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4Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 2, pp. 187-198. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd190208
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 34C15, 34C27, 34C37
Global Dynamics of Systems Close to Hamiltonian Ones Under Nonconservative Quasi-periodic
Perturbation
A. D. Morozov, K. E. Morozov
We study quasi-periodic nonconservative perturbations of two-dimensional Hamiltonian systems. We suppose that there exists a region D filled with closed phase curves of the unperturbed system and consider the problem of global dynamics in D. The investigation includes examining the behavior of solutions both in D (the existence of invariant tori, the finiteness of the set of splittable energy levels) and in a neighborhood of the unperturbed separatrix (splitting of the separatrix manifolds). The conditions for the existence of homoclinic solutions are stated. We illustrate the research with the Duffing-Van der Pole equation as an example.
Keywords: resonances, quasi-periodic, periodic, averaged system, phase curves, equilibrium states, limit cycles, separatrix manifolds
1. Introduction
Consider the system
dH(x,y) , ,
x =-^--hsg(x,y,ojit,...,ojmt),
dy (1.1) dH(x,y) .. .
+ £f(X,y,0Jit,...,0Jmt),
dx
Received April 14, 2019 Accepted June 20, 2019
This work has been partially supported by the Russian Foundation for Basic Research under grant no. 18-01-00306, by the Ministry of Education and Science of the Russian Federation (project no. 1.3287.2017/PCh) and by the Russian Science Foundation under grant no. 19-11-00280.
Albert D. Morozov [email protected] Kirill E. Morozov [email protected]
Lobachevsky State University of Nizhni Novgorod, prosp. Gagarina 23, Nizhni Novgorod 603950, Russia
where e is a small parameter, the Hamiltonian H and the functions g, f are sufficiently differ-entiable and uniformly bounded along with partial derivatives of order ^ 2 in a domain G c R or G c R1 x S1. Moreover, g,f are assumed to be sufficiently smooth and quasi-periodic in t uniformly with respect to (x,y) G G with incommensurable frequencies u1,u2,... ,um.
There are many works related to the topic. For instance, some specific examples of systems of form (1.1) were disscused in terms of complex dynamics in [1-3]. The Melnikov method is widely used in the literature as a criterion of chaos (for the Melnikov formula, see, e.g., [4-6]). A great number of papers refer to the existence of invariant tori in a weakly nonlinear case (see, e.g., [7-10]). There are also monographs concerning the averaging method (see [11-13]). In the present paper, we do not assume that (1.1) is quasi-linear and consider issues related to global dynamics along with local examination.
Suppose that the unperturbed system is nonlinear and has a region D c G filled with closed phase curves H(x,y) = h, h G [h-,h+]. The following condition is assumed to hold:
dg/dx + df/dy ^ 0, (1.2)
which implies that the system (1.1) is nonconservative. We also suppose that the system (1.1) with e = 0 has a separatrix loop of a saddle (xs,ys). Note that D contains neither a separatrix nor equilibria, nor their neighborhoods.
The global study of the system (1.1) involves examining the behavior of solutions both in D and in a neighborhood of the unperturbed separatrix. In turn, there appear resonance energy levels in D. A closed phase curve H(x, y) = hres with the frequency w(hres) is called a resonance one if the frequencies u(hres),u1,..., are commensurable.
Resonance levels are divided into passable, partly passable and impassable ones. If all the levels H(x,y) = hres are passable, then the qualitative behavior of (1.1) in D is determined by the autonomous system
dH(x,y)
x =-^-+ ego{x, y),
dy (1.3)
where
1
dx
f-2n r2n
9o(x,y) = -——- ... g(x,y,0i,...,0m)d9i...d0r>
(2n)
/0 JO
¡■2n r 2n
fo(x,y) = 1 I ... I f(x,y,01,...,0m)d01...d,0m. (2n)m J0 J0
It is assumed that the system (1.3) has only a finite number of limit cycles. The paper [14] has outlined approaches to solving the problem of the global solutions behavior. Here we implement these approaches and illustrate them using the following equation as an example:
x + ax + x3 = ef (x,y,t), (1.4)
where a = ±1, f = (p1 — x2)x + p2 sint sinvt, p1 ,p2,v are parameters and v is irrational.
