dx dy − l − n (30) = y, = sin x + t q κ y sin x + t q λ y, t 1, dt dt − l n ≥ where κ , λ = const, 0 F (x,y)∂ H (x,y) dl = 0, E (0, E ), l k 0.4 0.4 0.4 0.2 0.2 0.2 y 0.0 y 0.0 y 0.0 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 x x x n 1 n n 3 (a) = , λn = 1 (b) =1, λn = 1 (c) = , λn = 1 q 2 − q − q 2 − 2 1.0 0.5 1 0.5 y 0 y 0.0 y 0.0 -0.5 -1 -0.5 -1.0 -2 -2 -1 0 1 2 -1.0 -0.5 0.0 0.5 1.0 -0.5 0.0 0.5 x x x n 1 n n 3 (d) = , λn =0.3 (e) =1, λn =0.3 (f) = , λn =0.3 q 2 q q 2 Figure 2. The evolution of (x(t),y(t)) for solutions of (30) with q = 4, l = 1, κl = 1. The black points correspond to initial data (0.5, 0). unstable if λ > 0 and α + 3β > 0. • n m m Proof. The proof is similar to the proof of Theorem 1. In this case, we show that Lemma 3 is applicable with s = 2 and γm,s = (αm + 3βm)/2. Note that the change of the variables based on (27) leads to equation (9) with Λ (E) 0 for k dx dy − m − n (31) = y, = sin x + t q α x2y + t q λ y, t 1, dt dt − m n ≥ where αm, λn = const, 0 3 1.0 2 1 0.5 x 0 x 0.0 -1 -0.5 -2 -3 -1.0 0 100 200 300 400 0 100 200 300 400 t t (a) αm =0, λn =0.4 (b) αm = 2, λn =0.4 − Figure 3. The evolution of x(t) for solutions of (31) with q = 2, m = 1 and n = 2. The dashed lines correspond to tν/2, ν = (n m)/q = 0.5. ± − has the following eigenvalues: 1 n 2n − q 2 − q µ±(t)= λnt i 4 λnt . 2 ± q − Let λn > 0, then µ+(t) > 0 for all t 1. From Theorem 1 it follows that the equilibrium (0, 0) is unstable in the linearizedℜ system. On the≥ other hand, system (31) satisfies the conditions of Theorem 2 2 with l = m and Fm(x,y)= αmx y. From Corollary 2 it follows that the equilibrium is stable if λn > 0 and αm < 0 (see Fig. 3). Example 3. The system dx dy − m − n (32) = y, = sin x + t q δ x4y + t q α x2y, t 1 dt dt − m n ≥ 2 4 2 has the form of (1) with H0(x,y) = 1 cos x+y /2, Fm(x,y)= δmx y, Fn(x,y)= αnx y, Hi(x,y) 0 for i = 0, and F (x,y) 0 for j m,n− . It can easily be checked that ≡ 6 j ≡ 6∈ { } Z (E, ϕ) 0, Λ (E) 0, k (33) Fi(x,y) 0, 1 i 2 0.5 1 y 0 y 0.0 -1 -0.5 -2 -0.5 0.0 0.5 -2 -1 0 1 2 x x (a) δm = 0.3, αn = 0.3 (b) δm =0.3, αn =0.3 − − Figure 4. The evolution of (x(t),y(t)) for solutions of (32) with q = 2, m = 1 and n = 2. The black points correspond to initial data. which reduces system (1) to the following form: dv − n dϕ = t q Λ (v)+ R (v,ϕ,t), = ω(v)+ G(v,ϕ,t), dt n n+1 dt − n+1 − 1 R (v,ϕ,t)= (t q ), G(v,ϕ,t)= (t q ), t , n+1 O O →∞ 2 Λn(E) ω(E) Fn(X(ϕ, E),Y (ϕ, E))∂ϕX(ϕ, E) (λn µnE) ∂yH0 . ≡ h i≡ − h i Note that Λ (E)= λ E + (E2) as E 0. Hence, the equilibrium is asymptotically stable if λ < 0 n n O → n and the equilibrium is unstable if λn > 0. ′ If λn > 0 and µn > 0, the equation Λn(E) = 0 has the root Ec = λn/µn such that Λn(Ec) < 0. In this case, from Lemma 6 it follows that there is a stable limit cycle H0(x,y)= Ec. Remark 1. If λ < 0, γ < 0 and λ /µ < E , the fixed point (0, 0) is asymptotically stable and the n n | n n| 0 limit cycle H0(x,y) = Ec with Ec = λn/µn is instable. If Ec is the minimal nonzero root of Λn(E), then (x,y) : 0 H (x,y) < E is the domain of attraction of the fixed point (0, 0) . { ≤ 0 c} Example 4. The system 2 2 dx dy − n (x + y ) (34) = y, = x + t q y λn + κnx µn , t 1 dt dt − − 2 ≥ 2 2 with λn,µn, κn = const satisfies the conditions of Theorem 3 with H0(x,y) = (x +y )/2 and Fˆn(x,y)= κnxy. Therefore, if λn > 0 and µn > 0, the system has the stable limit cycle H0(x,y) = λn/µn (see Fig. 5). 7. Conclusion Thus, we have described possible bifurcations in asymptotically autonomous Hamiltonian systems in the plane. The important feature of non-autonomous systems is the inefficiency of the linear stability analysis: there are examples of nonlinear systems whose solutions behave completely differently than the solutions of corresponding linearized system. Here, through a careful nonlinear analysis based on the Lyapunov function method we have shown that depending on the structure of decaying pertur- bations the equilibrium of the limiting system can preserve or lose stability. Note also that in this paper perturbations preserving