New bifurcations in the simplest passive walking model

Abstract

This paper uncovers several new stable periodic gaits in the simplest passive walking bipedal model proposed in the literature. It is demonstrated that the model has period-3 to period-7 gaits beside the period-1 gaits found by Garcia et al. By simulations, this paper shows that each of these new gaits leads to chaos via period-doubling bifurcation and loses its stability by cyclic-fold bifurcation. This interesting phenomenon suggests a series of new bifurcation scenarios that have not been observed before. To confirm the new gaits and their bifurcations, this paper presents computer assisted proofs on the existence and stability of each periodic gait and its period-doubling gaits with the interval Newton method. To verify that the routes indeed lead to chaos, computer-assisted proofs are also given by means of topological horseshoes theory.

Dynamical Analysis of the Lorenz-84 Atmospheric Circulation Model

“…Theorem 11 (see [34][35][36] Here, the -shift is also called the Bernoulli -shif. The symbolic series space ∑ is compact, totally disconnected, and perfect.…”

“…As a basic and striking result in chaotic dynamics, topological horseshoe provides a powerful tool in the rigorous study of chaos and dynamical systems obtaining the topological entropy, verifying the existence of chaos, showing the structure of chaotic attractors, revealing the mechanism inside chaotic phenomena, and so on. Some definitions and properties about the topological horseshoe [33][34][35][36] are given at first.…”

“…The topological horseshoe is well recognized as one of the most rigorous approaches to study chaos with computer. It has been successfully applied in many chaotic systems and hyperchaotic systems [32][33][34][35][36][37][38][39][40]. As a basic and striking result in chaotic dynamics, topological horseshoe provides a powerful tool in the rigorous study of chaos and dynamical systems obtaining the topological entropy, verifying the existence of chaos, showing the structure of chaotic attractors, revealing the mechanism inside chaotic phenomena, and so on.…”

“…Definition 10 (see [33,36] → holds true for every connection of 1 and 2 , then ( ) is said to be acrossing and is denoted by ( ) ⊂ with respect to two pairs, ( 1 , 2 ) and ( 1 , 2 ).…”

“…Definition 9 (see [33,36]). For each , 1 ≤ ≤ , let 1 and 2 be two fixed disjoint compact subsets of contained in the boundary .…”