scholarly journals Isostable reduction of oscillators with piecewise smooth dynamics and complex Floquet multipliers

2019 ◽  
Vol 99 (2) ◽  
Author(s):  
Dan Wilson
Keyword(s):  
Piecewise Smooth ◽  
Floquet Multipliers ◽  
Smooth Dynamics

The Stick-Slip Motion of a Chaplygin Sleigh With a Piecewise Smooth Nonholonomic Constraint

2015 ◽  
Author(s):  
Vitaliy Fedonyuk ◽  
Phanindra Tallapragada
Keyword(s):  
Numerical Simulations ◽  
Nonholonomic Constraint ◽  
Nonholonomic Constraints ◽  
Piecewise Smooth ◽  
Chaplygin Sleigh ◽  
Stick Slip ◽  
Stick Slip Motion ◽  
Smooth Dynamics ◽  
Slip Motion

The Chaplygin sleigh is a canonical problem of mechanical systems with nonholonomic constraints, which arises due to the role of friction. The motion of the cart has often been studied under the assumption that the magnitude of friction is as high as necessary to prevent slipping. We relax this assumption by setting a maximum finite value to the friction. The Chaplygin sleigh is then under a piecewise smooth nonholonomic constraint and transitions between ‘slip’ and ‘stick’ modes. We investigate these transitions and the resulting non smooth dynamics of the system. Further more the piecewise smooth constraint offers the possibility of controlling the speed of the sleigh and we investigate this with numerical simulations.

scholarly journals Singularly Perturbed Oscillators with Exponential Nonlinearities

2021 ◽  
Author(s):  
S. Jelbart ◽  
K. U. Kristiansen ◽  
P. Szmolyan ◽  
M. Wechselberger
Keyword(s):  
Limit Cycles ◽  
Space Dimension ◽  
Blow Up ◽  
Exponential Convergence ◽  
Model System ◽  
Piecewise Smooth ◽  
Singularly Perturbed ◽  
Model Systems ◽  
Smooth System ◽  
Unique Limit

AbstractSingular exponential nonlinearities of the form $$e^{h(x)\epsilon ^{-1}}$$ e h ( x ) ϵ - 1 with $$\epsilon >0$$ ϵ > 0 small occur in many different applications. These terms have essential singularities for $$\epsilon =0$$ ϵ = 0 leading to very different behaviour depending on the sign of h. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $$\epsilon \rightarrow 0$$ ϵ → 0 limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen (Nonlinearity 30(5): 2138–2184, 2017), and prove—for both systems—existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

scholarly journals The primacy of rhythm: how discrete actions merge into a stable rhythmic pattern

2019 ◽  
Vol 121 (2) ◽  
pp. 574-587 ◽  
Author(s):  
Zhaoran Zhang ◽  
Dagmar Sternad
Keyword(s):  
Building Blocks ◽  
Periodic Pattern ◽  
Voluntary Movements ◽  
Rhythmic Pattern ◽  
Rhythmic Movements ◽  
Discrete Movements ◽  
Floquet Multipliers ◽  
Physiological Processes ◽  
Rhythmic Behavior ◽  
Virtual Target

This study examined how humans spontaneously merge a sequence of discrete actions into a rhythmic pattern, even when periodicity is not required. Two experiments used a virtual throwing task, in which subjects performed a long sequence of discrete throwing movements, aiming to hit a virtual target. In experiment 1, subjects performed the task for 11 sessions. Although there was no instruction to perform rhythmically, the variability of the interthrow intervals decreased to a level comparable to that of synchronizing with a metronome; furthermore, dwell times shortened or even disappeared with practice. Floquet multipliers and decreasing variability of the arm trajectories estimated in state space indicated an increasing degree of dynamic stability. Subjects who achieved a higher level of periodicity and stability also displayed higher accuracy in the throwing task. To directly test whether rhythmicity affected performance, experiment 2 disrupted the evolving continuity and periodicity by enforcing a pause between successive throws. This discrete group performed significantly worse and with higher variability in their arm trajectories than the self-paced group. These findings are discussed in the context of previous neuroimaging results showing that rhythmic movements involve significantly fewer cortical and subcortical activations than discrete movements and therefore may pose a computationally more parsimonious solution. Such emerging stable rhythms in neuromotor subsystems may serve as building blocks or dynamic primitives for complex actions. The tendency for humans to spontaneously fall into a rhythm in voluntary movements is consistent with the ubiquity of rhythms at all levels of the physiological system. NEW & NOTEWORTHY When performing a series of throws to hit a target, humans spontaneously merged successive actions into a continuous approximately periodic pattern. The degree of rhythmicity and stability correlated with hitting accuracy. Enforcing irregular pauses between throws to disrupt the rhythm deteriorated performance. Stable rhythmic patterns may simplify control of movement and serve as dynamic primitives for more complex actions. This observation reveals that biological systems tend to exhibit rhythmic behavior consistent with a plethora of physiological processes.

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.

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