Related papers: Slow Manifold Structure and the Emergence of Mixed…

Following the paper of K. Shimizu et al. (2011) we consider the Bonhoeffer-van der Pol oscillator with non-autonomous periodic perturbation. We show that the presence of mixed mode oscillations reported in that paper can be explained using…

We show that peculiar collective dynamics called slow switching arises in a population of leaky integrate-and-fire oscillators with delayed, all-to-all pulse-couplings. By considering the stability of cluster states and symmetry possessed…

Model order reduction in high-dimensional, nonlinear dynamical systems if often enabled through fast-slow timescale separation. One such approach involves identifying a low-dimensional slow manifold to which the state rapidly converges and…

In this paper, we report the bifurcations of mode-locked periodic orbits occurring in maps of three or higher dimensions. The `torus' is represented by a closed loop in discrete time, which contains stable and unstable cycles of the same…

We study a slow-fast system with two slow and one fast variables. We assume that the slow manifold of the system possesses a fold and there is an equilibrium of the system in a small neighbourhood of the fold. We derive a normal form for…

We consider the relation for the stochastic equilibrium states between the reduced system on a random slow manifold and the original system. This provides a theoretical basis for the reduction about sophisti- cated detailed models by the…

The phenomenon of slow switching in populations of globally coupled oscillators is discussed. This characteristic collective dynamics, which was first discovered in a particular class of the phase oscillator model, is a result of the…

This article provides an example of fast-slow system such that most orbits remain as close as possible to the unstable manifold of the fast dynamics for an arbitrarily long time.

We study a three-dimensional dynamical system in two slow variables and one fast variable. We analyze the tangency of the unstable manifold of an equilibrium point with "the" repelling slow manifold, in the presence of a stable periodic…

Mutual inhibition is a common motif in neural systems. Here, we establish that cusped singularities - folded singularities located at cusp points of critical manifolds - provide a universal organizing mechanism for mixed-mode oscillations…

We present a set of phase-space portraits illustrating the extraordinary oscillatory possibilities of the dynamical systems through the so-called generalized Landau scenario. In its simplest form the scenario develops in N dimensions around…

Slow-fast dynamical systems, i.e., singularly or non-singularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating…

We derive a mode-coupling theory for the slow dynamics of fluids confined in disordered porous media represented by spherical particles randomly placed in space. Its equations display the usual nonlinear structure met in this theoretical…

This article deals with invariant manifolds for infinite dimensional random dynamical systems with different time scales. Such a random system is generated by a coupled system of fast-slow stochastic evolutionary equations. Under suitable…

We investigate the emergence of complex dynamics in a system of coupled dissipative kicked rotors and show that critical transitions can be understood via bifurcations of simple states. We study multistability and bifurcations in the single…

Mixed-mode oscillations (MMOs) are complex oscillatory patterns in which large-amplitude relaxation oscillations (LAOs) alternate with small-amplitude oscillations (SAOs). MMOs are found in singularly perturbed systems of ordinary…

We revisit the classical Suarez-Schopf delayed oscillator. Special attention is paid to the region of linear stability in the space of parameters. By means of the theory of inertial manifolds developed in our adjacent papers, we provide…

The dynamics of a model, originally proposed for a type of instability in plastic flow, has been investigated in detail. The bifurcation portrait of the system in two physically relevant parameters exhibits a rich variety of dynamical…

The instability of mixing in the Kuramoto model of coupled phase oscillators is the key to understanding a range of spatiotemporal patterns, which feature prominently in collective dynamics of systems ranging from neuronal networks, to…

Systems such as fluid flows in channels and pipes or the complex Ginzburg-Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial…