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5Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 3, pp. 301-324. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd180302
MSC 2010: 37G15, 37G35
Features of Bifurcations of Periodic Solutions of the Ikeda Equation
We study equilibrium states and bifurcations of periodic solutions from the equilibrium state of the Ikeda delay-differential equation well known in nonlinear optics. This equation was proposed as a mathematical model of a passive optical resonator in a nonlinear environment. The equation, written in a characteristic time scale, contains a small parameter at the derivative, which makes it singular. It is shown that the behavior of solutions of the equation with initial conditions from the fixed neighborhood of the equilibrium state in the phase space of the equation is described by a countable system of nonlinear ordinary differential equations. This system of equations has a minimal structure and is called the normal form of the equation in the neighborhood of the equilibrium state. The system of equations allows us to pick out one "fast" and a countable number of "slow" variables and apply the averaging method to the system obtained. It is shown that the equilibrium states of a system of averaged equations with "slow" variables correspond to periodic solutions of the same type of stability. The possibility of simultaneous bifurcation of a large number of stable periodic solutions(multistability bifurcation) is shown. With further increase in the bifurcation parameter each of the periodic solutions becomes a chaotic attractor through a series of period-doubling bifurcations. Thus, the behavior of the solutions of the Ikeda equation is characterized by chaotic multistability.
Keywords: Ikeda equation, periodic solutions, bifurcation of multistability, chaotic multi-stability
Received July 19, 2018 Accepted August 29, 2018
Evgenii P. Kubyshkin kubysh.e@yandex.ru
Alena R. Moriakova alyona_moryakova@mail. ru P. G. Demidov Yaroslavl State University ul. Sovetskaya 14, Yaroslavl, 150003, Russia
E. P. Kubyshkin, A. R. Moriakova
1. Introduction
We consider the following delay-differential equation
x = j sin(x(t — t ) — c) — x. (1.1)
This equation was proposed in [1] to describe the dynamics of a passive optical resonator. In (1.1) the variable x(t) determines the phase shift of the electric field in the nonlinear medium of a ring resonator, t is the light propagation time in the ring resonator 0 ^ c < 2n — a constant phase shift, and j > 0 is the coefficient characterizing the intensity of the laser emission. Equation (1.1), called the Ikeda equation, is widely used in nonlinear optics to model nonlinear optical effects. Solutions of the Ikeda equation have been studied in [1-4], where the multiplicity of equilibrium states was shown and the existence of a variety of periodic solutions and chaotic oscillations was shown using numerical integration. In [5], the problem of reconstructing parameters of the equation from the observed oscillations is considered. In [6], an information transmission line with a chaotic carrier was proposed and studied using the Ikeda equation.
Depending on the values of the parameters c and /, Eq. (1.1) can have a large number of different stable and unstable equilibrium states. Some equilibrium states can arise in stable-unstable pairs. The loss of stability of an equilibrium state with changing parameters leads to a bifurcation of a certain number (depending on the behavior of the parameters) of stable (and unstable) periodic solutions, i.e., there is a bifurcation of multistability.
With further changes in the bifurcation parameters, each such periodic solution becomes a chaotic attractor through a series of period-doubling bifurcations. The dynamics of the solutions of the equation exhibits chaotic multistability. This paper is devoted to the study of these problems. The results obtained provide insight into the structure of the phase space of the equation. The method of uniform normalization, proposed in [7], is used as the research method. Some particular results in the study of Eq. (1.1) are outlined in the authors' note [8].
2. Mathematical formulation of the problem, analysis of equilibrium states
In real optical systems the delay t is substantially larger than the characteristic time of electric field oscillations. Passing to the dimensionless time t' = t/T in Eq. (1.1) and setting e1 = t-1 ^ 1, we get the equation
£\X(t) + x(t) = j sin(x(t — 1) — c), (2.1)
where e1 is a small parameter.
The equilibrium states x*(c,j) of Eq. (2.1) are determined by the roots of the equation
x = j sin(x — c) (2.2)
depending on c and j. We describe the equilibrium states and investigate their stability. The stability of x*(c,j) is determined by the location of the roots of the characteristic equation
P(A; e1) = e1\ + 1 — j cos(x*(/, c) — c) exp(—A) = 0. (2.3)
For \jcos(x*(j, c) — c)| < 1 and 0 < e1 ^ e0, where eo is small, all roots of Eq. (2.3) lie in the left complex half-plane. In this case the equilibrium state x* (c,j) is asymptotically stable.
For |scos(x*(s,c) — c)| > 1 and 0 < £\ ^ £0 Eq. (2.3) has roots lying in the right complex half-plane, i.e., the equilibrium state x*(c,s) is unstable [9]. The case |scos(x0(s,c) — c)| = 1 is a boundary case. This equality defines in the plane c, s the set of bifurcation points of periodic solutions from the equilibrium state x*(c, ¡).
For 0 <c< 2n Eq. (2.2) has a single solution x* (s,c), lim^0 x*(s, c) = 0, where —n < x*(s,c) < 0 if 0 < c < n and x*(s, n) = 0, 0 < x*(s, c) < n if n < c < 2n. For c = 0 Eq. (2.2) has the solution x*(s,c) = 0, and for ¡> 1 two solutions x*(s,c) > 0 (lim^i x*(s, c) = 0) and —x*(s, c).
Also, for every c there is a sequence 0 < ¡1(c) ^ ¡2(c) ^ ... for which Eq. (2.2) has multiple roots x-*(sk(c),c) = x+*(sk(c),c), sk(c)cos(x±(¡k(c),c)) = 1, which correspond to paired (stable and unstable) equilibrium states x-*(s, c) and x+*(¡, c) of Eq. (2.1) for s > (c). With further increase in the parameter s the equilibrium state x-*(s,c) loses stability for some ¡* and ¡* cos(x-(¡*,c)) = —1.
Figure 1 shows diagrams of equilibrium states of Eq. (2.1) for different values of c and 0 ^ s ^ 10. Stable equilibrium states are marked by thick lines, unstable ones, by thin lines. Equilibrium states x*(s*(c),c) losing stability under the conditions ¡*(c)cos(x*(s*(c),c)) = 1 and ¡*(c)cos(x*(s*(c),c)) = —1 are shown as dark and light circles, respectively. Further increase in the parameter s does not change the picture of the equilibrium states of Eq. (2.1) — a new pair (stable and unstable) of equilibrium states x-*(s,c) and x+*(sk(c),c) appears, the first of which later loses stability at some ¡* with ¡* cos(x-(s*,c)) = —1.
There are three types of stability loss of equilibrium states in Eq. (2.1): when the equilibrium state x*(s,c) loses stability for ¡* with ¡* cos(x-(s*,c)) = —1, when at c = 0 the equilibrium state x*(s, 0) = 0 at the point ¡* = 1 loses stability for ¡* cos(x-(s*, 0)) = 1, and at the point of appearance of the multiple root of Eq. (2.2) after the birth of paired equilibrium states x-*(s, c) and x+*(s, c).
We consider the first case of stability loss as the most frequent for the equilibrium state x*(s, c) (0 < c < 2n), lim^0 x*(s, c) = 0.
We perform in Eq. (2.1) the substitution
x(t)= x*(s,c)+y(t) (2.4)
and represent the nonlinear part of Eq. (2.1) with a Taylor series. As a result, in a neighborhood of the equilibrium state x*(s,c) the behavior of the solutions of Eq. (2.1) is determined by the following equation:
^iy(t)+y(t) — S cos(x* (¡,c) — c)y(t — 1) + f (y(t — 1); ¡,c) = 0, (2.5)
where f (y; ¡, c) = b2(s, c)y2 +b3(s, c)y3+o(y3) is an analytic function in a neighborhood of y = 0, where
b2(s, c) = x*(s, c)/2, b3(s, c) = scos(x*(¡, c) — c)/6.
We define the function s = ¡*(c) from the condition ¡*(c)cos(x*(s*(c),c) — c) = —1. The graph of the function ¡*(c) is shown in Fig. 2.
