CC BY
6Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 3, pp. 367-376. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd180307
MSC 2010: 34C15
On Synchronization of Quasiperiodic Oscillations
We study the role of quasi-periodic perturbations in systems close to two-dimensional Hamil-tonian ones. Similarly to the problem of the influence of periodic perturbations on a limit cycle, we consider the problem of the passage of an invariant torus through a resonance zone. The conditions for synchronization of quasi-periodic oscillations are established. We illustrate our results using the Duffing-van der Pol equation as an example.
Keywords: resonances, quasi-periodic, periodic, synchronization, averaged system, phase curves, equilibrium states
1. Problem statement
In 1930 A. Andronov and A. Witt investigated the problem of oscillation synchronization using the quasi-linear van der Pol equation as an example [1]. In other words, they found what is happening to a limit cycle when the system undergoes a periodic (harmonic) perturbation. Then a more general problem of synchronization in systems close to two-dimensional nonlinear Hamiltonian ones was considered [2, 3]. It was established that, in contrast to the quasi-linear case, when the synchronization interval with respect to a parameter is small along with the perturbation, in the nonlinear case it is of the order O(1). Recently, physicists found the effect of synchronization of quasi-periodic oscillations in the case of more sophisticated self-oscillating systems (see, for example, [4]).
Received May 11, 2018 Accepted June 22, 2018
This work was supported by the Russian Foundation for Basic Research under grants nos. 18-01-00306, 16-01-00364 and by the Russian Science Foundation under grant no. 14-41-00044, by the Ministry of Education and Science of the Russian Federation (project no. 1.3287.2017/PCh).
Albert D. Morozov [email protected]
Kirill E. Morozov
Lobachevsky State University of Nizhny Novgorod prosp. Gagarina 23, Nizhny Novgorod, 603950, Russia
A. D. Morozov, K. E. Morozov
In the present paper we shall focus on the problem of synchronization of quasi-periodic oscillations in nonlinear systems close to Hamiltonian ones. We shall follow the paper [5]. So, let us consider the system
dH(x,y)
x =----h eg(x, y, t),
y =
dy
dH{x, y) dx
(1.1)
+ ef (x,y,t),
where e is a small positive parameter, H is the Hamiltonian function and g, f are sufficiently smooth (analytic) and uniformly bounded in some domain G c R2 or G c R1 x S1 along with partial derivatives up to the second order. Besides, functions g, f are continuous and quasi-periodic (as functions of time) uniformly with respect to (x, y) £ G with incommensurable frequencies ui,u2,..., wm.
Let us assume that the unperturbed system is nonlinear and has a cell D c G filled with closed phase curves H(x,y) = h, h £ [h_, h+]. We also suppose that D does not contain any equilibria and separatrices. So D is a connected domain homeomorphic to a ring.
In D we change the variables from x, y to the action /-angle 6 variables. Then the system (1.1) takes the form
/ = eFi(/,d,di ,...,9m), 6 = w(/ )+eF2(/,6,6i,...,6m),
(1.2)
6k = Wk,
k = 1, 2,
, m,
where
(1.3)
Fi = f (x(/, 6),y(/, 6),6i,..., 6n)x'e - g(x(/, 6),y(/, 6),6i,..., 6n )y'd,
F2 = -f (x(1,6),y(/, 6),6i,..., On)^! + g(x(/, 6),y(/, 6),6i,..., 6n )y1,
the functions x(/, 6), y(/, O) define the change of variables.
System (1.2) is defined on the direct product of the interval K x Tm+i, where K = = [/_(h-),/+(h+)] and Tm+i is an (m + 1)-dimensional torus, where h-, h+ determine the boundaries of D. Here w(/) is a natural frequency of the unperturbed system. We shall suppose w(/) to be monotonic over the interval [/_,/+ ] .
Along with the system (1.1), let us consider the autonomous system
where
c)H — ± = ~dy+ Sg<yX'' ^
2n 2n
(1.4)
g{x,y) = J •••J g{x,y,9i,...,9m)d9i ...ddr,
0 0 2n 2n
f (x,y) =
1
(2n)m
f (x ,y,°°i, ..., 6m) d6i . . . d°n
00
Suppose that this system has a rough limit cycle. It implies that the generating Poincare-Pontryagin function B(h) has a simple zero h = h0 [3].
