Periodic Perturbations: Parametric Systems

We are not going to present the classical results on linear parametric systems, since they are widely discussed in literature. Instead, we shall consider nonlinear parametric systems and discuss the conditions of new motion existence in the resonance zones: the regular ones (on an invariant torus) and the irregular ones (on a quasi-attractor). On the basis of the self-oscillatory shortened system which determines the topology of resonance zones, we study the transition from a resonance to a non-resonance case under a change of the detuning. We then apply our results to some concrete examples. It is interesting to study the behavior of a parametric system when the ring-like resonance zone is contracted into a point, i.e., to describe the bifurcations which occur in the course of transition from the plain nonlinear resonance to the parametric one. We are based on article, and we follow a material from the book.


Introduction
Consider the following system: where ε > 0 is a small parameter, ν is perturbation frequency, and g, f are continuous periodic functions of period 2π with respect to φ ¼ νt. The Hamiltonian H as well as f and g will be assumed to be sufficiently smooth in a domain G ⊂ R 2 Â S 1 (or G ⊂ R 1 Â S 1 Â S 1 ¼ R 1 Â T 2 ).
Also, we shall assume that the unperturbed (ε ¼ 0) Hamiltonian system is nonlinear and has at least one cell D filled with closed phase curves.
We especially emphasize the following condition.
We also assume the following condition.
Condition B. System (2) has a finite set of rough limit cycles (LCs) in cell D.
Changing the variables x, y to the action I and angle θ, we obtain the system in the form where are periodic of period 2π with respect to θ and φ. System (3) is defined on the direct product Δ Â S 1 Â S 1 ¼ Δ Â T 2 , where T 2 is two-dimensional torus, Δ ¼ I À ; I þ À Á , I AE ¼ I h AE À Á .
The definition of resonance. We say that in system (3) a resonance takes place if where p, q are relatively prime integer numbers.
The energy level I ¼ I pq (H x; y ð Þ ¼ h pq ) of the unperturbed system is called the resonance.
The behavior of solutions in the neighborhoods U μ ¼ I; θ ð Þ : I pq À Cμ < I < I pq þ Cμ; 0 ≤ θ ≤ 2π; C ¼ const È É , μ ¼ ffiffi ffi ε p of individual resonance levels I ¼ I pq H x; y ð Þ ¼ h pq À Á can be derived, up to the terms O μ 2 À Á , from the pendulum-type equation [1,2] where is the unperturbed solution on the level I ¼ I pq . For nondegenerate resonance zones we consider here, it holds that b 6 ¼ 0. Functions From Eq. (7) follows.

Theorem 1
If the divergence of the vector field of Eq. (6) depends on v, then the divergence of the vector field of the original system (1) contains terms which depend on both the time t and the spatial coordinates.
In many cases the converse is also true. For example, it holds for the system The terms mentioned in Theorem 1 are called nonlinear parametric terms. Our goal is to study systems of the form (1) with such terms. The existence of those leads to new motions in resonance zones [1][2][3]. We shall demonstrate these motions on examples.

