Existence and Bifurcation of Periodic Solutions in Second Order Nonlinear Systems: Brouwer Equivariant Degree Method

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2019-08

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Abstract

We discuss the existence of periodic solutions in two types of symmetric second order nonlinear systems: (a): Γ-symmetric autonomous system x¨(t) = f(x(t)), x ∈ V = R n , where V is an orthogonal Γ-representation and f : V → V is a Γ-equivariant map. (b): The Γ-symmetric second order system of nonlinear difference equations: △2xn−1 + f(n, xn) = 0, xn ∈ R k , n ∈ Z, where w is an orthogonal Γ-representation and f : Z × W → W a Γ-equivariant map. Under some additional assumptions, we establish for (a) and (b) the existence of periodic solutions with fixed period. For (b), we also consider bifurcation problem for which we provide the topological classification of various symmetric types of solutions. The applied method is the usage of the equivariant Brouwer degree to associate (a) and (b) with appropriate equivariant topological invariant. Several concrete examples provide illustrations for the abstract results.

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Difference equations, Nonlinear systems

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©2019 Shi Yu. All Rights Reserved.

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