In complex dynamics, we construct a so-called nice set (one for which the first return map is Markov) around any point which is in the Julia set but not in the post-singular set. This simplifies the study of absolutely continuous invariant measures. We prove a converse to a recent theorem of Kotus and Świa̧tek, so for a certain class of meromorphic maps the absolutely continuous invariant measur...

This paper is devoted to a stability study of a walking gait for a biped. The walking gait is periodic and it is composed of a single-support phase, a passive impact, and a double-support phase. The reference trajectories are described as a function of the shin orientation versus the ground of the stance leg. We use the Poincaré map to study the stability of the walking gait of the biped. We on...

We propose a model-based reinforcement learning algorithm for biped walking in which the robot learns to appropriately modulate an observed walking pattern. Viapoints are detected from the observed walking trajectories using the minimum jerk criterion. The learning algorithm modulates the via-points as control actions to improve walking trajectories. This decision is based on a learned model of...

A proof of existence of stationary dark soliton solutions of the cubic-quintic nonlinear Schrödinger equation with a periodic potential is given. It is based on the interpretation of the dark soliton as a heteroclinic on the Poincaré map.

The discreet mathematical models are gotten directly via scientific experiences, or by the use of the Poincaré section for the study of a continuous model. One of these models is the Henon map. Many papers have described chaotic systems, one of the most famous being a two-dimensional discrete map which models the original Henon map [3, 4, 5, 7, 8]. Moreover, it is possible to change the form of...

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the wellknown notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using ε-chains, and the minimal lengths of these ε-chains give a way to measure recurrence time (...

We show that continuous-time chaos can be defined using linear dynamics and represented by an exact analytic solution. A driven linear differential equation is used to define a low-dimensional chaotic set of continuous-time waveforms. A nonlinear differential equation is derived for which these waveforms are exact analytic solutions. This nonlinear system describes a chaotic semiflow with a ret...

Abstract In this work, the Poincaré map numerical method was successfully developed to solve fourth-order differential equation that describes flexural vibrations of a beam, within Timoshenko beam theory. The Euler-Bernoulli continuity conditions were considered, which are valid for frequencies smaller than critical frequency. As an example, used design complex elastic structure, characterized ...