Investigation of Eq. (6)
The following representations hold: where B I ð Þ is the generating function of the autonomous system (2) and B 1 I ð Þ is the derivative of B I ð Þ. We shall focus on the case when σ is sign-alternating. In this case, from Eq. (9) follows the inequality: When studying the pendulum Eq. (6), we shall distinguish two cases: (I) B I pq À Á 6 ¼ 0 and (II) In case II system (2) has a rough limit cycle (LC) in a neighborhood of the level H x; y ð Þ ¼ h pq . There is no such cycle in case I.
Case I. Neglecting terms of order μ in Eq. (6), we arrive at the integrable equation If |B I pq À Á | > max v |A * v; I pq À Á |, then Eq. (11) has no equilibrium states. The resonance level I ¼ I pq is then referred to as passable. Note that the term "passable" has its origin in the topology of the resonance zone, as opposed to the same term used in physics, where "passing" stands for a change in perturbation frequency ν. In the case under consideration, there are no periodic solutions in the vicinity of the resonance level. The most interesting case is when Eq. (11) has equilibrium states, i.e., when the condition is satisfied. The resonance level I ¼ I pq is then said to be partly passable.
Under condition (10), Eq. (6) may have limit cycles. In order to find them, one must construct the Poincaré-Pontryagin generating function.  (10) and (12), and p ¼ 3. On the period 2π=3, there is a single limit cycle (note that on the period 2π (which is the period of the unperturbed solution) there are three limit cycles). If the cycle lies outside the neighborhood of the separatrix loop of Eq. (11), then there is a corresponding two-dimensional invariant torus in the original system. Since the period of the limit cycle of Eq. (11) is of order O 1=μ À Á , we then have a long-periodic beating regime in the original system (6) (the generatrices of the torus are of different order).
However, if the limit cycle lies in the neighborhood of the separatrix loop, then the twodimensional invariant torus in the original system (1) is destroyed. The bifurcation scene in which the cycle is caught into the separatrix loop is shown in Figure 1(b). Taking into account the nonautonomous terms, which were discarded in deriving Eq. (6), leads to the homoclinic structure. Such a structure is shown in Figure 1(c) for the Poincaré map with p ¼ 3. Because of the presence of non-compact separatrices, in this case we merely have an irregular transition process.
Case II. Now, Eq. (6) always possesses equilibrium states, and we have the third kind of resonance zone, namely, an impassable zone. In order to better understand the structure of such a zone, we introduce in Eq. (6) the detuning γ between the level I ¼ I pq and the level I ¼ I 0 , near which the autonomous system (2) has a limit cycle: Then, Eq. (6) can be rewritten as In Eq. (14) we change the variables from u; v ð Þ to the action J and the angle L (in both the oscillatory and the rotational zones) and average the resulting system over the "fast" angular variable L. As a result, we arrive at the equation where Φ J ð Þ is the Poincaré-Pontryagin generating function [2] and it is discontinuous at J ¼ J c when γ 6 ¼ 0. Here, J c corresponds to the contour in the "unperturbed" system formed by the saddle and two separatrix loops embracing the phase cylinder.
We shall therefore use Melnikov's formula [4] to determine the relative position of the separatrices which in the shortened system (15) constitute the contour formed by the outer separatrix loops: Here, v 0 , u 0 is the solution of Eq. (15) on the contour consisting of the saddle and the outer separatrix loops.
In virtue of Eq. (10), the second case is the most interesting. The case |a| < 1 is somewhat special since system (14) may then have limit cycles in both the oscillatory and rotational domains, which have no generating counterparts in system (2). Limit cycles in Eq. (14) can result from the following phenomena [5]: (a) from a degenerate focus, (b) from a separatrix loop (contour), and (c) from a condensation of trajectories. However, if the number of limit cycles does not matter, it suffices to consider the case when there is no more than one limit cycle in the oscillatory domain. Then, we can make a general conclusion on the change of qualitative dynamics of Eq. (14) under variation of the detuning. However, beforehand, we should study the problem for the case when f and g are trigonometric polynomials of degree N in φ. Then, A * and σ * are also trigonometric polynomials of degree N 1 ≤ N: From the definition of functions A * v ð Þ and σ v ð Þ (see Eq. (7)), it follows that, in general, different harmonics in the perturbation contribute to A * and σ. This means that different harmonics can dominate in Eq. (17). We count only these main harmonics in Eq. (17) (for A * ) 1 and σ * ) n). We then derive from Eq. (6) the equation The generating function Φ J ð Þ for Eq. (18) can be presented as [3] ) corresponds to the upper half of the cylinder, the minus to the lower half, and δ is the Kronecker delta. This enables us to find all the bifurcation sets (except the one corresponding to a contractable separatrix loop) explicitly [6].
We shall first consider the case when γ ¼ 0. In this case Eq. (18) is identical to the standard equation [2], and Φ r ð Þ is continuous at r ¼ 1. Thus, it determines the limit cycles up to the separatrix. This case was considered in Figure 2(a-e) that the rough topological structures are shown for n ¼ 1. Note that the limit cycles can "disappear at infinity" only when B 1 ¼ 0. This is impossible when Condition B is satisfied. Figure 2(e) shows the bifurcation when the limit cycle "clings" to the separatrix contour (Φ r ð Þ has the simple root r ¼ 1). Figure 2(f) shows the corresponding behavior of the invariant curves (separatrices) of the Poincaré map for the original system with p ¼ 3. The neighborhood of the homoclinic contour is attracting. Moreover, a complicated structure exists in the neighborhood [7], and, consequently, we have a quasi-attractor, i.e., a nontrivial hyperbolic set, and stable points can exist in it.  When γ 6 ¼ 0 the generating function Φ r ð Þ is discontinuous at r ¼ 1. The bifurcation of the cycle clinging to the separatrix must, therefore, be considered separately.
Using Melnikov's formula, we compute Δ AE 1 , which measures the split of the unperturbed separatrix for Eq. (18). One can see that equation Then, using Eq. (19) and assuming (for concreteness) n ¼ 1, we find the bifurcational values It is observed that the transformation of the phase portrait of Eq. (18) for r ffi 1 involves the creation of a contractable separatrix loop. By Condition B, we have a 6 ¼ 0, which implies that the saddle number is nonzero. The separatrix loop can, therefore, give rise to one limit cycle only [5]. The corresponding bifurcational set γ AE 1 a ð Þ in the parameter plane can be found numerically.
We thus obtain a partition of the parameter plane a; γ ð Þ into domains corresponding to different topological structures for Eq. (18), as well as the structures themselves (they are shown in Figure 3) for n ¼ 1. The structures corresponding to cases 8-12 are not shown in Figure 3, since they can be obtained from structures 5, 6, 3, 2, and 14, respectively, by the directions of the coordinate axes.
Note that, along with a non-contractable separatrix loop, Eq. (18) has either a stable limit cycle, or a stable equilibrium state, or the stable "point at infinity." This means that no quasi-attractor can exist in the original nonautonomous system when γ 6 ¼ 0. Remark that the homoclinic structure exists for a small range of γ values jγ À γ AE j ≃ exp À1=μ À Á À Á .
Those limit cycles of Eq. (18) which do not lie in the neighborhood of the unperturbed separatrix contour correspond to the two-dimensional invariant tori in the original system (like in the case B 6 ¼ 0). Unlike when B 6 ¼ 0, two kinds of such tori may exist in Eq. (18) corresponding to the limit cycles in the oscillatory and rotational domains. The tori corresponding to the cycles in the rotational domain (with one exception) have no generating "Kolmogorov torus" in the perturbed Hamiltonian system, while the (asymptotically stable) tori corresponding to the limit cycles in the oscillatory domain are images of the tori occupying the next level in the hierarchy of resonances.
Remark The cases of odd and even n should be considered separately. When n is even, an unstable cycle clings to the separatrix loop. For odd n the same thing happens to a stable cycle.
Only the case of odd n is therefore interesting when one studies the problem of existence of a quasi-attractor.
According to the bifurcation diagram (Figure 3), it is convenient to break the case |a| < 1 into three sub-cases: (a) À1 < a < a * , (b) a * < a < 0, and (c) 0 < a < 1, a * ¼ 1= 1 À 4n 2 À Á . Let n be odd. Then considering the solutions on the original cylinder v mod2π ð Þ ; u f g , we derive the following theorem.
1. Let a ∈ À1; a * ð Þ. Then, (1) when γ > γ þ 1 > 0, Eq. (14) has exactly one stable limit cycle (LC) in the rotational domain and no more than p n À 1 ð Þ LCs in the oscillatory domain (OD); (2) when γ þ 1 < γ < γ þ , there are p additional LCs in the OD, which are born from the separatrix loops at γ ¼ γ þ 1 ; (3) when γ ¼ γ þ , the stable LC in the rotational domain clings to the separatrix contour Γ þ p consisting of p saddles and their outer separatrices going from one saddle to another, while the "free" unstable separatrices approach an LC in the OD; (4) when γ À < γ < γ þ , there are no LCs in the rotational domain and no more than pn LCs in the OD; (5) when γ ¼ γ À , there appears a separatrix contour Γ À p which consists of p saddles and their outer separatrices but has orientation and location different from those of Γ þ p ; (6) when γ À 1 < γ < γ À , there are no more than pn LCs in  the OD and one stable non-contractible LC; and (7) when γ < γ À 1 , Eq. (14) has one stable noncontractible LC which lies in the lower half-cylinder u < 0 and no more than p n À 1 ð Þin the OD.

