Hyperbolicity and averaging for the Srzednicki–Wójcik equation

Abstract

For the Srzednicki-Wójcik equation, the planar nonautonomous ODE parameterized by κ ∈ R,using averaging we show how the region of hyperbolicty grows with |κ|.Based on this we give bounds on the sizes of bounded orbits.

Averaging Methods for Second-Order Differential Equations and Their Application for Impact Systems

“…Now we prove the second estimate in (1). Observe that the derivativeż t is bounded; its bound depends only on the initial value b and is independent of and t. This follows easily from an equation similar to (14). Hence the estimate (12) implies the uniform Lipschitz continuity of x .…”

“…The third estimate in (1) can be proven similarly to the second one. First, one has to prove the uniform Lipschitz continuity of z by using an equation similar to (14). Then the desired estimate follows from (12) and the first inequality in (1).…”

“…It has broad ranges of approaches, as is demonstrated in recent works [5,6]. An averaging method, as a part of those approaches, has many applications and it is well-developed for ordinary, partial, delay and stochastic differential equations; see, e.g., [3,4,[7][8][9][10][11][12][13][14] for more motivations and applications. One can find another applications in the field of non-smooth dynamical systems; see, e.g., [15].…”