In [5] the solutions behavior in neighborhoods of individual levels of energy (both resonance and nonresonance ones) was established. Here we shall study the solutions' behavior in a neighborhood of the level h = h0 (such that B(ho) = 0) when it is close to the resonance one. Actually, we consider the synchronization problem in the case of quasi-periodic perturbations. As an example we take the following system:
x = y,
3 2 (1-5)
y = -x - x + e[(1 - Pix )y + p2F(i)],
which is equivalent to the Duffing-van der Pol equation. Here p i > 0, are parameters, F(t) = SinWit Sin UJ2t, UJ1 = 1, UJ2 = VE.
2. Behavior of solutions in neighborhoods of resonance levels
For e = 0, the (m + 2)-dimensional phase space of system (1.2) is foliated by (m + ^-dimensional tori Tm+i, the motion on which is conditionally periodic with frequencies ui,..., wm. For e = 0, the invariant tori are destroyed owing to the nonconservativity of the perturbation and/or the presence of an integer combination of the frequencies u,ui,..., :
nw(I) - (k, Q)=0, k = (ki ,...,km), n = (wi,...,wm). (2.1)
For given Q and fixed k, n, relation (2.1) 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 = Ink (the closed phase curve H(x,y) = hnk of the unperturbed system) will be called a resonance level.
Let us proceed to the study of the behavior of solutions in the neighborhoods U^ = = {(/, 9): Ink -Cn < I < Ink + Cn, 0 ^ 9 < '2tt, C = const > 0, ¿t = v^} °f the individual resonance levels I = Ink. In system (1.2), we make the change of variables
m
9 = ip + YI kjdj/n, I = Ink + fiW, At = \/i- (2-2)
j=i
As a result, system (1.2) becomes
W = flFi^ Ink, ^ + ( m kj j/n,0i,...,9m^j +
+ fj2 dFi(^ Ink, ^ + ( m kj 9^/n,9i,...,9my dI W + O(p3), = ¡ib{W + fx2F2^ Ink, ^ + ( m kj n,9i,..., 9m) + O(^3),
(2.3)
9j = Uj, j = l,...,m,
where the functions F1, F2, dF1/dI are 2-^n-periodic in 91,92,...,9m. Note that since the function u(I) is monotone, one has b1 = u'(Ink) = 0 (the nondegeneracy condition for the resonance). The variables in system (2.3) are divided into the slow variables W,^ and the fast variables 91,..., 9m. One can represent this system in the standard form of the averaging method
and apply the Bogolyubov theorem ([6, p. 379]; see also [7, p. 210]). Indeed, by substituting 6j = Wjt into the first two equations in system (2.3), we obtain
n = ¡F(n, Wit,..., Wmt; ¡),
where n = (W,^), F = (Fi + ¡(dFid/)W + O(s2),biW + ^ + O(s2)). Then the averaged system acquires the form £ = ¡F0(£; ¡), where £ = (u, v), u = W + O(s), v = ^ + O(s),
Fo = tSLT f = j ... j F(c,e1,...,em]ß)de1...den
0 0 0
owing to the incommensurability of the frequencies. The two-dimensional averaged system becomes
u' = A(v, /nk) + ¡¡P0(v, Ink)u, v = biu + ¡(b2u2 + Q0(v, Ink)),
where the prime stands for the derivative with respect to the slow time t = ¡t and
Inn Inn m
1
(2.4)
A = J ... J Fi (j„k, v + ( jr kjd^j j n, 0!,..., O^j Mx... d0m,
0 0 j=1
Po 1
2nn 2nn m
(2nn)
m
J ... j dFi^i Ink, v + ( m kj j/n,01,...,0m) dl
0 0 j=1
Inn 2nn
d0i... d0m
(2.5)
Qo = j ... j F2 V + (Jp kjd^J j n, 0U..., 0m^j dÖ!... d0n
0 0 j=1
bi = du(Ink )/dI, b2 = (d2u(Ink )/dI2 )/2.
