scholarly journals Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations

Entropy ◽  
2014 ◽  
Vol 16 (4) ◽  
pp. 1949-1968 ◽  
Author(s):  
John Taborda ◽  
Ivan Arango
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Piecewise Smooth ◽  
Topological Classification

Discontinuity-induced bifurcations of piecewise smooth dynamical systems

2010 ◽  
Vol 368 (1930) ◽  
pp. 4915-4935 ◽  
Author(s):  
M. di Bernardo ◽  
S. J. Hogan
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Limit Cycles ◽  
State Of The Art ◽  
Piecewise Smooth ◽  
Hybrid Dynamical Systems ◽  
Current State ◽  
Relevant Class

This paper presents an overview of the current state of the art in the analysis of discontinuity-induced bifurcations (DIBs) of piecewise smooth dynamical systems, a particularly relevant class of hybrid dynamical systems. Firstly, we present a classification of the most common types of DIBs involving non-trivial interactions of fixed points and equilibria of maps and flows with the manifolds in phase space where the system is non-smooth. We then analyse the case of limit cycles interacting with such manifolds, presenting grazing and sliding bifurcations. A description of possible classification strategies to predict and analyse the scenarios following such bifurcations is also discussed, with particular attention to those methodologies that can be applied to generic n -dimensional systems.

scholarly journals Dynamics of Bose-Einstein Condensates: Exact Representation and Topological Classification of Coherent Matter Waves

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Leilei Jia ◽  
Qihuai Liu ◽  
Shengqiang Tang
Keyword(s):  
Angular Momentum ◽  
Dynamical Systems ◽  
Bifurcation Theory ◽  
Topological Classification ◽  
Exact Representation ◽  
Matter Waves ◽  
Existence And Multiplicity ◽  
Bose Einstein ◽  
The Waves

By using the bifurcation theory of dynamical systems, we present the exact representation and topological classification of coherent matter waves in Bose-Einstein condensates (BECs), such as solitary waves and modulate amplitude waves (MAWs). The existence and multiplicity of such waves are determined by the parameter regions selected. The results show that the characteristic of coherent matter waves can be determined by the “angular momentum” in attractive BECs while for repulsive BECs; the waves of the coherent form are all MAWs. All exact explicit parametric representations of the above waves are exhibited and numerical simulations support the result.

Baire's Category Theoretic Classification of Compact Expansive Dynamical Systems

2002 ◽  
Vol 09 (04) ◽  
pp. 315-323 ◽  
Author(s):  
Kazutaka Sakai
Keyword(s):  
Dynamical Systems ◽  
Classification Method ◽  
Topological Classification ◽  
Finite Dimensional ◽  
Theoretic Classification

A revised version of the estimation inequality of Akashi [2] is given, and this result is applied to Baire's category theoretic classification of ∊-expansive dynamical systems. Moreover, this classification method is applied to topological classification of shift dynamical systems on finite-dimensional compact domains.

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 Wavefront dislocations reveal the topology of quasi-1D photonic insulators

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Clément Dutreix ◽  
Matthieu Bellec ◽  
Pierre Delplace ◽  
Fabrice Mortessagne
Keyword(s):  
Local Density ◽  
Topological Defects ◽  
Wave Functions ◽  
Topological Classification ◽  
Local Density Of States ◽  
Topological Phase Transition ◽  
Interference Fringes ◽  
Classical Wave ◽  
Phase Singularities

AbstractPhase singularities appear ubiquitously in wavefields, regardless of the wave equation. Such topological defects can lead to wavefront dislocations, as observed in a humongous number of classical wave experiments. Phase singularities of wave functions are also at the heart of the topological classification of the gapped phases of matter. Despite identical singular features, topological insulators and topological defects in waves remain two distinct fields. Realising 1D microwave insulators, we experimentally observe a wavefront dislocation – a 2D phase singularity – in the local density of states when the systems undergo a topological phase transition. We show theoretically that the change in the number of interference fringes at the transition reveals the topological index that characterises the band topology in the insulator.

Topological classification of nodal-line semimetals in square-net materials

2021 ◽  
Vol 103 (16) ◽  
Author(s):  
Inho Lee ◽  
S. I. Hyun ◽  
J. H. Shim
Keyword(s):  
Nodal Line ◽  
Topological Classification

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

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