Consider system (8) which is equivalent to the equation [3]
where P i , i ¼ 1; 2; 3; 4 ð Þare parameters. Here, we focus only on the effects which are due to the nonlinear parametric term x _ xsin νt ð Þ. Let us assume ν ¼ 4. Then, for small P i i ¼ 1; 2; 3; 4 ð Þ system (20) can have only two "splittable" resonance levels: H x; y ð Þ ¼ h 11 , H x; y ð Þ ¼ h 31 and h 31 < h 11 . The corresponding autonomous system (P 3 ¼ P 4 ¼ 0) has at most one LC. The passage of this LC through the resonances under a change of parameter P 2 was considered in [2]. If this LC lies outside the neighborhoods of resonance levels H x; y ð Þ ¼ h 11 , H x; y ð Þ ¼ h 31 , then in the original nonautonomous system (20), there is a two-dimensional invariant torus T 2 corresponding to the cycle. There is a generating "Kolmogorov torus" in the Hamiltonian A computer program was developed by the author for a simulation of Eq. (20). The results of such simulation are presented in Figures 4-6. In the numerical integration, the Runge-Kuttatype formulae are used with an error of order O h 6 À Á per integration step h. In Figure 4(a) we present the Poincaré map for P 1 ¼ 0:0472, P 2 ¼ À0:008, and P 3 ¼ 0:018, which determines the structure of the main resonance zone p ¼ 1; q ¼ 1 ð Þ . Along with the separatrices of the saddle fixed point S, a closed invariant curve encircling the unstable fixed point O is shown, which corresponds to a stable LC in the oscillatory domain of Eq. (6). This closed invariant curve appears for P 3 ≈ 0:014 when the fixed point O loses its stability. As P 3 increases, so does the size of the closed invariant curve, and for P 3 ≈ 0:0487 the curve clings to the separatrix of the saddle point S, forming a contour (see Figure 4(b)). As P 3 increases further, two closed invariant curves appear, shown in Figure 5 for P 3 ¼ 0:15. The structural changes of the resonance zone observed in the experiment are in good agreement with the theoretical results for γ ¼ 0. The observations for γ 6 ¼ 0 are consistent with the theory, too.  In the case presented in Figure 6, the transversal intersection of the separatrices of S cannot be detected visually. We, therefore, increased P 4 to obtain a better picture of the homoclinic structure. When P 4 ¼ 8, the structure can be seen clearly (Figure 6(a)). The corresponding quasi-attractor is the only attracting set (Figure 6(b)). Stable periodic points with long periods can exist inside the quasi-attractor itself. However, they are extremely difficult to detect numerically.