Assuming that c = c* is fixed, from the equation
(¡*(c*) + s) cos(x*(¡*(c*) + S, c*) — c*) = —1 — £2, £21 < 1,
with ((¡*(c) + s)cos(x*(¡*(c) + ¡, c) — c))M|M=o = —(1 + x*(s*(c),c)2/(2s*(c)) = 1(c) > 0, we get s = s(£2). Hence, we have
S = S*(c*) + ¡(£2) = ¡*(c*) + Si(c* )£2 + o(£2 ). (2.6)
10
5
h 0 -5 -10
c = 0
0 2 4 6 8 10 C=27T/3
0 2 4 6 8 10 C = 4TT/3
10
5
h 0 -5 -10
10
5
h 0 -5 -10
10
5
h 0 -5 10
c=7r/3
<r
........■............
0 2 4 6 8 10
C = 7T
*
0 2 4 6 8 10
c = 5?r/3
...............i..............
«s*
0 2 4 6 8 10 M
Fig. 1. Equilibrium states of Eq. (1.1) depending on parameter c.
a. 1.7
Fig. 2. / (c).
Substituting (2.6) in (2.5), we get the equation
eu/(t) + y(t) + (1+ e2 )y(t — 1) + f (y(t — 1); e2) = 0, (2.7)
where f (y(t — 1);e2) = f (y; j*(c*) + jfo),c*) = &2(e2)y2 + 63^2)y3 + o(y3), and
b2(e2) = x*(j*(c*) + j(e2),c*)/2, b3(e2) = —(1 + e2)/6. (2.8)
The stability of the zero solution of Eq. (2.7) is determined by the behavior of the solutions of its linear part
e1 y(t) + y(t) + (1+ e2)y(t — 1) = 0, (2.9)
which is determined by the location of the roots of the characteristic equation
P(A; e) = e1A + 1 + (1+ e2) exp(—A) = 0 (e = (e1, e2)) (2.10)
of Eq. (2.9).
3. The structure of the solutions of equation (2.9)
Let us study the location of the roots of the characteristic equation (2.10) and the structure of the solutions of Eq. (2.9) for \e\ ^ e0 (\e\ = (ef + e2)1/2). In this case the characteristic equation (2.10) has no roots lying on the real axis, therefore, we will take the area Im A > 0. Equation (2.10) in the form
eA(e1A + 1) = —(1 + e2) is equivalent to the sequence of equations
eA+in(1+ei A) = ein (1+£2)+ink k = 135
where In z = In + i arg z (—tt < arg z < tt), i = 1-
To determine the roots of Eq. (2.10), we consider the sequence of equations
A + ln(1+e1A) = ln(1+ e2) + ink, k = 1,3, 5,.... (3.1)
Note that Eq. (2.10) can have only a finite number of roots of finite multiplicity in any bounded domain, since the left-hand side of Eq. (2.10) is an analytic function of the complex variable A. In this case, the chain of roots of the form Ap(e) = Yp(e) + iap(e) (ap(e) > 0), \Ap(e)\ ^ to (p = 1,2,...) can exist in Eq. (2.10) only under the condition Yp(e) = —to,
limp^^\cp(e)/jp(e)\ = to (see, for example, [9]). Therefore, there exists ^>0 > 0, which allows us to determine the region of existence of the roots of the equation as
Avo = {A: ^o < arg A < n — po}.
Consider the equation
F (A; e1) = A + ln(1+e1A)= w, (3.2)
where A e A^, w G WXQ = {w: — xo ^ Rew ^ .To, Imw ^ 7r, xq > 0 being a small fixed number}. Since \e1 \ ^ e0 < sinand A £ \e1\/\1 + e1A\ ^ q < 1, the inequality m0 < \Fa(A;e1)\ < m^ holds uniformly with respect to e1 and A, where m0, m^ are fixed
positive constants. Thus, Eq. (3.2) for |£11 ^ £0 has a single solution A(w; £1), which can be obtained through the iterative process
Ap + ln(1+£1Ap-1) = w, p = 1,2,..., Ao = w. (3.3)
The iterative process converges uniformly with respect to £1 for |£1| ^ £0 and w from any limited subdomain WXQ. The limit function
A(w; £1) = w + A1 (w; £1),
/ (3.4)
A1(w; £1) = — ln (1 + £1(w — ln (1 + £1(w — ln (1 + £1(w — ...))))))
will be continuous in the aggregate of variables, analytic in w £ Wx0 for each fixed £1 (£ | ^ £0) and analytic in ti for each fixed w £ WXQ. Note that
A(w; £1) — A1(w; £1) = — ln (1 + £1 w*) +ln (1+ £1w) = £?(1 + £1A*)-1 ln (1 + £1w*), (3.5)
where according to (3.3)-(3.4) A1 (w; £1) = w — ln(1 + £1w), w* = w — ln(1 + £1(w — ln(1 + + £1(w — ...)))), the point A* is on a straight line connecting points w and w*. Taking into account that the expression |(1 + £1A*)-1 ln(1 + £1w*)| is limited uniformly with respect to £1
(Iill ^ to) and w £ WXQ, we have the estimate
|A(w;£1) — A1(w;£1 )| <Ke2 (K > 0). (3.6)
This enables us to formulate the following assertion.
Theorem 1. There is a £0 > 0, that when £| < £0 the set of roots of Eq. (2.10) is defined by the formula
Ak(£) = ink + ln(1+£2) +Al(ink + ln(1+£2); £1), A-k(£) = Xk(£), k = 1,3, 5,.... (3.7)
Hence, on the basis of (3.6) the following equality holds uniformly relative to k: Ak (£) = Yk (£) + i(nk + ak (£)),
Yk (£) = ln (1 + £2) — ln ((1 + £1 ln (1 + £2))2 + £2n2k2 )/2 + o(|£|), (3.8)
ak(£) = — arccos((1 + £1 ln (1 + £2))/((1 + £1 ln (1 + £2))2 + £21n2k2)1/2) + o(£|).
Thus, the problem of the stability of the solutions of Eq. (2.9) reduces to analyzing the functions Yk(£), which are analytic at the point £ = 0 and have the radius of convergence rk = 0(k-1). In this case we have
Yk (£)= £2 — £?(nk)2/2 — £1£2 — £2/2 + o(|£|3), k = 1, 3, 5,..., (3.9)
i.e., for small £ and £2 > £2(nk)2/2 the kth root of the characteristic equation (2.10) has a positive real part.
Let us now describe the structure of the solutions of Eq. (2.9). The phase space of Eq. (2.9) is the space of continuous real-valued functions C[—1; 0], where the norm is defined in a standard way and denoted as || • ||c. The solution of Eq. (2.9) (Eq. (2.5)) with t ^ 0 and initial condition y0(s) £ C[—1;0] is the function y(t + s; £) (—1 ^ s ^ 0), which turns Eq. (2.9) (Eq. (2.5)) into an identity equation and satisfies the initial condition y(t0 + s; £) = y0(s).
Introduce the Cauchy function K(t; e) of Eq. (2.9)
eiK(t; e) + K(t; e) + (1+ e2)K(t - 1; e) = 0, t> 0
_! ........ „ (3.10)
K(0; e) = e—i, K(t; e) = 0, -1 < t< 0.
Applying the Laplace transform to (3.10) and using the inversion formula, we have the expression
x+ioo
1 „ <x
K(t]s) = -±- / p-\\;s)extd\ = T(l+s1+s1\k(s)r1eK{£)t, t > 0,
_ / (3.11)
X — l<X
K(0; e) = e—1 ,K(t; e) = 0, t< 0.
In (3.11) x > 0 is a fixed number. The series (3.11) converges uniformly for every 0 ^ t ^ i.
Applying the Laplace transform to Eq. (2.9) and using the function (3.10) taking into account its representation in the form (3.11), the solution of Eq. (2.9) with initial condition yo(s) can be written as
y(t + s; e) = T(t; e)yo(s) =
o
eiK(t + s)yo(0) - (1+ e2) J K(t + s - 1 - si)yo(si) dsi, t + s ^ 0, (3.12)
— i
k yo(t + s), t + s < 0.
Formula (3.12) defines the family T(t;e), (t ^ 0) of linear completely continuous operators acting in the space C[-1,0] and forming a semigroup T(ti + t2; e) = T(ti; e)T(t2; e) = = T(t2; e)T(ti; e) (T(0) = I — unit operator) of class Co[10]. According to (2.9) and (3.12), the generating (infinitesimal) semigroup operator T(t; e) has the form
. dv/ds, - 1 ^ s < 0,
A(e)v ^ i (3.13)
^ - e—i(v(0) + (1 + e2)v(-1)), s = 0
with domain D(A) = {v(s) e Ci[-1,0], eiv'(0) + v(0) + (1 + e2)v(-1) = 0}.
