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Register a new userDiscretized Fast-Slow Systems near Transcritical Singularities
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We extend slow manifolds near a transcritical singularity in a fast-slow system given by the explicit Euler discretization of the corresponding continuous-time normal form. The analysis uses the blow-up method and direct trajectory-based estimates. We prove that the qualitative behaviour is preserved by a time-discretization with sufficiently small step size. This step size is fully quantified relative to the time scale separation. Our proof also yields the continuous-time results as a special case and provides more detailed calculations in the classical (or scaling) chart.
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Motivated by the normal form of a fast-slow ordinary differential equation exhibiting a pitchfork singularity we consider the discrete-time dynamical system that is obtained by an application of the explicit Euler method. Tracking trajectories in the vicinity of the singularity we show, how the slow manifold extends beyond the singular point and give an estimate on the contraction rate of a transition mapping. The proof relies on the blow-up method suitably adapted to the discrete setting where a key technical contribution are precise estimates for a cubic map in the central rescaling chart.
We study the problem of preservation of canard connections 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 imply a similar result as in continuous time, guaranteeing the occurrence of canard connections between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a non-canonical Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem.
Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the fast subsystem) and associated fast dynamics, parametrized by the slow variables. Using a distinguished parameter approach we are able to classify bifurcations for one fast and one slow variable. Some of these bifurcations are associated with the critical set losing manifold structure. We also conjecture a list of generic bifurcations of the critical set for one fast and two slow variables. We further consider how the bifurcations of the critical set can be associated with generic bifurcations of attracting relaxation oscillations under an appropriate singular notion of equivalence.
Although the spatially continuous version of the reaction-diffusion equation has been well studied, in some instances a spatially-discretized representation provides a more realistic approximation of biological processes. Indeed, mathematically the discretized and continuous systems can lead to different predictions of biological dynamics. It is well known in the continuous case that the incorporation of diffusion can cause diffusion-driven blow-up with respect to the $L^{infty}$ norm. However, this does not imply diffusion-driven blow-up will occur in the discretized version of the system. For example, in a continuous reaction-diffusion system with Dirichlet boundary conditions and nonnegative solutions, diffusion-driven blow up occurs even when the total species concentration is non-increasing. For systems that instead have homogeneous Neumann boundary conditions, it is currently unknown whether this deviation between the continuous and discretized system can occur. Therefore, it is worth examining the discretized system independently of the continuous system. Since no criteria exist for the boundedness of the discretized system, the focus of this paper is to determine sufficient conditions to guarantee the system with diffusion remains bounded for all time. We consider reaction-diffusion systems on a 1D domain with homogeneous Neumann boundary conditions and non-negative initial data and solutions. We define a Lyapunov-like function and show that its existence guarantees that the discretized reaction-diffusion system is bounded. These results are considered in the context of three example systems for which Lyapunov-like functions can and cannot be found.
We show that every (invertible, or noninvertible) minimal Cantor system embeds in $mathbb{R}$ with vanishing derivative everywhere. We also study relations between local shrinking and periodic points.
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