Example 2
As opposed to Example 1, this one pursues a different goal, namely, to study the transition from the classical parametric resonance to the nonlinear resonance. One of the problems for which this can be done is that of the pendulum with a vibrating suspension.
The pendulum with vibrating suspension is a classical example of a problem in which a parametric resonance can be observed. A large number of publications (see, e.g., [8,9]) are devoted to this problem. Other problems of this sort include the bending oscillations of straight rod under a periodic longitudinal force [10], the motion of a charged particle (electron) in the field of two running waves [11], etc. The parametric resonance in this kind of systems appears when a fixed point of the corresponding Poincaré map loses its stability and is, therefore, usually described by the linearization near this point.
It is interesting to study the behavior of a parametric system when the ring-like resonance zone is contracted into a point, i.e., to describe the bifurcations which occur in the course of transition from the plain nonlinear resonance to the parametric one. This paragraph is devoted to the solution of this problem in the case of a nonconservative pendulum with a vertically oscillating suspension. The motion of the pendulum with vertically oscillating suspension (under some simplifying assumptions) is described by the equation [13] where p 1 , p 2 , β are parameters.
Let us now complicate the model even more and consider the equation with the phase space R 1 Â S 1 Â S 1 . The term p 3 _ xcosx appears, for example, in the case of the pendulum in which the force of resistance is created by a vertical plate perpendicular to the plane of oscillations. Consider Eq. (22) when it is close to integrable, i.e., for small values of parameters Eq. (23) in the conservative case, when C 2 ¼ C 3 ¼ 0, is considered in many publications. For instance, for small angles of the deviation x, the case β ffi 2 is studied in [8]. The criterion of resonance overlap is applied in [11] to estimating the width of the "ergodic layer." The existence of homoclinic solutions is discussed in [12] without the assumption on smallness of parameter ε.
Phase curves of the unperturbed mathematical pendulum equation are determined by the integral H Þin the oscillatory domain and h > 1 in the rotational domain. The peculiarity lies in the way period τ depends on h in the oscillatory domain.
We have Here, K ¼ K k ð Þ is the complete elliptic integral of the first kind, k being its modulus. From Eq. (24) it follows that the period τ changes noticeably only for h close to 1, i.e., in the neighborhood of the separatrix. Therefore, small intervals of period τ, which determines the width of resonance zones, correspond to fairly large intervals of variable x.

Structure of resonant zones
In the investigation of the perturbed equation, we first focus on the structure of resonance zones in domains The resonance condition τ h pq À Á ¼ p=q In our case the divergence of the vector field of Eq. (23) contains no terms explicitly depending on t; hence, σ does not depend on v, i.e., σ ¼ const.
The functions A s ð Þ 0 and σ s ð Þ in an explicit form were obtained in [13]. It is also found that the width of the resonance zone decreases rapidly with the increase of p when q ¼ 1.
In addition, in all three cases, we assume C 2 þ C 3 < 0.