Ck[-1,0] is the space of fc-times continuously differentiable real-valued functions, the norm in which is defined by the standard method. We denote by C[-1,0] the complex expansion of space C[-1,0], where ||v(s)||g = ||(vi(s)2 + v2(s)2)i/2\\c, v(s) = vi(s) +iv2(s).
The eigenvalues of the operator A(e) are Xk = Xk (e), and the corresponding eigenfunctions are ek (s; e) = eA* /P'(Xk (e); e) = eA* (e)s/(1 + ei + eiAk (e)), k = ±1, ±3,..., \\ek (s; e)||C - 1 for fc —^ TO.
Along with (3.13) we introduce the operator
dh/ds, 0 < s ^ 1,
e—i (h(0) + (1 + e2)h(1)), s = 0
A*(e)h ^ ...................(3.14)
in space C[0,1] with domain D(A*) = {h(s) e Ci[0,1], -eih'(0) + h(0) + (1+ e2)h(1) = 0}. The operator (3.14) is conjugate to (3.13) with respect to the scalar product of S.N. Shimanov [11],
which has the following form for Eq. (2.9):
0
(v(s), h(s)) = sMOMO) - (1 + e2) I w+lMO (3.15)
-1
The eigenvalues of the operator A* (e) are -Ak(e), and the corresponding eigenfunctions are hk = = hk(s; e) = e-Xk(e)s. The following orthogonality conditions are satisfied between the functions ek(s; e) and hk(s; e):
{ek! (s; e),hk2 (s; e)) = 5^, (3.16)
where 5klk2 is the Kronecker symbol. The expressions
0
Pk(v; e) = {v(s),hk(s,e)) = J v(s) drk(s,e), p-k(v; e) = pk(v; e),
1
r (s-e) = i ^^ s = 0,
rk(s; 1 - (ei + A-1(e))e-Xk(e)s, - 1 < s< 0, k = 1,3,...,
define a sequence of linear continuous functionals in the space C[-1; 0], the norms of which are
0
\\Pk(v; e)|| = sup \pk(v, e)| < V\rk(s; e)| = 0(1), k (3.17)
lk(s)lle=i -1
Theorem 2. A system of functions ek (s,e) is complete in C [—1,0] in the following sense: there is no function q(s) £ C[—1,0], q(s) ^ 0, for which {q(s), hk(s)) =0, k = ±1, ±3,...
Proof. The operator A-1(e) acting in C[—1,0] has the following form:
0
A-1(e)/(s) = y f (s1) dsiR(s,s1; e), (3.18)
-1
where
(1 + e2)(s + 1), - 1 < s1 <s, 1+ e2 - s1, s ^ s1 1 - e1 + e + 2, s1 = 0,
R(s,si]e) = 2 { 1 +e2 ~ si, s ^ si < 0
is a completely continuous operator having single eigenvalues \k 1(e), k = ±1, ±3,...,
| A—1 (e) | > |A-1(e)| > ..., which correspond to the eigenfunctions ek (s; e), k = ±1, ±3,____
It follows from A- 1(e) that A- 1(e)f (s) = 0 if and only if /(s) = 0. For the operator (3.18) in C[-1,0] there is an equivalent norm W Wj^, where ||A—1(e)W^ = sup||/1|*,=1 11A 1 (e)f (s)||
c =
= maxk |A—1 (e)| = |A—1 (e)| (see, for example, [12, pp. 15-16]).
Suppose we have found q(s) £ C[-1,0], q(s) ^ 0, {q(s),hk(s)) =0 k = ±1, ±3,... We normalize q(s) ^ q(s)/1|q(s)||C, i.e., ||q(s)||C = 1. Denote ||A—1(e)q(s)||j = 5(e) > 0. Consider all eigenvalues A— 1(e) for which |A—1 (e)| ^ 5(e)/2, |A—+L1(e)| < 5(e)/2, and there can be only a finite number of them k = ±1, ±3,...,±l. Let Ci+1[-1,0] denote the subspace of functions C [-1,0] for which {v(s),hk (s,e)) =0, k = ±1, ±3,... ,±l. In this case, sup||/(s)||£=1,/(s)ecI+i[-1,0] ||A—1(e)f(s)||C = |Ar+1 (e)| < 5(e)/2, but q(s) £ Ci+1[-1,0] and || A-1 (e)q(s) ||
c ~ have obtained the contradiction. The theorem is proved. ■
Let l2 denote the subspace of sequences of the form z = (zi,z—i,z3, z—3,...), zk e C, z—k = zk,
X
||z||2 = \zk|2 < to. Let li denote the subspace of l2 sequences z = (zi,z—i,z3,z—3,...) for k=i
which ||z||2 = Y, \^k(e)|2\zk\2 < to and || ^ zk\k(e)ek(s;e)||c < to. Denoteby D*(A)
2 k=i,3,... k=±i,±3,...
the subspace C[-1,0] of functions of the above-mentioned kind.
Theorem 3. Every solution y(t + s;e) of Eq. (2.9), determined for t > 0 with initial condition y(s; e) = yo(s) e C[-h, 0], for t ^ to (to ^ 3) can be represented in the form
y(t + s;e) = ^ zk(t;e)ek(s;e), z(t;e) = (zi(t;e),z— i(t;e),...) e l^, (3.19)
k=±i,±3,...
where z(t; e) is a .solution on the equation
z = A(e)z, A(e)z = (Ai(e)zi ,X—i(e)z— i,...), (3.20)
in the space l2 with initial condition z(to;e) = (zi(to;e),z— i(to;e),...), where
zk (to; e) = (y(to + s; e),hk (s; e)), fc = ±1, ±3,.... (3.21)
Conversely, if z(t; e) with t ^ to is a solution of Eq. (3.20) with initial condition z(to; e) e l then the expression (3.19) determines for t ^ to the solution of Eq. (2.9).
Proof. For to ^ 3 the solution of Eq. (2.9) is y(to + s; e) e C3[-1,0] and eiy/"(to)+y"(to) + + (1 +e2)y//(to - 1) = 0. Hence, integrating by parts equation (3.21) taking into account (3.15) and (3.17), we have
zk (to; e) = (y/// (to + s; e),hk (s; e))X—3 (e), \zk (to; e)| = O(fc—3), fc — to.
Thus, z(to; e) e li and zk (t,e) = eXk(s)(t—t0 )zk (to; e), fc = ±1, ±3,.... Substituting zk (t; e) in (3.19), we note that the series and its derivative with respect to t converge uniformly for -1 ^ s ^ 0 and satisfy Eq. (2.9) for t ^ to. Conversely, if z(t; e) is the solution of (3.20) with initial condition z(to; e) e l^, then the function defined by the series (3.19) y(to + s) e D*(A) and for t > to satisfies Eq. (2.9), in which it is easily verified by direct substitution. The theorem is proved. ■
4. The structure of solutions of Eq. (2.7)
in a neighborhood of the zero equilibrium state
Let us study the structure of solutions y(t + s; e) of (2.7), determined for t > 0 with initial conditions y(s; e) = yo(s) G S(R0) = {y(s) G C[—1,0], ||y(s)||c < Ro}, i.e., belonging to a sphere of fixed radius R0 with the center at the origin of the phase space. For t > t0 (t0 ^ 3) y(t + s;e) G C3[-1,0] and
ei"y/.(t; e) + y(t; e) + (1+ e2)y(t - 1; e)+ f "(y(t - 1; e); e2)y2(t - 1; e) +
+ f'(y(t - 1; e); e2)y(t - 1; e) = 0. (4.1)
Taking this into account and integrating by parts the scalar product (3.15) for t ^ t0, we
have
{y(t + s; e), hk(s; e)) = A-1(e)f (y(t - 1; e); e2) + A-2(e)f'(y(t - 1; e); e2)y(t - 1; e) + + A-3(e)(f"(y(t - 1; e); e2)y2(t - 1; e)+ f '(y(t - 1; e); e2)y(t - 1; e)+ (4.2)
+ {'y'(t + s; e), hk(s; e))) = A-1(e)f (y(t - 1; e); e2) + Wk(t; e), k = ±1, ±3,.... Since
A-1 (e)ek (s; e) = A-1(e)(1 + e1 + e1Ak (e))-1 eAk (e)s =
= \k (e)-1 (1 + ei + eiAfc (e))-1(1 + eAk (e))(1 + e2)-leXk (£)(s+1) =
= -(1 + e2)-1(e1 + A-1(e))(1 + e1 + e1 Ak(e))-1 eAk(e)si, 0 < s1 < 1, it follows from (3.11) and (3.17) that the series A-1(e)ek(s; e) converges uniformly
k=±1,±3,...
for -1 ^ s ^ 0 and w(t;e) = (w1(t;e),w-1 (t;e),...,) £ l^, where wk(t;e) are determined in accordance with (4.2). This implies that, when t ^ t0, the following representation holds for the solution y(t + s; e) of Eq. (2.7):
y(t + s; e)= ^ (wk(t; e)+A-1(e)f(y(t - 1; e); e2)ek(s; e)). (4.3)
k=±1,±3,...
