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We are still far from a comprehensive theory of bifurcation in dynamical systems with
multiple time scales. However, systematic application of geometric methods and study
of examples have produced d...
Bifurcations of Relaxation Oscillations near Folded Saddles
Folded Saddles Relaxation Oscillations
2015/8/25
Relaxation oscillations are periodic orbits of multiple time scale dynamical systems
that contain both slow and fast segments. The slow-fast decomposition of these orbits is
de ned in the singular l...
A Geometric Model for Mixed-Mode Oscillations in a Chemical System
mixed-mode oscillations bifurcations kneading theory
2015/8/25
This paper presents a detailed analysis of mixed-mode oscillations in the “autocatalator,” a threedimensional,
two time scale vector field that is a chemical reactor model satisfying the law of mass
...
Infinite-Period Phase Transition versus Nucleation in a Stochastic Model of Collective Oscillations
Infinite-Period Phase Transition versus Nucleation Stochastic Model Collective Oscillations
2011/7/7
A lattice model of three-state stochastic phase-coupled oscillators has been shown by Wood et al. [Phys. Rev. Lett. 96, 145701 (2006)] to exhibit a phase transition at a critical value of the coupling...
Global return maps for mixed-mode oscillations with one fast and two slow variables
Dynamical Systems (math.DS) Classical Analysis and ODEs (math.CA) Chaotic Dynamics (nlin.CD)
2010/11/10
Alternating patterns of small and large amplitude oscillations occur in a wide variety of physical, chemical, biological and engineering systems. These mixed-mode oscillations (MMOs) are often found i...
Mixed-Mode Oscillations in a Stochastic, Piecewise-Linear System
Dynamical Systems (math.DS) Chaotic Dynamics (nlin.CD)
2010/11/10
We analyze a piecewise-linear FitzHugh-Nagumo model. The system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist. The addition of small noise induces mixed-m...
Characterizing mixed mode oscillations shaped by noise and bifurcation structure
mixed mode oscillations underlying dynamical structure extrinsic noise
2010/4/2
Many neuronal systems and models display a certain class of mixed mode oscillations (MMOs) consisting of periods of small amplitude oscillations interspersed with spikes. Various models with different...
Relaxation oscillations and negative strain rate sensitivity in the Portevin - Le Chatelier effect
Relaxation oscillations strain rate Portevin - Le Chatelier effect
2010/11/2
A characteristic feature of the Portevin - Le Chatelier effect or the jerky flow is the stick-slip nature of stress-strain curves which is believed to result from the negative strain rate dependence ...