scholarly journals On the construction of smooth ergodic skew-products

1988 ◽  
Vol 8 (2) ◽  
pp. 311-326 ◽  
Author(s):  
Mahesh G. Nerurkar
Keyword(s):  
Dynamical System ◽  
Coefficient Matrix ◽  
Small Divisor ◽  
Almost Periodic ◽  
Qualitative Behaviour ◽  
Skew Products ◽  
Ergodic Properties ◽  
Periodic Approximation ◽  
Linear Differential ◽  
Smooth Dynamical System

AbstractIn this paper we prove results about lifting dynamical and ergodic properties of a given smooth dynamical system to its skew-product extensions by smooth cocycles. The classical small divisor argument shows that in general such results are not possible. However, using the notion of the ‘fast periodic approximation’ introduced by A. Katok, we will show that if the dynamical system admits such a ‘fast periodic approximation’ then indeed a certain qualitative behaviour which is prohibited by small divisor type conditions is now in fact generic. The techniques are also applied to show that ‘recurrent-proximal’ behaviour of solutions of linear differential equations with almost periodic coefficients is generic under suitable conditions on the coefficient matrix.

scholarly journals (ω, c)- Pseudo almost periodic distributions

2020 ◽  
Vol 7 (1) ◽  
pp. 237-248 ◽  
Author(s):  
Mohammed Taha Khalladi ◽  
Abdelkader Rahmani
Keyword(s):  
Almost Periodicity ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Differential Systems ◽  
Schwartz Distributions ◽  
Pseudo Almost Periodic Solutions ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
Pseudo Almost Periodicity ◽  
Pseudo Almost Periodic

AbstractThe paper is a study of the (w, c) −pseudo almost periodicity in the setting of Sobolev-Schwartz distributions. We introduce the space of (w, c) −pseudo almost periodic distributions and give their principal properties. Some results about the existence of distributional (w, c) −pseudo almost periodic solutions of linear differential systems are proposed.

scholarly journals The vibro-impact capsule system in millimetre scale: numerical optimisation and experimental verification

Meccanica ◽  
2020 ◽  
Vol 55 (10) ◽  
pp. 1885-1902
Author(s):  
Yang Liu ◽  
Joseph Páez Chávez ◽  
Jiajia Zhang ◽  
Jiyuan Tian ◽  
Bingyong Guo ◽  
...  
Keyword(s):  
Dynamical System ◽  
Mathematical Formulation ◽  
Path Following ◽  
Piecewise Smooth ◽  
Experimental Results ◽  
Scale Down ◽  
Speed Up ◽  
Small Bowel Endoscopy ◽  
Numerical Optimisation ◽  
Smooth Dynamical System

Abstract The vibro-impact capsule system has been studied extensively in the past decade because of its research challenges as a piecewise-smooth dynamical system and broad applications in engineering and healthcare technologies. This paper reports our team’s first attempt to scale down the prototype of the vibro-impact capsule to millimetre size, which is 26 mm in length and 11 mm in diameter, aiming for small-bowel endoscopy. Firstly, an existing mathematical model of the prototype and its mathematical formulation as a piecewise-smooth dynamical system are reviewed in order to carry out numerical optimisation for the prototype by means of path-following techniques. Our numerical analysis shows that the prototype can achieve a high progression speed up to 14.4 mm/s while avoiding the collision between the inner mass and the capsule which could lead to less propulsive force on the capsule so causing less discomfort on the patient. Secondly, the experimental rig and procedure for testing the prototype are introduced, and some preliminary experimental results are presented. Finally, experimental results are compared with the numerical results to validate the optimisation as well as the feasibility of the vibro-impact technique for the potential of a controllable endoscopic procedure.

scholarly journals On choosing state variables for piecewise-smooth dynamical system simulations

2018 ◽  
Vol 95 (2) ◽  
pp. 1165-1188 ◽  
Author(s):  
Jin-Song Pei ◽  
Joseph P. Wright ◽  
François Gay-Balmaz ◽  
James L. Beck ◽  
Michael D. Todd
Keyword(s):  
Dynamical System ◽  
Piecewise Smooth ◽  
State Variables ◽  
Smooth Dynamical System

PERIOD-DOUBLING SCENARIO WITHOUT FLIP BIFURCATIONS IN A ONE-DIMENSIONAL MAP

2005 ◽  
Vol 15 (04) ◽  
pp. 1267-1284 ◽  
Author(s):  
V. AVRUTIN ◽  
M. SCHANZ
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Symmetry Breaking ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Period Doubling Bifurcation ◽  
Coexisting Attractors ◽  
One Dimensional ◽  
Poincaré Return Map ◽  
Smooth Dynamical System

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

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