From (4.3) for || ^ wk(t; e)ek(s; e)||c ^ r0, where r0 is small, we get
k=±1,±3,...
y(t - 1; e) = wk (t; e)ek (-1; e)+g I ^ wk (t; e)ek (-1; e); el. (4.4)
k=±1,±3,... yk=±1,±3,... J
Here g(y; e) = g2(e)y2 + 0(y2) is an analytic function in a neighborhood of the point y = 0.
Denote by S(r0) = {z = (z1,z_1,...) £ I2, Wz^ < r0}, S1 (r0) = {z = (^1,^-1,...) £ l^ ||z||ii < r0}. It follows from (4.4) that, when w(t; e) = (w1(t; e),w-1 (t; e),...) £ S 1(r0),
(y(t + s; e),hk(s; e)) = Wk(t; e)+A-1(e)^i ^ wj(t; e)ej(-1; e) +
j=±1,±3,...
( \ \ (^ + Y1 Wj(t; e)ej(-1; e); ej ; e^ , k = ±1, ±3.
Considering f (y(t — 1);e2) in (2.5) as an inhomogeneous term of the linear equation (2.9), we write the solution y(t+s; e) of Eq. (2.5) for t ^ t0 with initial condition yo(s; e) = yo(to + s; e), according to (3.12) and using the function K(t;e), in the following form:
0
e1K(t — to + s; e)y(to; e) — (1 + e2) J K(t — to + s — 1 — s1)y(to + s1; e)ds1 +
-1
t
+ j K(t + s — t1; e)f(y(t1 — 1; e); e2) dt1, t + s ^ to,
y(t + s; e) = <
to
yo(t + s; e), t + s<to.
(4.6)
Hence, taking (3.10)-(3.11) and (4.4), (4.5) into account, we have the sequence of equalities Wk(t; e) + A"1(e)/( ^ Wj (i; e)ej (-1; e)+9( ^ Wj (t; e)ej (-1; e); ej; e2) =
^ j=±1,±3,... ^ j=±1,±3,... ' '
t
= (y(to + s; e),hk (s; e))eXk (s)(t"to> - f eXk £ Wj (ti; e)ej (-1,e) + (4.7)
to Vj=±1,±3,..
+ 9i Wj (ti; e)ej (-1,e); ^ ; e^ dti, k = ±1, ±3,....
S=±1,±3,... ' '
Differentiating Eqs. (4.7) for t,we obtain a system of differential equations in the space l2 of the form
Wk + A"1(e)//i y1 Wjej(-1;e2)+^( y1 Wjej(-1;e);e);e0 x
^ j=±1,±3,... ^ j=±1,±3,... ' '
x (1+9\ E Wjej(-1;e); e)) E Wjej(-1; e) = xk(e)Wk,
^ j=±1,±3,... ' ' j=±1,±3,...
k = ±1, ±3,...,
whose solution for t ^ t0 is provided by the functions Wk (t; e) (k = ±1, ±3,...), defined in (4.2) with corresponding initial conditions Wk(to; e). The opposite is also true. If W(t; e) = = (W1(t; e),W-1 (t; e),...) £ S 1(r0) with t ^ t0 is a solution of the system of differential equations (4.8), then according to (4.4), (4.5) for t ^ t0 the series (4.3) converges uniformly in the area {-1 ^ s ^ 0, t0 ^ t ^ t} and, taking (4.2) into account, is a solution of (2.5).
Let us solve (4.8) for Wk. Multiply (4.8) by ek(-1; e) and sum by k. This allows us to obtain in explicit form an expression for ^ Wkek(-1; e). Note that ek(-1; e) = e"Xk(e)(1 + e1 +
k=±1,±3,...
+ e1Ak(e))"1 = (1 + e1Ak(e))(1 + e1 + e1Ak(e))-1(1 + e2)-1 - 1 for k ^to.
As a result, we obtain a system of differential equations equivalent to (4.8):
Wk = Ak(e)Wk - Ak1(e)f( £ Wjej (-1; e)+gi £ Wjej (-1; e); ej ; e2j x
^ j=±1,±3,... ^ j=±1,±3,... ' '
x (1+9^( y1 Wjej(-1;e);e)) y1 Ap(eHep(-1;^u + v1^ x x/\ Wjej(-1;e)+g( Wjej(-1;e);e);e0 x
j=±1,±3,... j=±1,±3,...
x (1+9^ E Wjej(-1;e);e^ , k = ±1, ±3,....
(4.9)
Note the following property of the function Ak (e).
Theorem 4. There are e0 > 0, c0 > 0 such that, when 0 ^ |e| ^ e0, the following inequalities hold true uniformly with respect to kj = ±1, ±3,... (j = 1,..., 4) :
| Ak! (e) + Ak2 (e) + Ak3 (e) |, | Akx (e) + Ak2 (e) + Ak3 (e) + Ak4 (e) | ^ C0, (4.10)
the second inequalities are true under the condition k1 + k2 + k3 + k4 = 0.
Proof. The proof follows immediately from (3.7), (3.8). We first consider the first inequality (4.10). According to (3.8), |Akl(0) + Ak2(0) + Ak3(0)| ^ n. Consider the case of equality, and let k1, k2 > 0, and k3 = -k1 - k2 + 1. For e = 0, according to (3.8), nkj - n/2 < Im Aj(e) < nkj and Im Akj ^ nkj - n/2 for kj ^ to (j = 1,2). But then infk1,k2,\e\^e0(Im(Ak1 (e) + Ak2(e) + + Ak3(e)))2 > 0. Thus, there exists c0 > 0 with the properties specified in (4.10). Strict inequality and other combinations of k1, k2, k3 are treated similarly. For the second inequality of (4.10) for k1 + k2 + k3 + k3 = 0 \Akl(0) + Ak2(0) + Ak3(0) + Ak4(0)| ^ 2n. In accordance with (3.8) we have infk1,k2,k3,k4,\s \<£0(Im(Ak1 (e) + Ak2 (e) + Ak3(e) + Ak4(e)))2 ^ n2. The theorem is proved. ■
Taking into account inequalities (4.10), we can perform the normalizing change of variables in the system of equations (4.9) w = z + c2(z)+ c3(z): S1(r0) ^ S 1(r0) of the form
wk = zk + ^ Cck1k2 (e)zk1 zk2 + ^ Ck1k2k3 (e)zk1 zk2 zk2, z-k = Zk,
(k1,k2)en2 (k1 ,k2,k3)enk (4.11)
k = ±1, ±3,...,
where Q2 = {(k1 ,k2): k1,k2 = ±1, ±3, ...,k ^ k2}, ^k = {(k1 ,k2, k3): k1,k2, k3 = ±1, ±3,..., k1 + k2 + k3 = k, k1 ^ k2 ^ k3}, leading it to a system of equations in the space l2 of the form
zk = Ak(e)zk + ^ dk1 k2k3(e)zk1 zk2zk3 + Z*k(z; e) = Zk(z; e), k = ±1, ±3,..., (4.12)
(k1,k2,k3)enk
where Qk = {(k1 ,k2, k3): k1,k2, k3 = ±1, ±3, ...,k1 + k2 + k3 = k, k1 ^ k2 ^ k3}, is the smooth operator with respect to the set of variables Z(z;e) = (Z1(z;e),Z-1 (z;e),...) (Z(0;e) = 0):
S 1(r0) ^ S(r0) and\\Z*(z; e)\\h = o(\\z\\f1).
l2
To build a replacement (4.11) and Eqs. (4.12), substitute (4.11) in (4.9) with respect to (4.12) and equate coefficients of the same degree of zk1 zk2, zk1 zk2 zk3, k1, k2, k3 = ±1, ±3,... in the resulting equations successively for each k = ±1, ±3,.... As a result, we obtain a recursive sequence of algebraic equations, from which the coefficients ck1k2 (e), ck1k2k3 (e), dk1k2k3 (e) with the indicated properties are uniquely determined. From the conditions of final identities, for each k, the functions Zk(z; e) with the indicated properties are uniquely determined.
