Asymptotic description of separatrix crossing near a saddle-center point

O.M.Kiselev, S.G.Glebov,
Institute of Mathematics
of Ufa Sci. Centre of RAS,
Ufa State Petroleum University

We consider a formal asymptotic solution of the main resonance equation:

We investigate a saddle-center bifurcation for the slowly varying equilibriums of this equation and construct a matching asymptotic solution uniformly as e®0 before, inside and after the bifurcation layer.

This problem may be considered as a separatrix crossing in confluent point. The passing through a separatrix of the second order equations in a general position was considered by A.V. Timofeev in 1979, A.I. Neishtadt in 1986. The separatrix crossing for the second order equations in the confluent point was considered in a preliminary fashion by R.Haberman in 1979 and D.C.Diminnie and R.Haberman in 2000 in more detail. The asymptotic solution crossing the separatrix in the confluent point was constructed by O.M.Kiselev in 1999,2001 for the Painlevé-2 equation.

Using our approach one can construct the uniform asymptotic solution crossing the separatrix in the confluent point for second order equation in general case.

Algebraic analysis

Let us seek an asymptotic solution of (1) in the form of the formal asymptotic series:

To obtain an answer on the precision question one can consider an autonomous equation with a "frozen" coefficient T:

	iV'+(|V|^2-T)V=1. This equation has three equilibrium positions T > t*. There are U1 < U2 < U3, where U1 is a saddle, U2 and U3 are centers.

At T = t* the saddle-node bifurcation takes place and there exist center U3 and confluent saddle-center point U*.

	When T < t* there exists along center U3.

We will construct the formal asymptotic solution of the equation (1) in the interval t Î [t_*-C,t_*+C] where C = const > 0 uniform on e . We suppose that the solution in the domain t > t_* has the form

The qualitative analysis shows that this asymptotic solution oscillates when t < t_*. Our problem is to study the transition layer between the nonoscillating asymptotics when t > t_* and the oscillating asymptotics when t < t_*.

	In the domain (t-t*)e-4/5 >> 1 the asymptotics has the form: U(t,e) = ån = 0¥enU(t). Here U(t) = U2(t) and corrections U(t) are algebraic functions of t.

In the domain |t-t*| << 1 the asymptotics is defined by four various expansions of different types. First of them is:

In the neighborhood of t = t₀ the coefficients of the asymptotic expansion depend on one more fast time scale q = (t-t₀)e^-1/5. Denote by

namely: w(q0) = [(-2)/((q0-iU*)2)]. In the domain -q0 >> 1 the asymptotic solution is defined by a sequence of two alternating asymptotics. Let us call them by "intermediate" and "separatrix" asymptotics. To obtain the intermediate  asymptotics let us introduce one more slow variable:

An asymptotic solution in the intermediate domain for not too large values k << e^-1/7 has the form:

U(t,e) = U_*+e^1/3å_{n = 0}^¥e^i/30(A_k+ie^1/6 B_k). The leader term satisfies to the equation :

A_k = -2Ã(T_k;0,g₃(k)),  g₃(k) = [1/56](g₃(k-1)+p/2).

Here g3(0) = [(a4)/56], a4 is the coefficient as (t-t0)4 in the Laurent expansion of a(t). The intermediate expansion with the leader termis valid in the domain between two poles Tk = 0 and Tk = Wk of theÃ-function: -e-1/6Tk >> 1,  e-2/15(Tk+Wk) >> 1.

At the large values of k the intermediate asymptotics are constructed in the form

U(t,e) = U_*+e^1/3å_{n = 0}^¥e^n/6(A_k + ie^1/6 B_k).

The main term of the asymptotics is:

g₃(k,e) = g₃(k)+å_{n = 1}^¥e^n/30g₃(k).
The intermediate expansion with the leader term is valid in the domain between the poles of the Weierstrass function as

The separatrix expansions are valid in a small neighborhood of the Weierstrass function poles. Denote:

q_k = (T_k+W_k-¹/₄å_{n = 1}^¥e^n/30x_k⁺)e^-1/6,    k = 1,2,.... When |q_k|e^1/6 << 1 the formal asymptotic solution of equation (1) has the form :

The leader term of the asymptotics W(qk) is a separatrix solution of the autonomous equation (5): W(qk) = [(-2)/((qk-iU*)2)]. The sequence of the alternating intermediate expansions and separatrix asymptotics is valid as e-1/6(t*-t) << 1.

In the domain (t_*-t)e^-2/3 >> 1 the asymptotic solution breaks into oscillation. The amplitude of the stimulated oscillations in the solution of (1) oscillates fast. The form of the solution is:

U(t,e) = U(t₁,t,e) + eU(t₁,t,e) + e²U(t₁,t,e),

where t1 is a new fast variable t1 = S(t)/e+f(t). The main term of the asymptotics U lies on the curve G(t): 1/2|y|4 -t |y|2 - (y + [`y]) = E(t), and satisfies to the Cauchy problem for the equation

References

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Kiselev O.M., Hard loss of stability in Painlevé-2 equation. J. of Nonlin. Math. Phys., 2001, v.8, n1, pp.65-95.

Kiselev O.M., Glebov S.G., Asymptotic description of nonlinear resonance. http://arxiv.org/abs/math/0105011