the equilibrium of a Hamiltonian system have been considered. If the equilibrium disappears in the perturbed equations, instead of the equilibrium, a particular solution of a perturbed system should be considered. STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS 17 2 3 2 1 1 y 0 y 0 -1 -1 -2 -2 -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 x x (a) λn = µn =0.5 (b) λn = µn = 0.5 − Figure 5. The evolution of (x(t),y(t)) for solutions of (34) with n/q = 1 and κn = 0 as t 1. The black points correspond to initial data. The black solid (dashed) line ≥ corresponds to the stable (unstable) limit circle H0(x,y) = 1. Acknowledgments The research is supported by the Russian Science Foundation (Grant No. 20-11-19995). References [1] J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer, New York, 1983. [2] P. A. Glendinning, Stability, instability and chaos: an introduction to the theory of nonlinear differential equations, Cambridge University Press, Cambridge, 1994. [3] H. Hanßmann, Local and semi-local bifurcations in Hamiltonian systems - Results and examples, Lecture Notes in Mathematics, 1893, Springer, Berlin, 2007. [4] L. Markus, Asymptotically autonomous differential systems. In: S. Lefschetz (ed.), Contributions to the theory of nonlinear oscillations III, Ann. Math. Stud., vol. 36, pp. 17–29, Princeton University Press, Princeton, 1956. [5] J. S. W. Wong, T. A. Burton, Some properties of solutions of u′′(t)+ a(t)f(u)g(u′)=0. II, Monatsh. Math., 69 (1965) 368–374. [6] R. C. Grimmer, Asymptotically almost periodic solutions of differential equations, SIAM J. Appl. Math., 17 (1968) 109–115. [7] H. R. Thieme, Convergence results and a Poincar´e-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992) 755–763. [8] H. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math., 24 (1994) 351–380. [9] J. A. Langa, J. C. Robinson, A. Su´arez, Stability, instability and bifurcation phenomena in nonautonomous differential equations, Nonlinearity, 15 (2002) 887–903. [10] P. E. Kloeden, S. Siegmund, Bifurcations and continuous transitions of attractors in autonomous and nonautonomous systems, Internat. J. Bifur. Chaos., 15 (2005) 743–762. [11] M. Rasmussen, Bifurcations of asymptotically autonomous differential equations, Set-Valued Anal., 16 (2008) 821–849. [12] C. P¨otzsche, Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach, Discrete Contin. Dynam. Systems B, 14 (2010) 739–776. [13] A. S. Fokas, A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, Painlev´etranscendents. The Riemann-Hilbert approach, Mathematical Surveys and Monographs, vol. 128, Amer. Math. Soc., Providence, 2006. [14] L. A. Kalyakin, O. A. Sultanov, Stability of autoresonance models, Differ. Equat., 49 (2013) 267–281. 18 STABILITY AND BIFURCATION PHENOMENA IN ASYMPTOTICALLY HAMILTONIAN SYSTEMS [15] O. A. Sultanov, Stability of capture into parametric autoresonance, Proc. Steklov Inst. Math., 295, suppl. 1 (2016) 156–167. [16] A. Pikovsky, M. Rosenblum, J. Kurths. Synchronization: a universal concept in nonlinear sciences, Cam- bridge University Press, Cambridge, 2001. [17] L. A. Kalyakin, Synchronization in a nonisochronous nonautonomous system, Theoret. and Math. Phys., 181 (2014) 1339–1348. [18] W. Wasоw, Asymptotic expansions for ordinary differential equations, John Wiley and Sons, Inc., New York, 1966. [19] O. Sultanov, Stability and asymptotic analysis of the autoresonant capture in oscillating systems with combined excitation, SIAM J. Appl. Math., 78 (2018) 3103–3118. [20] O. Sultanov, Lyapunov functions and asymptotic analysis of a complex analogue of the second Painlev´e equation, J. Physics: Conf. Ser., 1205 (2019) 012056. [21] O. A. Sultanov, Bifurcations of autoresonant modes in oscillating systems with combined excitation, Studies in Appl. Math., 144 (2020) 213–241. [22] M. M. Hapaev, Averaging in stability theory: a study of resonance multi-frequency systems, Kluwer Aca- demic Publishers, Dordrecht, Boston, 1993. [23] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, Springer, Berlin, 2006.