The system of equations (4.12) is called the normal form of Eq. (2.5) in the neighborhood of the zero equilibrium state normalized to cubic terms inclusive.
Note that the normalization of Eqs. (4.9) can be performed up to any odd power in zk. In this case, for the functions Ak(e), the relations of higher order similar to (4.10) are valid, and the normalized equations will not contain even powers of zk.
Substituting the expressions (4.11) in (4.3) with (4.4), we get the final representation of the solutions of Eq. (2.5) with the initial conditions from the neighborhood of the zero equilibrium state.
Theorem 5. There are R0, r0,e0 > 0 such that, when 0 < \e\ < e0, every solution y(t+s; e) of Eq. (2.5), determined for t > 0 with initial condition y0(s; e) £ S(R0), for t ^ t0(t0 ^ 3) and t as long as y(t + s; e) £ S(R0), can be represented in the form
y(t + s; e) = (zk (t; e)+gk (z(t; e); e))ek (s; e), (4.13)
k=±1,± 3,...
where the operator, which is smooth in the set of variables, has the form g(z; e) = = (91 (z;e),9-1(z;e),...)(g(0;e) = 0): S1 ^ S^0) and ||g(z;e)|| = o(||z||ji), z(t;e) = = (z1 (t; e),zk1 (t; e),...) £ S 1(r0), and zk (t; e) (k = ±1, ±3,...) are solutions of the system of equations (4.12) with initial conditions zk(t0;e) defined by (y(t0 + s;e),hk(s;e)) = zk(t0;e) + + 9k(z(t0; e); e). Conversely, if z(t; e) £ S 1(r0) for t ^ t0 are solutions of the system (4.12), then the function y(t + s; e) defined according to (4.13) is a solution of Eq. (2.5).
Thus, the question of analyzing the bifurcations of periodic solutions of Eq. (2.5) from the zero equilibrium state with changing parameter e reduces to solving an analogous problem for the system of equations (4.12).
In what follows, the main part of the normal form (4.12) will be called the system of equations containing on the right-hand side terms that are not higher than those cubic in zk.
The calculation of the main part of the normal form (4.12) can be performed directly from Eq. (2.5). Let us proceed from Eq. (2.5) to the equivalent initial boundary value problem
du _ du dt ds'
du ds
(4.14)
= -u(0,t) - (1 + e2>u(-1,t) - / (u(1,t); e2), u(s, 0) = y0(s) (4.15)
s=0
in the band -1 ^ s ^ 0, t ^ 0, putting u(s, t) = y(t + s). Introduce the function
(s,z;e) = £ zk ek(s; e)+ £ zki zk2 Uki k2 (s; k=±1,±3,... (k1,k2)=H2
+ ^ ^ zki zk2 zk3 Ukik2k3 (s; e),
(ki,k2,k3) = ^3
(4.16)
where Q2 is defined in (4.11), Q3 = {(k1,k2,k3): k1,k2,k3 = ±1, ±3, ...,k1 ^ k2 ^ k3}, which operates from s1(r0) ® {|e| < e0} in C[-1,0] and which depends smoothly on its variables, and the main part of the normal form (4.12) of Eq. (2.5)
zk = Ak (e)zk + £ dki k2 k3 (e) zki zk2 zk3 ■ (4.17)
(kik2k3)enk
Functions ukik2(s;e), ukik2k3(s;e), dkik2k3(e) should be determined. The condition for the trajectories of the system of equations (4.17) to belong to the boundary value problem (4.14)-(4.15) by virtue of (4.16) takes the form
^ (s z; e) x ^ d^M (s z' e)
E dzj* + E dklk2k3(£)zklzk2zk3) = ' , (4.18)
k=±1,±3,... (kik2k3)enk
du(s, z; e)
ti
ds
= -u(0, z; e) - (1 + e2)u(-1, z; e) - /(u(-1, z; e); e2). (4.19)
s=0
Relations (4.18)-(4.19) determine the identities that should be satisfied uniformly over |e| < e0 modulo o(|z|32). At the first degrees zk they are satisfied by the definition of functions ek(s; e) and Ak(e). Relations (4.18)-(4.19) allow one to sequentially define the functions u*(s; e) and d*(e) by equating the coefficients on the right and on the left for the same powers zk.
We equate the coefficients of (4.18)-(4.19) with zk1 zk2. As a result, we obtain the boundary value problem
(Xkl(e) + Ak2(e))uklk2(S]e) = , (4.20)
duklk2(s; t) ds
duklk2 (s; e)
ds
= -Ukik2(0; e)- (1 + e2)ukik2 (-1; e) - Pkk^2)eki (-1; e)ek2 (-1; e) (4.21)
to determine uklk2(s;e). In (4.21) pklk2 = 1 for fci = k2, and pklk2 = 2 for ki = k2. The solution (4.20)-(4.21) is defined uniquely
Ukik2 (s; e) = -Pkik2 b2(e2 )e(Xki (e)+Xk2 (^))s/P (e) + Ak2 (e); e). Equating in (4.18)-(4.19) coefficients for zki zk2 zk3 yields the boundary value problem
efc ( s ; t ) 4i fc2 fc3 ( t ) ( Afci (t ) + Afc2 (t ) + Afc3 ( t ) ) ukl k2k3{s] e) = dUklk^s, j k = ki+k2+k3, (4.22)
tr
ds
dukik2k3 (s; e)
ds
= -ukik2k3 (0; e) - (1 + e2)ukik2k3 (-1; e) + fkik2k3 (e), (4.23)
s=0
fkik2k3(e) = -Pkik2k3b3(e2)eki(-1; £)ek2 (-1; e)ek3(-1; e) - 262^2) x
{eki (-1; e)uk2k3 (-1; e), if ki = k2 = k3,
ekj(-1;e)ukmkp(-1;e) +ekm(-1;e)ujkp(-1;e), if kj = km = kp,j,m,p = 1,2,3, eki (-1; e)uk2k3(-1; e) + ek2(-1; e)ukik3(-1; e) + ek3(-1; e)ukik2(-1; e), if ki = k2 = k3,
Pkik2k3 = 1, if ki = k2 = k3, Pkik2k3 = 3, if ki = k2 = k3, or ki = k3 = k2, or k2 = k3 = ki, Pkik2k3 = 6, if ki = k2 = k3.
The general solution of Eq. (4.22) has the form
ukik2k3(s; e) = eXkik2k3(e)s(c + dkik2k3(e)(1 + ei + eiAk(e)) x x (1 - e(Ak(-)-Xkik2k3(-))s)(Ak(e) - Akik2k3(e))-i),
where we use the notation Akik2k3(e) = Aki(e) + Ak2(e) + Ak3(e) and c is an arbitrary constant. Substituting (4.24) into the boundary condition (4.23), we get
dkik2k3(e) = (ei + (1+ e2)(e-Xkik2k3- e-Xk^/(Ak^(e) - Ak(e)))- x 2g)
x (1 + ei + Ak(e))fkik2k3 (e).
The condition for the function ukik2k3(s; e) to be continuous in e with 0 ^ |e| ^ |eo| gives c = 0, which uniquely defines it by equality (4.24).
When constructing the normal form (4.12) up to order 2m + 1, where m > 1, of the corresponding order, one takes expansions (4.16) and (4.17). The algorithmic part repeats the foregoing.
5. Analysis of the normal form of Eq. (2.7)
We transform the system of equations (4.12) to polar coordinates by setting zk = pkeiTk (pk ^ 0, 0 ^ Tk < 2n). As a result, we obtain the following system of equations:
Pk = 7k(e)pk + R<i) (p, t; e) + R* (p, t; e), (5.1)
fk = nk + ak (e) + (Tk3)(p,T; e)+T(*)(p,T; e))/pk, k = 1, 3,... , (5.2)
where 7k(e) and ak(e) are defined in (3.8), p = (p1,p3,...), Hp!2 = k2pk < o, t =
2 k=1,3,...
