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

This paper proposes a homoclinic bifurcation control method in a planar system of nonlinear differential equations (x ˙ = f (x) , x ∈ R 2 , f : U ⊂ R 2 ⟶ R 2). The feedback control law is formulated within the framework of Melnikov theory and L p spaces, and it will be called as L p control. Here it is proved that if γ 0 (t) is the homoclinic orbit of the planar system, then f (γ 0 (t)) ∈ L q 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) ∈ L p space) have been developed, where each of them can be found by choosing one particular L q space to f (γ 0 (t)) , such that p and q are conjugate exponents, that is, 1 p + 1 q = 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. [ABSTRACT FROM AUTHOR]