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On the derivation of spatially inhomogeneous periodic solutions from limit cycles of autonomous systems of ordinary differential equations perturbed by small diffusion terms. (Russian) Zbl 0770.35033

In some oxidization problems and other chemical reactions using catalysts, several authors used a mathematical model containing a “small” diffusion term. As the “small” parameter approaches zero the system of partial differential equations reduces to one of ordinary differential equations. The perturbed system: \[ (1)\qquad\quad \partial z/\partial t-\mu B(\partial^ 2z/\partial x^ 2)-(A_ 0+\mu^ 2A_ 1)z-N(z,z)=0 \] branches from the limit cycle \(z=z_ 0(t)\) for small values of \(\mu\). Zero initial-boundary conditions for derivatives are assumed. Here \(A_ 0,A_ 1,B\) are \(n\times n\) matrices while \(N(z,z)\) is an analytic vector function.
The author introduces an operator \(J\) linearizing (1) for \(\mu=0\). \(J\) is Fredholm and so is its adjoint \(J^*\). Solutions of (1) for small values of \(\mu\) are written in terms of elliptic integrals and expanded in infinite series of a small parameter. He proves the following theorem. For sufficiently small values of \(\mu\) there exists a solution which is a \(T\)-periodic function of \(t\) and which has a representation: \(z(t,x,\varepsilon)=z_ 0(t)+\varepsilon\dot z_ 0(t)p(x)+\sum^ \infty_{k=2}\varepsilon^ k\xi_ k(t,x)\) with \(\mu(\varepsilon)=\varepsilon\Sigma\varepsilon^ k\mu_ k\). Here \(z_ 0\) is periodic with period \(T\). The series converge in some neighborhood of \(\varepsilon=0\).

MSC:

35K55 Nonlinear parabolic equations
35K57 Reaction-diffusion equations
35C10 Series solutions to PDEs
35B10 Periodic solutions to PDEs
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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