= (t1,t3,...), the functionals R^(•), R^O, Tfc3^), Tk*)(-) smoothly depend on incoming
(3) (3)
variables and parameters and are 2n-periodic in Tj, with Rk (•) and Tk (•) being cubic forms in p, and R0)|, T(•) = 0(||p||fi).
The structure of the system (5.1)-(5.2) allows one to introduce one "fast" variable and a countable number of "slow" variables. How this can be done will be shown first by the example of a "truncate" system. Consider the main part of the normal form (4.17), setting zk = 0, k = ±5, ±7,... As a result, we have the system of equations
z1 = A1(e)z1 + d_111 (e)z_1 z2 + d_313(e)z_3z1z3 + d_1_13 (e)z_1z3, (5.3)
-¿3 = A3(e)z3 + d_113 (e)z_1 z1z3 + d_333 (e)z_3 zf + dm (e)z3. (5.4)
Equations for z_1, z_3 are obtained by conjugation of (5.3)-(5.4) taking into account the equalities Ak(e) = A_k(e), Ak(e) = z_k(e). Denoting d*(e) = a*(e) + ib*(e), A*(e) = d(e)|, /3*(e) = = argd*(e) £ [0,2n), we change in (5.3)-(5.4) to polar variables p1, p3, T1, t3. This yields
p1 = (71(e) +a_m (e)p1 + a_313 (e)p2)p1 + A_1_13(e)cos(-3T1 + T3 + ^_1_13 (e))p1p3, (5.5)
-33"
7-1 = n + a1(e) + b_ 111 (e)p1 + b_313(e)p2 + A_1_13(e) sin(-3T1 + T3 + ^_1_13(e))p1 p3, (5.7) T3 = 3n + a3(e) + b_1_13(e)p1 + b_333(e)p3 + Am(e) sin(3T1 - T3 + (e))p1 /p3. (5.8)
Let us pass in (5.5)-(5.8) to variables p1, p3, 01 = -3t1 + t3 and t = T1. As a result, we obtain the system of equations
p1 = (71(e) + a_m (e)p2 + a_313(e)p3)p1 + A_1_13 (e)p1 cos(6>1 + ^_1_13 (e))p2 p3, (5.9)
P3 = (73(e) + a_1_13(e)p1 + a_33_3(e)p3)p3 + Am (e)p1 cos(-01 + (e))p3, (5.10)
O1 = ¿1(e) + (—3b_111 (e) + b_1_13(e))p2 + (-3b_313(e) + b_333(e))P2 - (5.11) - 3A_1_13(e) sin(6»1 + /3_1_13(e))p1 p3 + Am(e) sin(-^1 + (e))p3/p3,
for "slow" variables, where ¿1(e) = -3a1(e) +a3(e), and an equation for the "fast" variable
T = n + a1(e)+ b_ 111 (e)p1 + b_313(e)p2 + A_1_13(e) sin(6l1 + ^_1_13(e))p1p3. (5.12)
We note that the right-hand side of (5.9)-(5.11) does not depend on t.
Suppose now that in (4.17) all zk(t) = 0. We can introduce the variables dk in more than one way, but they are all related to each other by linear relations. One of the ways to introduce dk is as follows. As a "fast" variable t we take t1 and consider the "truncated" finite-dimensional
P3 = (73(e) +a_1_13(e)p1 + a_33_3(e)p3)p3 + Am(e)cos(3T1 - T3 + A11 (e))p3, (5.6)
systems, consistently assuming in (4.17) zk = 0, k = ±5, ±7,..., then k = ±7, ±9,..., etc. The first case has been considered above. In the second case, two new equations are added to the system (5.3)-(5.4) for variables z5 and z-5. In this case, a resonant monomial appears on the right-hand side of the equation for zi z5z-3z-:1(in the equation for z-i the resonant monomial z-5z3z1 appears). When moving to polar coordinates, this leads to the appearance of terms that depend on the expression t5 — t3 — 2t1 = d3, which we take as a new "slow" variable. As a result, we have two additional "slow" variables p5, d3. The system of equations for p1, p3, p5, d1, d3 will have a form similar to (5.9)-(5.11), the right-hand side of which will not depend on t either. On the right-hand side of Eq. (5.12), new terms appear for t. When considering the next "truncated" system on the right-hand side of the equation for z1, we obtain a resonant monomial z7z-5z-1, and changing to polar coordinates, we obtain a term depending on the expression t7 — t5 — 2t1 = Q5. We have two new "slow" variables p7, d5. In the general case, in the next step, adding equations for zko and z-ko, we obtain in the equations for z1 and z-1 new monomials zkoz-ko+2z-1 and z-kozko-2z1, respectively, and moving to the polar coordinates, we obtain terms depending on Tko — Tko-2 — 2t1 = dko-2. We have two new "slow" variables pko, dko-2. Continuing this process, we change in (4.17) to the "slow" variables p = = (p1,p3,...), 9 = (d1,d3,...) and the "fast" variable t. The corresponding transformation to "slow" and "fast" variables in (5.1), (5.2) gives the following system of differential equations for their determination
pk = Yk(e)pk + Rk(p, 9; e) + Rk(p, 9, t; e), (5.13)
Ok = Sk(e)+8k(p, 9; e) + ©k(p,9,T; e), k = 1,3,..., (5.14)
T = n + ^(e) + T(p, 9; e) + T*(p, 9, t; e), (5.15)
where the functionals Rk(-), Rk(•), ©k(•), ©k(•), T(•), T*(-) — 2n are periodic in 9k, the other properties and properties of the function 5k (e)(Sk (0) = 0) are determined by the properties of the functions and functionals appearing in (5.1)-(5.2).
The phase space of the system of equations (5.13)-(5.15) is the product of spaces l2 ® c x R, here I2 = {p = (p1,p3,...), pk > 0, \\p\\22 = E pk < c = {9 = (91,93,...), ||0||c =
k=1,3,...
= supk \9k\ < to}. The domain of the right-hand side of (5.13)-(5.15) is a product of spaces l1 ® co x R, where l1 = {p e I2, pk > 0, \\p\\21 = E k2p2k < to}, co = {9 = (91,93,...) e c,
2 k=1,3,...
0 < 9k < 2n}.
We introduce variables Z ^ 0 and n/2 < ^ < n/2 in the domain {(e1,e2),e1 > 0, \e\ < e0}, setting
Z = (e1 + \e2 \)1^2, e1 = Z cos ^, e2 = Z2 sin2 ^ sign ^. (5.16)
Substituting (5.16) into (5.13)-(5.15) and normalizing pk Zpk, we get the system of equations
pk = Yk(^, Z)pk + Z2Rk(p, 9; 4, Z) + Z3Rk(p, 9, T; ^, Z), (5.17)
9k = Sk (4,Z )+Z 2©k (p, 9; 4,Z)+Z 3©k (p,9,T; 4,Z), k = 1,3,..., (5.18) f = n + 01(^,Z) + Z2T(p, 9; Z) + Z3Tk(p, 9, t; Z), (5.19)
in which the properties of functions and functionals are determined by the properties of the functions and functionals of (5.13)-(5.15).
Since, according to (3.8) and (5.16), the equation Yk(4,Z) = O(Z2), Sk(4,Z) = O(Z3) is satisfied uniformly with respect to k, Eqs. (5.17) and (5.18) justify the name of equations of "slow" variables, and (5.19), the name of the equation of a "fast" variable.
Consider the main part of the system of equations of "slow" variables (5.17)-(5.18). Rescal-ing time by t ^ t/Z2, we get
pk = 7k(0,Z)pk + Rk(p,0; 0,Z), (5.20)
Ok = Sk ((0,Z)+©k (p,0; 0,Z), k = 1,3,..., (5.21)
where, according to (3.8) and (5.16), 7k(0,Z) = 7k(0,Z)/Z2, ¿k(0,Z) = ¿k(0,Z)/Z2 are continuous functions n/2 ^ 0 ^ -n/2, 0 ^ Z ^ Zo. In this case, 7k(0, 0) = 7k(0) = sin2 0 sign 0 -- n2k2 cos2 0/2, Sk(0, 0) = 0.