By analogy with the case of periodic perturbations, system (2.4) can be reduced modulo O(ß2) terms to the form (see [3, 9])
u' = A(v,Ink) + ßau, v' = b1u + ßb2u
2 (2.6)
where
2nn 2nn
dQ0 1
a = Po +
dv (2nn)m
00
I ... j (gX + fy )d01 ...d0m. (2.7)
Here x = x(Ink,v + E™=i kj6j)/n), y = y(Ink,v + E™=i kj6j)/n). The functions A(v,Ink) and a(v,Ink) are sufficiently smooth and periodic with least period 2n/n [8, p. 158]. The phase space of system (2.6) is the cylinder {u,v mod (2n/n)}.
Associated with a simple stable equilibrium of the averaged system (2.6) is a stable quasi-periodic resonance solution with frequencies kiwi/n,..., kmwm/n [6].
Let us represent A(v,Ink) in the form 1
A(v, Ink) = A(v, Ink) + B(Ink),
2n/n
n
where B(Ink) = — / A(v,In]^)dv is the mean of A(v,In-k). Suppose that the level I = In^ 2n J 0
is an impassable one and it is close to the level I = I0, in the neighborhood of which the system (1.4) has the rough limit cycle.
3. Passage of the (m + 1)-dimensional torus through the resonance zone
Let us introduce in (2.6) the "detuning" 7 which measures the deviation of the resonance level I = Ink from the level I0. Following [8] and [9], we observe the transition from the exact resonance to the nonresonance case, as we change 7. In order to do that, we consider the system on a cylinder
du
— = A{v, Ink) + n{cr{v, Ink)u + 7),
dV (3.1)
dv 2
— = bu + /j,biu , dr
obtained from (2.6) by letting
B(Ink) = (dBo(Io)/dI)(Ink - Io) = №. Along with (2.6) consider the conservative system
= A{v,InVi)
dT (3.2)
dv
— = bu.
dT
We call the separatrices of the saddles of (3.1) embracing the phase cylinder (v mod 2n,u) external separatrices.
Let us describe the relative positions of the external separatrices when the terms O(u) are taken into account. We make use of the formula = /j,Ai + O(^2) for the distance between the corresponding separatrices of (2.6) [10]. Let us consider (3.2) as the unperturbed system and substitute in (3.1)
v = £ + vo - Vy/A (vo), u = n, (3.3)
where vo is the coordinate of the saddle of (3.2).
As a result, (3.1), up to the terms O(^2), assumes the form
^ = + vo) + /47 + + vo)v - + vo)/A'(vo)] di = br1 + ^r1\ ( M
dT
*^An example of computation of the function A(v,Ink) is given below in Section 4. _ RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2018, 14(3), 367 376.
Evidently, the right-hand sides of (3.4) vanish when £ = n = 0. Therefore, according to [10], we
A1 = - [bl,fTdT +
+ v0)V + 7 - 7 s
A (vo)
dr. (3.5)
dr
Here (£(t), n(T)) is the solution of (3.2) on the separatrix. From the integral of (3.2) we obtain the following relation between n and
n = ±
2
-(V(i,vo)-V(0,vo)
1/2 f „
, V (£,vo)= A(£ + vo)d£. (3.6)
Substituting (3.6) in (3.5), we find
2-K/p
Af = / <7(£ + t*>)
o
l(V(Z,vo)-V(0,vo))
1/2
d£. (3.7)
According to [8, 9], the following prepositions hold.
Preposition 1. For the exact resonance (i.e., for y = 0) when the function a(v) has constant sign the external separatrices are split, i.e. AM = 0.
Preposition 2. When function a(v) has constant sign, system (3.1) with y = 0 has no limit cycles.
When a(v) has constant sign, from the conditions A^ = 0 we find the (bifurcational) values of the "detuning" y = Y± corresponding to the appearance in (3.1) of a separatrix going from a saddle to a saddle. The plus in y± corresponds to the domain n > 0 and the minus — to n < 0. As y goes past y±, a noncontractible limit cycle is born in (3.1).