In the case of periodic perturbations, the problem of the global solutions behavior for systems of type (1.1) was considered in [11] (see also [15]). The periodic analogue of (1.4) is studied in [16, 17].
2. Behavior of solutions in D
In the action I -angle 0 variables the system (1.1) takes the form
I = eFi (I,0,0i,...,0m),
0 = u(i) + eF2 (I,0,0i,...,0m), (2.1)
0k = Uk, k = 1,2,...m,
where
Fi = f (x(1,0),y(1,0),0i,..., 0n)x'e - g(x(1,0),y(1,0),0i,..., 0n)y'e, F2 = -f (x(1,0),y(1,0), 0i,..., 0n)x1 + g(x(1,0),y(1,0),0i,..., 0n)y'I,
(2.2)
the functions x(I,0),y(I,0) define the change of variables, u(I) is the natural frequency of the unperturbed system. Suppose that u(I) is monotone on the interval (I_,I+).
The phase space of the system (2.1) is [I_(h_),I+ (h+)] x Tm+i, where Tm+1 is an (m + + 1)-dimensional torus and h_,h+ determine the boundaries of D. The functions Fi, F2 are sufficiently smooth in I,0,0i,..., 0m. For e = 0, the (m + 2)-dimensional phase space of the system (2.1) is foliated by (m + 1)-dimensional invariant tori Tm+i. For e = 0, the tori can be destroyed by the nonconservativity of the perturbation and/or the presence of an integer combination of the frequencies ,..., um:
nu(I) - (k, Q) = 0, k = (ki ,...,km), Q = (ui,...,um). (2.3)
For given Q and fixed k,n, relation (2.3) can be viewed as an equation for I. If this equation has a real solution I = Ink on the interval [I_,I+ ], then the level I = In\bfk (the closed phase curve H(x, y) = hnk of the unperturbed system) will be called a resonance level.
Let us begin with consideration of the neighborhoods of individual resonance levels of energy.
2.1. Neighborhood of a resonance level
According to [14], the averaged system that describes the behavior of solutions in the neighborhood U^ = {{1,0) : Ink — C/x < I < Ink + Cn, 0 ^ 6 < '2tt, C = const > 0, ¿t = sfe} of resonance level I = Ink (H(x,y) = h(Ink)) has the form
u' = A(v; Ink) + ^(v; Ink)u v' = b1u + /ib2 u
2 (2.4)
where the prime denotes the derivative with respect to the slow time t = ft, b\ = w'(Ink), b2 =
= ^"(Ink),
rZnn rZnn ""
Ink) = / ... F^In k, (v + (V kjQj)/n),Q 1,..., 9m)d9 xdB2 ... d,9m, (2.5)
(2nn) J 0 J 0 j=1
1 r 2nn r 2nn
a{V] Ink) = jMr Jo "Jo {g'x + f'y)d01''' d0m' (2'6)
where x = x(In]l, v + £ YJjLi kjOj), y = y(Ink, v + ± YJjLi kjOj). The functions A(v, I„k) and &(v,Ink) are sufficiently smooth and periodic with the least period equal to 2n/n. The phase space of (2.4) is a cylinder (u,v mod 2n/n). Simple equilibria of (2.4) correspond to quasi-periodic solutions with m frequencies in the initial system (i.e., m-dimensional invariant tori in the system (2.1)). The following theorem holds [14].
Theorem 1. Assume that the system (2.4) has a simple equilibrium (v0,0) and a = 0. Then the system (1.1) has a quasi-periodic solution x(t),y(t) with frequencies ui/n,... ,um/n for sufficiently small e > 0. The solution is asymptotically stable if bi A(v0) > 0 and a < 0 and unstable (saddle) if biA'(v0) < 0. Accordingly, the system (2.1) has an m-dimensional stable (or saddle) invariant torus Tm.
Let us represent the function A(v; Ink) in the form: A(v; Ink) = B0(Ink) + A(v; Ink), where B0 is the mean of A. We shall say that I = Ink determines a resonance energy level of the first type if max|A(v;Ink)| < \B0(Ink)|, B0(Ink) = 0; a resonance level of the second type if max|A(v;Ink)| > |Bo(Ink)|, B0(Ink) = 0; a resonance level of the third type if B0(Ink) = = 0, A(v, Ink) ^ 0. The qualitative behavior of (2.4) in the neighborhoods of levels of each type was established in [11]. If a is sign-preserving, then there exists enk > 0 such that a first (second, third) type level determined by I = Ink is passable (partly passable, impassable) if only
0 < e < enk.
