Journal of Nonlinear Science
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Published By Springer-Verlag

1432-1467, 0938-8974

2022 ◽  
Vol 32 (2) ◽  
Author(s):  
O. E. Omel’chenko

AbstractAbout two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence–incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott–Antonsen reduction technique.


2022 ◽  
Vol 32 (2) ◽  
Author(s):  
Lili Fan ◽  
Hongjun Gao ◽  
Haochen Li

2022 ◽  
Vol 32 (2) ◽  
Author(s):  
Maximilian Engel ◽  
Christian Kuehn ◽  
Matteo Petrera ◽  
Yuri Suris

AbstractWe study the problem of preservation of maximal canards for time discretized fast–slow systems with canard fold points. In order to ensure such preservation, certain favorable structure-preserving properties of the discretization scheme are required. Conventional schemes do not possess such properties. We perform a detailed analysis for an unconventional discretization scheme due to Kahan. The analysis uses the blow-up method to deal with the loss of normal hyperbolicity at the canard point. We show that the structure-preserving properties of the Kahan discretization for quadratic vector fields imply a similar result as in continuous time, guaranteeing the occurrence of maximal canards between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem.


2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Martin Kružík ◽  
Paolo Maria Mariano ◽  
Domenico Mucci

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Yu-Jhe Huang ◽  
Hsuan Te Huang ◽  
Jonq Juang ◽  
Cheng-Han Wu
Keyword(s):  

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Hongyong Cui ◽  
Arthur C. Cunha ◽  
José A. Langa

AbstractFinite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in $$L^2$$ L 2 is established by the squeezing approach and that in $$H_0^1$$ H 0 1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.


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