Theorem 1. Let a(v,Ink) have constant sign, and assume, for concreteness, that ea < 0. Then there exists Ynk = \y±| + O(y) such that:
1) for y > Ynk Eq. (3.1) has a noncontractible stable limit cycle;
2) for y = Ynk the limit cycle "sticks" to the separatrix contour r+ formed by n saddles and external separatrices connecting one saddle to another, while the other unstable separatrices of the saddles approach stable focuses as t ^ to;
3) for —Ynk < Y < Ynk there are no noncontractible limit cycles;
4) for y = -Ynk there appears a contour r—, formed by n saddles and their external separatrices, whose direction and position differ from those of r+;
5) for y < -Ynk there is a stable noncontractible limit cycle.
Figure 1 illustrates this theorem.
In fact, the theorem describes the "passage" of the (m + 1)-dimensional torus through a resonance zone. Indeed, the limit cycle of (3.1), embracing the phase cylinder, corresponds to the (m + 1)-dimensional torus in the initial (1.1) as well as the simple equilibrium corresponds to the m-dimensional torus (there is a quasi-periodic solution with the frequencies kiui/n,k2(¿2/n,... ,kmum/n) [5, 6].
2.4 1.6 0.8 u 0.0 -0.8 -1.6 -2.4
2.4 1.6 0.8 u 0.0 -0.8 -1.6 -2.4
0.0
v
(a)
2.4 1.6 0.8 u 0.0 -0.8 -1.6 -2.4
2.4 1.6 0.8 u 0.0 -0.8 -1.6 -2.4
0.0
v
(b)
2.4 1.6 0.8 u 0.0 -0.8 -1.6 -2.4
0.0
v
(c)
0.0
v
(d)
0.0
v
(e)
Fig. 1. Phase portraits of (3.1) under variation of 7.
4. Example
Let us study system (1.5). The unperturbed system (e = 0) admits the following integral of energy H(x, y) = y2/2 + x2/2 + x4 = h, h> 0. Denote the annular region of the phase plane 0 <h- ^ h ^ h+ < to by D.
The unperturbed system is equivalent to the second-order equation that has the following solution:
x(d,I)= Xicn(2Kd/n), 9 = ut, (4.1)
where u = n(1 + 4h):/4/(2K) is the frequency of motion on closed phase curves H = h(I), xi is a positive solution of the equation x2/2 + x4/4 = h, cn(-u) is the Jacobi elliptic function, K is the complete elliptic integral of first kind, and k = k(h) is its modulus.
4.1. Autonomous system
First let us consider the autonomous system
x = y, y = —x — x3 + e[(1 — pix2
(4.2)
The Poincare-Pontryagin generating function has the form (up to a positive factor) [9] B(p) = [(1 - p)( 1 - 2p)K - (1 - 2p)2E] + '^-[(p - 1)(2 - p)K + 2(p2 -p+ 1)E], (4.3)
where E is the complete elliptic integral of the second kind, p = k2 = —1 + 1 + —. For pi > 0
2 a/1 + 4/?.
it has a unique zero (denote it by p* <E (0,1/2)). As is well known [9], the simple zeros of the generating function determine the energy levels, in the neighborhood of which system (4.2) has a rough limit cycle for small enough e > 0. It is not hard to see that p* — 0 as p1 — oo and p* — 1/2 as p1 — 0+. Since a = B'(p*) < 0, the cycle is stable, where
a = 1 - pi
2
(1 - 2p)K
[E - (1 - p)K].
(4.4)
From the condition B(p,p1) = 0 we find
2(p_i)(2-p)K + 2(p2-p + l)E
(4.5)
Figure 2 shows the fragments of the dependence pi(p) in neighborhoods of resonances with n = 1 (a) and n = 3 (b).
0.46 0.458 0.456 0.454 0.452 0.45 0.448 0.446
0.4 Pi (a)
0.8
0.12 0.11 0.1 p 0.09 0.08 0.07 0.06
16 17 18 19 20 21 22 Pi (b)
Fig. 2. Fragments of the dependence pi(p).
4.2. Resonance zones and synchronization
Let us consider the resonance level I = In,k1k2 determined by the condition nu(h(I)) = = ki^i + ^2^2.