If the perturbed autonomous system (1.3) has a limit cycle in the neighborhood of the level H(x,y) = hnk, then this level is of the third type (i.e., impassable). Indeed, B0(I) is the Poincare-Pontryagin generating function for the system (1.3), therefore, B0(Ink) = 0. The passage of the limit cycle through the resonance zone was described in [18]. If the condition B0 = 0, |B0(Ink)| > max )| holds, the averaged system (2.4) does not have equilibria
and the resonance level I = Ink is passable for e small enough.
2.2. Neighborhood of a nonresonance level
Let us now consider a nonresonance energy level I = I* and make the change of variable
1 = I* + iW in (2.1). As a result, we obtain
W = ixFi (I*,9, 9i,..., em) + n2[dFi{-)/dI] W + O(M3), 9 = u(I*) + iibiW + II2 (b2W2 + F2 (I*,d, di,..., em)) + O(v3), (2.7)
Qk = uk, k = 1,2,...m,
where b\ = ci/(/*),&2 = Then we make the following change of variable:
u = W + in £ + if £ kfc(/*) - hWk0^1 ^ (2.8)
where Fik are the Fourier expansion coefficients for the function Fi(I*,d,di,...,dm), k = = (k0, ki,..., km),9 = (9,9i,..., 9m), u = (u(I*),ui,..., um). The system takes the form
u = iiB0(I*) + i2Bi (I*)u + O(i3),
9 = u(I*) + ibiu + i2Q(u, 9) + O(i3), (2.9)
9k = uk, k = 1,2,...m,
where
B0 = F10 =
r2n r2n
r2n r2n
/ ... Fi(I*,9,9i,...,9m)d9d9i ...d9n 00
(2n)m+1 J0 J0
B' ^ dB"/dI ^ (2^ C -T . . . Mm.
Neglecting terms of order O(|3), we obtain the equation
U = /j.Bo(L) + f2B i(I*)u,
(2.10)
which coincides with the averaged one. Small denominators (k,ui) may appear in (2.8). This requires an estimate of the frequencies which would provide the series convergence. For example, it will be sufficient if the frequencies satisfy the following condition:
for all (k0, k]_,..., km) G Zm+1 \{0}. C is a positive constant1.
If B0(I*) = 0, then the level I = I* is passable. If B0(I*) = 0 and B1 (I*) = 0, then the simple root u = 0 of (2.10) corresponds to a quasi-periodic solution with the frequencies w(I*),w1,..., um in (1.1) (or an (m+1)-dimensional invariant torus in (2.1)). Thus, the following theorem holds.
Theorem 2. Assume that the autonomous system (1.3) has a rough limit cycle generated from a nonresonance level I = I* for sufficiently small e > 0 (i.e., B0(I*) = 0, B1 (I*) = 0) and condition (2.11) is satisfied. Then the system (1.1) has a quasi-periodic solution with the frequencies u(I*),u1 ,...,um. The solution corresponds to an (m + 1)-dimensional invariant torus in the system (2.1), which is asymptotically stable if B1 (I*) < 0.
2.3. Global behavior of solutions in D
The averaged system (2.4) is similar to the one obtained in the case of periodic perturbations. It was stated that there is only a finite number of splittable resonances for small e [11] (see also [15]). The statement is naturally transferred to the quasi-periodic case. Indeed, replacing scalars by vectors in the appropriate formulas in [11, pp. 188-191], we obtain the following theorem.
Theorem 3. There is only a finite number of resonance levels such that max \A(v,Ink)\ >
This implies that the system u' = A(v; Ink), v' = b1u has simple equilibria only for a finite number of resonance levels I = Ink if B0(Ink) = 0.
Since the system (1.3) has only a finite number of limit cycles by the assumption, there is only a finite number impassable nonresonance levels. This fact and Theorems 1-3 let us establish the global behavior of solutions in D. Indeed, the number of the neighborhoods of splittable resonance levels is finite and we can make their y^-neighborhoods not intersect. Outside these neighborhoods, all the levels H(x,y) = const are passable for e small enough. Note that attraction basins of invariant tori can have complex structure due to heteroclinic orbits [11]. If all the levels H(x,y) = const in D are passable, then the qualitative behavior of solutions to (1.1) in D is determined by the autonomous system (1.3).