Consider the system of nonlinear equations in l2 ® c
7k(0)pk + Rk(p, O) = 0 (Rk(p,O) = Rk(p,O; 0, 0) = 0), (5.22)
©k (p, 0) = 0 (8k (p,O) = ©k (p,O; 0,0)=0), k = 1,3,..., (5.23)
where (p, 0) £ E0 = 12, ® c0. In (5.22) the operator 7(0)p = (71 (0)p1,73(0)p3,...) is considered on the basis of (5.20) as symmetrically extended to 11.
Let (p *(0), 0*(0)) £ l1 ® co be the solution of the system (5.22)-(5.23). We introduce the infinite matrix
R,n (7k(0)skj + dRk/dpj dRk/d0j\
B (0)^ 988k/dPi O8k/d0j) (k,j = 1,3,...), (5.24)
calculated at the point p*(0), 0*(0), where Skj is the Kronecker symbol. The matrix (5.24) defines the linear operator
B(0)v (v = (p, 0)): E1 = l1 ® co ^ E = I2 ® c, (5.25)
HvHb = ||p||i2 + ||0||c, ||v||ei = ||p||ji + ||0||c.
Let pk(0) (B(0)vk(0) = pk(0)v(0)) be the eigenvalue of the operator (5.25). We show that the operator (5.25) has a countable number of eigenvalues pk(0) which can be numbered in increasing order of their moduli and that any bounded domain of the complex plane contains only a finite number of eigenvalues of finite multiplicity. The limit can only be an infinity point, with limk^^ Repk(0) = -o. Indeed, the functions
z1(t; 0,Z) = Zz* (t ; 0) = Zp1(0)eiT, z3(t; 0,Z) = Zz** (t ; 0) = Zp3(0)ei(3T+di W),
z5(t; 0,Z) = Zz* (t ; 0) = Zp5(0)ei(5T+0i W) ,...,z_k (t; 0,Z) = Ak (t; 0,Z) (5.26)
k = 1, 3,... ,
f = n(1 - Z cos 0), (5.27)
determine the periodic solution of the following system of differential equations: zk = (ink(1 - Z cos 0) + Z27k(0))zk + £ dkik2k3 (0)zki zk2zk:i, k = ±1, ±3,... (5.28)
(ki,k2,k3)enk
(Q§ and dkik2k3(0) are determined, respectively, in (4.12) and (4.25)), which can be easily checked by direct substitution of (5.27) into (5.28), taking into account the substitutions for p* (0), 0*(0)
and Eqs. (5.22), (5.23) for p*(0), O*(0). The system of equations (5.28) is abstractly parabolic in the complex space l2 with the domain l^. For such systems of equations the monodromy operator constructed for the periodic solution (5.27) is completely continuous, containing a countable number of eigenvalues (multipliers) pk(0), limk^^ \pk(0)\ = 0. The operator (5.25) determines the characteristic exponents of the periodic solution (5.27). This leads to all the listed properties of ¡k(0). Substituting p*(0), O*(0) into (5.19), we obtain by virtue of (3.8) and (4.25)
t = n - Z cos 0 + Z2 cos2 0 + O(Z3). (5.29)
Theorem 6. Suppose that for some 0 the system of equations (5.22)-(5.23) has a solution (p*(0),O*(0)) £ l^ ® c0 and the matrix (5.24) defines the operator (5.25), which has no eigenvalues lying on the imaginary axis of the complex plane. Then there exists a Z0 > 0 such that for 0 < Z ^ Z0 Eq. (2.5), where e1 and e2 are defined according to (5.16), has a periodic solution y*(t + s; e) = y*(t + s; 0, Z) of the form
V*(t; 0,Z ) = Z £ Z*(T; 0) - Z 2b2(0)/2 £ Pk!k2 z*kl (t ; 0)z*2 (t ; 0) + O(Z3), (5.30)
k=±i,±3 (ki
where b2(e2), Q2, pklk2, zk(t; 0) and t are, respectively, defined in (2.8), (4.11), (4.20), (5.27), and (5.29).
The solution y*(t + s; 0,Z) is asymptotically orbitally stable if all eigenvalues of the operator (5.25) lie in the left complex half-plane and are unstable if m roots, with multiplicities taken into account, belong to the right complex half-plane. In the latter case, the periodic solution has an m-dimensional unstable invariant periodic manifold.
Proof. Let us use the new time variable t in (5.17)-(5.18) according to (5.19). As a result, we obtain a system of differential equations in space l2 ® c, which we write in the operator form
% = c2(7*('0; op + r(p, 0; 0 + zrw(p, o, t; 0, 0), (5.31)
f = z\s*(t, z) + B(p, 9; 0, z) + c©v(p, 9, r; 0, ()), (5.32)
The operators 7*0) = (n-17*0),n-17-i(0,...), S* (•) = (n-1S* (-),n-15-i(-),...), R(-) = = (n-1Ri (•),n-1R-i(^),...), ©(•) = (n-1©i(),n-1 ©-i(0,...), and the operators RV(0 = = (RV(),R— 1 (•),...), ©V(0 = (©V(), ©-1 (•),...), are constructed according to functionals RV( ) and ©—(•), whose properties are determined by the properties of the functionals R*k(•), ©k(•), T*(0, T(•).
Denote by
2n 2n
R%(p, 9; 0, Z) = t^J Rk(p, 0, r; 0, Q dr, e%(p, 9; 0, Q = ± J 0^(p, 9, r; 0, Q dr (5.33) 00
the mean of the function R— (•) and ©—(•) and consider in E the system of nonlinear operator equations
Y*(0; Z )p + R(p, 0; 0, Z) + ZRV(p, 0; 0, Z) = 0, (5.34)
S*(0; Z )P + ©(P, 0; 0, Z) + Z © V(P, 0; 0, Z) = 0, (5.35)
with domain E1, where the operators Rv(-), 8v() are defined according to (5.33). For Z = 0 the system (5.34)-(5.35) has the solution p*(0), 0*(0). In this case, the eigenvalues pj(0) of the operator (5.25), constructed from the partial derivatives matrix (5.24) corresponding to this solution, satisfy the condition | Re pj(0) > po > 0. Consequently, by the implicit function theorem for operator equations in a Banach space (see, for example, [12, pp. 304]) there is a Zo such that, when 0 ^ Z ^ Zo, the system of equations (5.34)-(5.35) has the solution p*(0; Z), 0*(0; Z) (p*(0;0) = p*(0), 0*(0;0) = 0*(0)) smoothly dependent on the parameter Z.
Linearizing (5.34) and (5.35) on the solution p*(0; Z), 0*(0; Z), we obtain an infinite matrix that is similar to the matrix (5.24) and turns into it when Z = 0, and defines a linear operator
B(0,Z)v (v = (p, 0), B(0, 0)v = B(0)v): E1 — E. (5.36)
Note that the eigenvalues pj(0,Z) (Pj(0,0) = Pj(0)) of operator (5.36) for 0 < Z < Zo satisfy the condition | Re pj(0,Z)| > po > 0.
We introduce the Banach spaces C2n, of, respectively, continuous and continuously differentiable 2n-periodic functions f (t) (-o < t < o) with values in E by defining the norms as Hf ||c2. = supt Hf (t)||e, Hf ||ci = Hf H^ + Uf'^.
Denote by E+(0, Z) and E_(0, Z) the root subspaces of the operator (5.36) (dim E+(-) = m), which correspond, respectively, to eigenvalues with positive and negative real parts, E+(0,Z) © ©E_(0, Z) = E1. We denote the corresponding projections by P+ (0, Z) and P_(0, Z)(P+ (0, Z) + + P_(0,Z) = I), P±(0, Z)E+1 = E±(0; Z).
Consider in E the linear differential equation
ft =B(0,()v. (5.37)
Let T(t; 0,Z)(t ^ 0) be a semigroup of bounded linear operators T(•): E1 — E1 (T(t1 + + T2; •) = T(T1; OT(t2; •) = T(72; OT(71; •)) which defines the solutions (5.37) for t ^ To with initial condition vo = v(to): v(t; 0,Z) = T(t; 0,Z)vo. We leave the previous notation for the continuous expansion T(•), P+(-), P_(^) on E (E+ (•) © E_(^) = E).
We introduce the Green's operator defined on E of the equation
C(T; 0,0 = ( T+ (T:0-C>= T(T; 0'Z)P+ <0-<>- t< 0 (5.38)
Z \T_ (t ; )= T(t ; 0]()P_ (0,0, t > 0.