The pendulum type equation describing the topology of the resonance zone has the form
= (4.6)
Just as before, we represent the function A(v) in the form
A(v; Inkik2) = B(Inkik2) +P2S(v; Inkk), (4-7)
where
2nn 2nn
s{v, \„u 1,. ) = J J x'd{0, Inklk,J sm6>i sin 9-2 d9i d,62.
0 0
0.0
x
(a)
16
8
y 0 -8 -16
0.0
x
(b)
0.0
x
(c)
-4 -2 0 2 4
x
(d)
-4 -2 0 2 4
x
(e)
Fig. 3. Phase portraits of (4.6) in the (x, y)-plane in the neighborhood of the level h311: (a), (b), (c) and hiii: (d), (e).
Computing the function S(v), we obtain the following expressions:
an/2
A(v; In
nki k21
p2\/2(wi + u2) 1a+ sinnv + B(Inklk2);
if n is odd, k1 = k2 = 1, B(Ink1k2); if n is even or ki = 1,k2 = 1;
(4.8)
where a = exp —
yrKiVf - A'2) KJk
From the conditions oj = (1 + y/%)/n and oj > 1 it follows that there can be only two splittable resonance levels (n = 1, n = 3).
Figure 3 shows the phase portraits of Eq. (4.5) in the (x; y)-plane under variation of p1 in the neighborhood of resonance levels of energy.
The intervals of synchronization of quasi-periodic oscillations with respect to p1 (see on Fig. 2) could be obtained similarly to the case of periodic perturbations [8, pp. 184-187].
Rough equilibria of (4.6) correspond to the resonance quasi-periodic solutions with frequencies w1/n, w2/n, n = 1, n = 3 of the initial system (1.5). Such a solution for p1 = 18, p2 = 1, e = 0.05 and n = 3 is presented in Fig. 4a. The projection of this solution onto the (x, y)-plane is shown in Fig. 4b.
y
References
[1] Andronov, A. A. and Witt, A. A., Zur Theorie des Mitnehmens von van der Pol, Arch. fur Elek-trotech, 1930, vol.24, no. 1, pp. 99-110.
[2] Morozov, A. D. and Shil'nikov, L. P., To Mathematical Theory of Oscillatory Synchronization, Dokl. Akad. Nauk SSSR, 1975, vol.223, no. 6, pp. 1340-1343 (Russian).
[3] Morozov, A. D. and Shil'nikov, L. P., On Nonconservative Periodic Systems Close to Two-Dimensional Hamiltonian, J. Appl. Math. Mech, 1983, vol.47, no. 3, pp. 327-334; see also: Prikl. Mat. Mekh, 1983, vol.47, no. 3, pp. 385-394.
[4] Anishenko, V. S. and Nikolaev, S. M., Experimental Research of Synchronization of Two-Frequency Quasiperiodic Motions, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 2007, vol.15, no. 6, pp. 93-101 (Russian).
[5] Morozov, A. D. and Morozov, K.E., Quasiperiodic Perturbations of Two-Dimensional Hamiltonian Systems, Differ. Equ., 2017, vol.53, no. 12, pp. 1557-1566; see also: Differ. Uravn., 2017, vol.53, no. 12, pp. 1607-1615.
[6] Bogoliubov, N.N. and Mitropolsky, Yu. A., Asymptotic Methods in the Theory of Non-Linear Oscillations, New York: Gordon & Breach, 1961.
[7] Mitropolsky, Yu. A. and Lykova, O.B., Integrated Manifolds in the Nonlinear Mechanics, Moscow: Nauka, 1973 (Russian).
[8] Morozov, A. D., Quasi-Conservative Systems: Cycles, Resonances and Chaos, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol.30, River Edge, N.J.: World Sci., 1999.
[9] Morozov, A. D., Resonance, Cycles and Chaos in Quasi-Conservative Systems, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).
[10] Mel'nikov, V. K., On the Stability of a Center for Time-Periodic Perturbations, Tr. Mosk. Mat. Obs, 1963, vol. 12, pp. 3-52 (Russian).