3. Splitting of separatrix manifolds
Without loss of generality, we suppose that the saddle (xs,ys) of the system (1.1) at e = 0 lies at the origin. (xs, ys) corresponds to the saddle quasi-periodic solution in the extended phase
1 This also requires F1(I*, 0,01,..., 0m) to be sufficiently smooth with respect to 0,01,..., 0m.
m
(2.11)
\B0(Ink) \ •
space. Let us denote the coinciding stable and unstable integral manifolds of the unperturbed saddle quasi-periodic solution by W£, W0W. The solution persists under the perturbation when e > 0 small enough and has invariant manifolds WW, W££ close to the unperturbed ones [5]. The problem is to analyze the distance between separatrix manifolds W££ and WW, which coincide in the unperturbed system. For periodic perturbations, one can use the Melnikov formula to determine the distance [4]. In [5], the applicability of the Melnikov formula was extended to sufficiently smooth systems. In [6] the quasi-periodic Melnikov function was derived and a general discussion of the discrete maps construction from the trajectories of time-dependent vector fields was presented. Following [5], we briefly describe the derivation of the Melnikov function in our case.
Let us find the distance A between the manifolds W£ and WW of the system (1.1). We assume
f (0,0, ui t,..., um t) = g(0,0, ui t,..., umt) = 0
for all t. It stands for the manifolds W£'W(t) adjacent to the t-axis in the (x, y, t)-space.
According to [5], for sufficiently small e we have
sup II W£(t) - Ws(t) ||= O(e),
T
sup || W0w(t) - W£w(t) ||= O(e).
Then there exist solutions x£'W,y£'W of (1.1) such that
lim x£(t) = lim xW(t) = 0, lim y£(t) = lim y£w(t) = 0. (3.1)
t^<x t—y—^o t^<x t—y—^o
According to [5], for each x'0'Wy'S'W C WS'W we can find x%,ys£ and xW,yW such that
sup \\xs0'w(t) - x£(t)|| = O(e(1 - e—Clt)), sup \\y£'W(T) - y£(T)|| = O(e(1 - e—Clt)),
T G[t,^] T €[t,^]
sup \\x0'W(T) - xW(T)\\ = O(e(1 + e—Clt)), sup \\y£W(T) - yW(T)\\ = O(e(1+ e—Clt))
for certain Cl > 0 (Cl depends only on the unperturbed system) and for all t G R. Then we define rf, i = s,u by
x\(t) - xs0'W(t) = e{i(t, e), yi(t) - y£'W(t) = eni(t; e), sup \\C£(t;e)\\ = O(1 + e—Clt)), sup \\V£(t;e)\\ = O(1 + e—Clt)),
t e[t,<^] T G[t,^]
sup \\eW(T;e)\\ = O(1 + eClt)), sup \\VW(t;e)\\ = O(1 + eClt)). Set x'S'W(ts) = xs,y£'W(ts) = ys and define
Ap-(t0,x0,y0) = W^rf^e) + (3.2)
where t0 determines the plane (x, y, t0) on which we measure the distance. Then the value that determines the distance between the manifolds W£(t) and W£W(t) takes the form
A£(ts,xs,ys) = A'W(ts,xs, ys) - A££(ts,xs,ys). . RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2019, 15(2), 187 198_
Consider the Taylor series at e = 0: As(t0) = A0(t0) + eA1(i0)+O(e2) = eA1(i0) + O(e2). Following [5], we find
/<x
[f (x0(t -t0),V0(r -t0),T)x0(t -1) -g(x0(T -h),y0(r -t0),r)yo(r -t0)\dr, (3.3)
x0 (r ),y0(r) is the separatrix solution of the unperturbed system, f ^ f (x, y, u1r+010,..., umr+ + 0m0), g ^ g(x, y, u1 r + 010,..., umr + 0m0), 010,..., 0m0 are constants. Since the functions f,g are quasi-periodic in t, it follows that A1(t0) is quasi-periodic as well. Note that A1 depends on 010,..., 0m0. This leads to a variety of geometrical interpretations of the stable and unstable manifolds intersections [6].