In (5.38) the operator T+(-) is finite-dimensional, and therefore it can be continued to the domain t < 0 in a unique way. Also, the following estimates are valid:
||T+ (t;0,Z)||* < ke^0T(t < 0), HT_(t;0,Z)H* < ke_^°T(t > 0), (5.39)
where ^(^)|* = sup|H|B=1 ||T^Me.
Consider in E the differential equation
^ = (2B(0, Qv + /(r), /(r) G C2w. (5.40)
Equation (5.40) has the single 2n-periodic solution
oo
v(t ; 0,Z))= j G(Z2(t - T1)(0; Z))f(n) dn, (v(t,0,Z) £ C^ (5.41)
and, by virtue of (5.39),
Mr ; 0,Z)\\aL <
+
I T-(Z2(t - ti); 0,Z)f (ti) dn
-x
Z-22k/po\\f (Ti)\\o2n.
cL
J T+ (Z2(t - Ti); 0,Z)f (ti) dTi
<
cL
¿IX
(5.42)
The formula (5.41) is checked by a direct substitution into Eq. (5.40). The stability of the 2^-periodic solution (5.41) is determined by the location of the eigenvalues of the operator (5.36). For m = 0 the solution (5.41) is asymptotically stable, and for m > 0 the solution (5.41) has an m-dimensional invariant unstable manifold.
We show that the system of equations (5.31)-(5.32) has a 2^-periodic solution of the form
v * (t ; 0,Z )= v * (0,Z)+v *(t ; 0,Z ) =
= (p *(0,Z),O*(0,Z)) + (p 1(t;0,Z),O*(t;0,Z)), v*(t;0,Z) G Cj«. (5.43)
Assuming v1(t) = v1(t; •) in (5.42) as the required function (we omit the sign *), we substitute into (5.31), (5.32). As a result, we obtain an equation for determining v1(t) of the following form
^ = Z2(B(tP,Z)v1 + F(v1]tP,Z) + ZFv(T,v1]tP,Z)+Zf*(r,t,Z)),
(5.44)
in which the function f *(t; Z) E C2n and the operators F(•), Fv(-), smoothly depending on the parameters and variables, satisfy the conditions \\F(v\; 0,Z)\\e = 0(||vi\\2Ei), Fv(t, 0; 0,Z) = 0, with the averages f* (0; Z) = 0 and (v1; 0,Z) = 0.
The problem of the existence of a solution v1(t) E of Eq. (5.44) based on (5.40) and (5.41) is equivalent to the solvability of the following operator equation:
oo
vi(T) = Z2 / G(Z 2(t - Ti)(F (vi (ti); 0, Z) + ZF v(ti,vi(ti); 0, Z) + Zf * (ti ; 0, Z)) dTi (5.45)
in space . Based on (5.42), we can apply the implicit function theorem for operator equations in a Banach space [12, pp. 304] to the operator equation (5.45). As a result, we obtain the solution v*(t; 0,Z)(v(t; 0, 0) = 0) of Eq. (5.45), which smoothly depends on the incoming parameters. Performing inverse substitutions, we have a periodic solution y*(t + s; 0,Z). Formula (5.30) was obtained using (4.13) and (4.16) (with s = 0), (5.19) by taking into account only (p*(0),d*(0)) — the "main part" of the solution (5.43). The stability of the periodic solution y* (t + s; 0,Z) is determined by the stability of the periodic solution of Eq. (5.40). The theorem is proved. ■
6. Numerical investigation of the normal form of Eq. (2.7)
Here are some of the results of numerical analysis of stable equilibrium states (p * (0), d* (0)) of the system of differential equations (5.20)-(5.21) (for Z = 0), which are solutions of the system of nonlinear algebraic equations (5.22)-(5.23) and define, according to (5.26), (5.29)-(5.30), stable periodic solutions of equation (2.1), which bifurcate from an equilibrium state x*(p*(c*),c*) for the case c* = n/3. In this case, x*(p*(c*),c*) ~ -1.33, p*(c*) w 1.56.
Fig. 3. Picture of D— partitions.
Figure 3 presents a picture of D-partitions [13] of the parameter plane (e1,e2). It shows the curves on which Eq. (2.10) has roots lying on the imaginary axis. These curves are the boundaries of domains of D-partitions. For values of e from the area D0 all roots of the characteristic equation (2.10) lie in the left complex half-plane, for e £ D2 one pair of complex conjugate roots is in the right complex half-plane, and the remaining pairs are in the left one, for e £ D4 two pairs of complex-conjugate roots are in the right complex half-plane, and the remaining pairs are in the left one, and so on. The dotted line denotes the boundaries of subsequent regions of D-partitions, which are similar to those described above. We note that the reduced picture of D-partitions of the parameter plane (e1 ,e2) can be easily mapped, according to (2.6), to the parameter plane (e1,i).
Fix the values of the parameters 0 = 1.51 in (5.16). This value of 0 in Fig. 3 corresponds to a curve in the domain Dg, marked by points. For this value of 0, stable equilibrium states of the system of differential equations (5.20)-(5.21) (for Z = 0) were defined numerically. Calculations were carried out according to the following scheme. A sequence of truncated finite-dimensional systems (5.20)-(5.21) (with Z = 0), where k = 1,3,... ,N was considered. The finite-dimensional system of differential equations was numerically integrated under various initial conditions. For numerical integration we used the software complex [14], which allowed us to simultaneously build a large number of integral trajectories (with an automatic choice of the integration step) for initial conditions randomly "abandoned", for example, according to the uniform law. There was a stable state of equilibrium (p*N(0),O*N(0)). This equilibrium state was refined in the system of equations (5.20)-(5.21) (for Z = 0) for k = 1,3,...,N + 1 etc. until jjpN(0) —
— PN+i(0)\\i1 /\\pN(0)\\l < A (A = 10-2).
For the specified value of 0 five stable equilibrium states of the system of differential equations (5.20)-(5.21) (for Z = 0) were identified. There were no other stable equilibrium states. For each of the found equilibrium states a periodic solution of Eq. (2.1) was constructed according to (2.4), (5.26), (5.29)-(5.30). Figure 4 shows for Z = 0.1 (in Fig. 3, a bold point in area Dg) graphs of these solutions (the thick line) and their projections onto the plane (x(t),x(t — 1)). The figures also show graphs of the solutions of Eq. (2.1), which were obtained by its numerical integration by the Euler method with constant step selection, with the same parameter values. The integration step was reduced until a complete stabilization of the computational results was achieved. The periodic functions (solutions) obtained in accordance with (2.4), (5.26), (5.29)-(5.30) were taken as initial functions.
Fig. 4. Periodic solutions of Eq. (2.1) and the projection of phase portraits onto the plane (x(t), x(t — 1)) for Z = 0.1, 0 = 1.51.
Thus, for the selected value of the parameter 0 with changing parameter Z > 0 in Eq. (2.1), in which si and p vary according to (2.6) and (5.16), a simultaneous bifurcation from the state (2.4) of five periodic solutions is observed. Such a phenomenon is called a bifurcation of multistability.
For other values of the parameter 0, the system of equations (5.20)-(5.21) (for Z = 0) also has a large number of stable equilibrium states to which the periodic solutions of Eq. (2.1) correspond. For example, for 0 = 1.55, seven stable equilibrium states were identified. The number of equilibrium states increases with increasing 0.
Now fix the value of the parameter 0 = 1.51 and increase Z. In Fig. 3, this case is represented by the dotted line in region Ds. Let us consider how the resulting periodic solutions will change as Z changes. As the parameter Z increases, every periodic solution turns into irregular (chaotic) oscillations through a series of period doubling bifurcations. The maximum characteristic exponent was calculated for chaotic oscillations and turned out to be positive. In this case, there exists a range of values of the parameter Z for which there are simultaneously five different chaotic oscillations (attractors). This phenomenon can be called chaotic multistability. For numerical integration of Eq. (2.1) and calculation of the maximum characteristic exponent, the software complex [14] was used. Figure 5 shows the projections of phase portraits of periodic solutions to the plane (x(t),x(t — 1)) corresponding to the transition of the first two periodic solutions presented in Fig. 4 from regular oscillations to chaotic values for Z = 0.2, 0.4, 0.8.
Fig. 5. Projections of phase portraits of the first and second periodic solutions of Eq. (2.1) onto the plane (x(t), x(t - 1)) for the values ^ = 1.51, Z = 0.2, 0.4, 0.8.
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