226 research outputs found

    The Turing bifurcation in network systems: Collective patterns and single differentiated nodes

    Get PDF
    We study the emergence of patterns in a diffusively coupled network that undergoes a Turing instability. Our main focus is the emergence of stable solutions with a single differentiated node in systems with large and possibly irregular network topology. Based on a mean-field approach, we study the bifurcations of such solutions for varying system parameters and varying degree of the differentiated node. Such solutions appear typically before the onset of Turing instability and provide the basis for the complex scenario of multistability and hysteresis that can be observed in such systems. Moreover, we discuss the appearance of stable collective patterns and present a codimension-two bifurcation that organizes the interplay between collective patterns and patterns with single differentiated nodes

    A sequence of order relations, encoding heteroclinic connections in scalar parabolic PDE

    Get PDF
    We address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations ut = uxx + ƒ (x, u, ux), 0 < x < 1 on a bounded interval with Neumann conditions. Introducing a sequence of order relations, we prove a new and simple criterion for the existence of heteroclinic connections, using only information about nodal properties of solutions to the stationary ODE problem. This result allows also for a complete classiffication of possible attractors in terms of the permutation of the equilibria, given by their order at the two boundaries of the interval

    Oscillatory instability in systems with delay

    Get PDF
    Any biological or physical system, which incorporates delayed feedback or delayed coupling, can be modeled by a dynamical system with delayed argument. We describe a standard oscillatory destabilization mechanism, which occurs in such systems

    Bifurcations in the Sakaguchi--Kuramoto model

    Get PDF
    We analyze the Sakaguchi-Kuramoto model of coupled phase oscillators in a continuum limit given by a frequency dependent version of the Ott-Antonsen system. Based on a self-consistency equation, we provide a detailed analysis of partially synchronized states, their bifurcation from the completely incoherent state and their stability properties. We use this method to analyze the bifurcations for various types of frequency distributions and explain the appearance of non-universal synchronization transitions

    Slow motion of quasi-stationary multi-pulse solutions by semistrong interaction in reaction-diffusion systems

    Get PDF
    In this paper, we study a class of singularly perturbed reaction-diffusion systems, which exhibit under certain conditions slowly varying multi-pulse solutions. This class contains among others the Gray-Scott and several versions of the Gierer-Meinhardt model. We first use a classical singular perturbation approach for the stationary problem and determine in this way a manifold of quasi-stationary NN-pulse solutions. Then, in the context of the time-dependent problem, we derive an equation for the leading order approximation of the slow motion along this manifold. We apply this technique to study 1-pulse and 2-pulse solutions for classical and modified Gierer-Meinhardt system. In particular, we are able to treat different types of boundary conditions, calculate folds of the slow manifold, leading to slow-fast motion, and to identify symmetry breaking singularities in the manifold of 2-pulse solutions

    Describing a class of global attractors via symbol sequences

    Get PDF
    We study a singularly perturbed scalar reaction-diffusion equation on a bounded interval with a spatially inhomogeneous bistable nonlinearity. For certain nonlinearities, which are piecewise constant in space on 푘 subintervals, it is possible to characterize all stationary solutions for small ε by means of sequences of 푘 symbols, indicating the behavior of the solution in each subinterval. Determining also Morse-indices and zero numbers of the equilibria in terms of the symbol sequences, we are able to give a criterion for heteroclinic connections and a description of the associated global attractor for all 푘

    Mode-locking in systems of globally coupled phase oscillators

    Get PDF
    We investigate the dynamics of a Kuramoto-type system of globally coupled phase oscillators with equidistant natural frequencies and a coupling strength below the synchronization threshold. It turns out that in such cases one can observe a stable regime of sharp pulses in the mean field amplitude with a pulsation frequency given by spacing of the natural frequencies. This resembles a process known as mode-locking in laser and relies on the emergence of a phase relation induced by the nonlinear coupling. We discuss the role of the first and second harmonic in the phase-interaction function for the stability of the pulsations and present various bifurcating dynamical regimes such as periodically and chaotically modulated mode-locking, transitions to phase turbulence and intermittency. Moreover, we study the role of the system size and show that in certain cases one can observe type-II supertransients, where the system reaches the globally stable mode-locking solution only after an exponentially long transient of phase turbulence
    corecore