The fact that A1(t0) is not sign-preserving implies that Ws(t0) and Wu(t0) intersect transversally and there exist doubly-asymptotic (homoclinic) solutions, i.e, solutions satisfying the condition lim x£(t) = lim y£(t) = 0. The behavior of solutions in the extended
neighborhood of a doubly asymptotic solution was studied in [21] and it relates to the existence of a nontrivial hyperbolic set ("irregular" dynamics). In some cases, a stable irregular invariant set of the so-called "quasi-attractor" may appear. Let us now consider Example (1.4).
4. Example
Consider Eq. (1.4)
x + ax + xs = e[(p1 - x2)x + p2 sin t sin vt)], (4.1)
v = \/5. If a = — 1, there are three regions in the phase space of the unperturbed equation filled with closed phase curves. If a = 1, there exists a unique such region. For simplicity, when we study the global behavior of the solutions to (4.1) in D, we take a = 1 (the behavior in three regions for a = -1 can be studied in the same way, e.g., the dissipative case was considered in [14]). For a = -1, we will investigate only the neighborhood of the unperturbed separatrix. Note that the equation of type (4.1) for a = 1 was considered in [18]. However, in [18] only the neighborhoods of resonance levels were studied.
4.1. Autonomous equation
The unperturbed equation (e = 0) admits the following energy integral:
H(x, y) = x2/2 + x2/2 + x4/4 = h,h> 0 (4.2)
and it has the solution
x(0,I)= xicn(2K0/n),0 = ut, (4.3)
where u = n(1 + 4h)1/4/(2K) is a natural frequency, x1 is the positive solution of x2/2 + x4/4 = = h, cn(u) is the Jacobi elliptic function, K is the complete elliptic integral of the first kind, and k = k(h) is its modulus. Denote the annular region of the phase plane {(x,y) : H(x,y) = = h, 0 <h- <h <h+ < by D.
The perturbed autonomous system (p2 = 0)
x = y, y = -x - x3 + e[(pi - x2)y] (4.4)
has a unique limit cycle if pi > 0. It follows from the analysis of the Poincare - Pontryagin generating function, which has the following form (up to the factor 4/(3^(1 — 2p)5/2)):
Bo(p) = Pi[(1 - p)(1 - 2p)K - (1 - 2p)2E]-
2 o
--[(p - 1)(2 - p)K + 2(p2 - p + 1)E],
(4.5)
where E is the complete elliptic integral of the second kind, p = k2 = 2 ~' B0(p,pi)
-10
Fig. 1. Bo (p; P13) and Bo (p; pn).
Let p* G (0,1/2) be a simple zero of Bo(p). One can see that p1 ^ 00 as p* ^ 0 and pi ^ +0 as p* ^ 1/2. It is well known that p* determines the level of the unperturbed system which generates a rough limit cycle in (4.4). If B'(p*) < 0 and £ > 0, the cycle is stable. The sign of B' (p*) coincides with the sign of a (p*), where
^(p) = Pi -
2
(1 - 2p)K
[E - (1 - p)K].
(4.6)
Figure 1 shows the function Bo (p; p1 ) for two values of p1, where P31 is found from the condition Bo(p(^3k);p) = 0 and pii is found from the condition Bo(p(h1k);p) = 0. Simple eqiulibria of the equation Bo(p; p1 ) = 0 correspond to rough limit cycles in (4.4) [15].
4.2. Nonautonomous equation
The behavior of solutions in the neighborhoods of individual resonance levels is described by the averaged system (2.4), where
A(v ) = B +
p2
>2nn r 2nn
4n2 n2
sin 01 sin 02x'e (v + (k101 + k2 02)/n)d01 d02 =
B +p2V2(kiUi + h2u2)j^ sin
nv
=
if n is odd, k1 = ±1, k2 = 1, Bo if n is even or k1 = ±1, k2 = 1,
&(Ink) is determined by formula (4.6).
0
0
From the conditions uo = (±1 + \/b)/n and uo > 1, one can find that only three "splittable" resonance levels H(x,y) = Н^ц, H(x,y) = h1—1д, H(x,y) = h111 (Нз11 < hi—i,i < h111) are possible.
