Control of homoclinic bifurcation in two-dimensional dynamical systems by a feedback law based on [formula omitted] spaces

Abstract

This paper proposes a homoclinic bifurcation control method in a planar system of nonlinear differential equations (x˙=f(x),x∈R2,f:U⊂R2⟶R2). The feedback control law is formulated within the framework of Melnikov theory and Lp spaces, and it will be called as Lp control. Here it is proved that if γ0(t) is the homoclinic orbit of the planar system, then f(γ0(t))∈Lq space (1≤q≤∞). To avoid the transverse intersection of the stable and unstable manifolds of the hilltop saddle, a lot of control laws (u(x)∈Lpspace) have been developed, where each of them can be found by choosing one particular Lq space to f(γ0(t)), such that p and q are conjugate exponents, that is, 1p+1q=1. Furthermore, a procedure to find the control gains is presented. Numerical results show the efficiency of the proposed method in avoiding the homoclinic bifurcation of a classical Duffing system.