Global recurrences of multi-time scaled systems

Previous Article
Variational inequality for the Stokes equations with constraint
PROC Home
This Issue
Next Article
Ellipticity of quantum mechanical Hamiltonians in the edge algebra

2011, 2011(Special): 430-436. doi: 10.3934/proc.2011.2011.430

Global recurrences of multi-time scaled systems

Jean-Pierre Francoise ^1, and Claude Piquet ^1,

1.	Université P.-M. Curie, Paris 6, Laboratoire Jacques-Louis Lions, 4 Pl. Jussieu, Paris 75252, France, France

Received July 2010 Revised April 2011 Published October 2011

Abstract
Related Papers
Under a Creative Commons license

We develop here the rst steps of a long-term investigation on multi-time scaled systems with some spatial heterogeneity. They can be characterized by a slow-variation of \actions" interspaced by local fast variations of \angles". The principle of construction of these systems is presented on a rst example where the actions are periodic solutions of a planar Hamiltonian system. Once the fast perturbation of the angles is added, the whole system displays kind of bursting oscillations (characterized by the alternate of quiescent phases interspaced by fast oscillations) although it is completely integrable. In this rst example, the full analysis of the underlying recurrence is possible. We then discuss a second example which has been discovered by Rossler in the 70s and which inspired this study. This example looks paradigmatic of the complex global recurrences these systems may have.

Keywords: axiom A attractors, bursting oscillations, Recurrences, fast-slow systems.

Mathematics Subject Classification: Primary: 34C05, 34A34; Secondary: 34C1.

Citation: Jean-Pierre Francoise, Claude Piquet. Global recurrences of multi-time scaled systems. Conference Publications, 2011, 2011 (Special) : 430-436. doi: 10.3934/proc.2011.2011.430

[1]	Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123
[2]	Chunhua Shan. Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021097
[3]	Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257
[4]	Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[5]	Ilya Schurov. Duck farming on the two-torus: Multiple canard cycles in generic slow-fast systems. Conference Publications, 2011, 2011 (Special) : 1289-1298. doi: 10.3934/proc.2011.2011.1289
[6]	Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621
[7]	Enoch H. Apaza, Regis Soares. Axiom a systems without sinks and sources on $n$-manifolds. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 393-401. doi: 10.3934/dcds.2008.21.393
[8]	C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603
[9]	Zhuoqin Yang, Tingting Guan. Bifurcation analysis of complex bursting induced by two different time-scale slow variables. Conference Publications, 2011, 2011 (Special) : 1440-1447. doi: 10.3934/proc.2011.2011.1440
[10]	Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3967-4002. doi: 10.3934/dcdsb.2017204
[11]	Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233
[12]	Seung-Yeal Ha, Dohyun Kim, Jinyeong Park. Fast and slow velocity alignments in a Cucker-Smale ensemble with adaptive couplings. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4621-4654. doi: 10.3934/cpaa.2020209
[13]	Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021
[14]	Renato Huzak. Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 171-215. doi: 10.3934/dcds.2016.36.171
[15]	Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069
[16]	Alexander Krasnosel'skii. Resonant forced oscillations in systems with periodic nonlinearities. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 239-254. doi: 10.3934/dcds.2013.33.239
[17]	Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure & Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219
[18]	Qunying Zhang, Zhigui Lin. Blowup, global fast and slow solutions to a parabolic system with double fronts free boundary. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 429-444. doi: 10.3934/dcdsb.2012.17.429
[19]	Renato Huzak, P. De Maesschalck, Freddy Dumortier. Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2641-2673. doi: 10.3934/cpaa.2014.13.2641
[20]	Langhao Zhou, Liangwei Wang, Chunhua Jin. Global solvability to a singular chemotaxis-consumption model with fast and slow diffusion and logistic source. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021122

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Jean-Pierre Francoise Claude Piquet
on Google Scholar
Jean-Pierre Francoise Claude Piquet

[Back to Top]