Thus, we have u' = Bo (Ink) + lau for almost all resonance levels. If Bo = 0, then the level I = Ink is passable. If Bo = 0 and a < 0 (the autonomous system (1.3) has a stable limit cycle in the neighborhood of I = Ink), then u = 0 is a stable equilibrium. This implies that there exists a stable invariant curve embracing the phase cylinder of the averaged system (2.4) and a 3-dimensional stable torus in the system (2.1) (see Theorem 2). By varying p1, one can see the passage of this torus through resonance. Figure 2 shows phase portraits of the averaged system for resonances with n = 3 and n =1. There is a stable limit cycle of the averaged system for e = 0.1, p1 = 0.005, n = 3 (Fig. 2a). The cycle corresponds to a 3-dimensional torus in the initial system (2.1). As p1 increases, the cycle comes into the neighborhood of the resonance level I = I311 (H(x,y) = Нз11). The passage through a resonance zone is described in [18]. In Fig. 2b, there are no cycles and the oscillations are synchronized at the frequencies u1 /3,u2/3. Then, as p1 increases further, a cycle outside the resonance occurs (Fig. 2c, where p1 = 0.06). The cycle is growing along with p1 and is getting closer to the resonance level H(x, y) = h1—11. Further, the cycle passes through this zone and, finally, the resonance zone with H(x,y) = h111 (see Figs. 2d-2f). The dynamics of the averaged system in the neighborhood of H(x, y) = h1—11 is similar to the one in the neighborhood of H(x,y) = h111.
4.3. The neighborhood of the unperturbed separatrix
We now consider the equation
x — x + x3 = £[(p1 — x2)y + p2 sin t sin vt].
The saddle of the unperturbed system lies at the origin. To fix the saddle solution, we make a change of variable x = £ + ex1 (t) + O(e2). Then, neglecting terms of order O(e2)), we obtain the following equation:
£ — £ + £3 = £[(pi — £2 )£ — 3£2 xi (t)], (4.7)
where
It follows from (3.3) that
X1 (t) =
p2
cos (1 - v)t cos (1 + V)t Li + (i-^)2 + i + (i + z/)2j
Ae(to ) = eAo (to ) + 0(s2 ),
where
Here
Ao (to ) = / [(p1
J OO
1 - g(t - to))is(t - to) - 3^2(t - to)X1 (t)]is(t - to)dt.
£s(t) = ±\/2/cosh t, is(t) = V2 sinh (i ) / ch2 (t)
(4.8)
(4.9)
is the separatrix solution to the unperturbed equation. Here the sign "+" corresponds to the right separatrix loop and the sign "-" corresponds to the left separatrix loop. The result is [20]
where
Ao(io) = 5pi - 4 + p2 — [B1 sin (1 + u)t0 + B2 sin (1 - z/)i0],
B1,2 =
(1 ± v )
cosh ((1 ± v )n/2)'
(4.10)
0.8 0.6 0.4 0.2 X2 0.0 -0.2 -0.4 -0.6 -0.8
-1.2 -0.8 -0.4 0.0 0.4 0.8 1.2
Xi 0.001 0.02575 0.0505 0.07525 0.1
(a) (b)
Fig. 3. Irregular repelling set (a) and chart of the senior Lyapunov exponent (b).
Note that the Melnikov function is quasi-periodic, as is the perturbation. From (4.10) one can conclude that if \pi\ < (3/8) V^I^Ia/^i + B\ is satisfied, then the function Ao(to) is sign-alternating and, therefore, WS H WU = 0. This implies that there exist homoclinic solutions
2
01020215010002020253235348485353532348535353535348234853535353
to the saddle solution. They correspond to the homoclinic orbits of the Poincare map. The structure of the neighborhood of these orbits was studied in [21].
Note that the unstable limit cycle of the autonomous perturbed system (p2 = 0) merges with the separatrix and forms an unstable separatrix loop if p1 = 4/5. In this case, the free term of Ao is neglected and a irregular repelling set2 can appear. Such a set is shown in Fig. 3a for e = 0.1,p1 = 0.8,p2 = 0.8 (the fractional part of the Lyapunov dimension equals & 0.826). Figure 3b shows a chart of the largest Lyapunov exponent X1 in the (e,p2)-plane for p1 = 0.8. There are regions with X1 < 0 that correspond to regular asymptotic behavior (t ^—<x>) in the neighborhood of the unperturbed separatrix (unstable quasi-periodic solutions).
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2One can make this set stable (the so-called quasi-attractor [15]) by the change t ^—t .
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