相关论文: On the approximation of the canard explosion point…
The canard explosion is the change of amplitude and period of a limit cycle born in a Hopf bifurcation in a very narrow parameter interval. The phenomenon is well understood in singular perturbation problems where a small parameter controls…
We analyze canard explosions in delayed differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow-fast system with delayed self-coupling. In the absence of delays,…
Classical canard explosion results in smooth systems require the vector field to be at least $C^3$, since canard cycles are created as the result of a Hopf bifurcation. The work on canards in nonsmooth, planar systems is recent and has thus…
In this paper we analyse the phenomenon of the slow passage through a transcritical bifurcation with special emphasis in the maximal delay $z_d(\lambda,\varepsilon)$ as a function of the bifurcation parameter $\lambda$ and the singular…
A sudden transition to a state of high amplitude limit cycle oscillations is catastrophic in a thermo-fluid system. Conventionally, upon varying the control parameter, a sudden transition is observed as an abrupt jump in the amplitude of…
In multiple time-scale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose…
Canard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singularly perturbed Ordinary Differential Equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically,…
We show that a nonlinear, piecewise-smooth, planar dynamical system can exhibit canard phenomena. Canard solutions and explosion in nonlinear, piecewise-smooth systems can be qualitatively more similar to the phenomena in smooth systems…
Hopf bifurcations in fast-slow systems of ordinary differential equations can be associated with surprising rapid growth of periodic orbits. This process is referred to as canard explosion. The key step in locating a canard explosion is to…
Geometrical Singular Perturbation Theory has been successful to investigate a broad range of biological problems with different time scales. The aim of this paper is to apply this theory to a predator-prey model of modified Leslie-Gower…
The aim of this work is to establish that the bifurcation parameter value leading to a canard explosion in dimension two obtained by the so-called Geometric Singular Perturbation Method can be found according to the Flow Curvature Method.…
This paper presents a new parameter estimation algorithm for the adaptive control of a class of time-varying plants. The main feature of this algorithm is a matrix of time-varying learning rates, which enables parameter estimation error…
In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in $\mathbb R^3$ through the folded saddle-node of type II and the singular Hopf bifurcation. In particular, we prove -- in the…
The main purpose of this paper is to study limit cycles in non-linear regularizations of planar piecewise smooth systems with fold points (or more degenerate tangency points) and crossing regions. We deal with a slow fast Hopf point after…
Torus canards are solutions of slow/fast systems that alternate between attracting and repelling manifolds of limit cycles of the fast subsystem. A relatively new dynamic phenomenon, torus canards have been found in neural applications to…
In this article, we study a system of reaction-diffusion equations in which the diffusivities are widely separated. We report on the discovery of families of spatially periodic canard solutions that emerge from {\em singular Turing…
This paper deals with the asymptotic study of the so-called canard solutions, which arise in the study of real singularly perturbed ODEs. Starting near an attracting branch of the "slow curve", those solutions are crossing a turning point…
Generic slow-fast systems with only one (time-scaling) parameter on the two-torus have attracting canard cycles for arbitrary small values of this parameter. This is in drastic contrast with the planar case, where canards usually occur in…
Recently, research on the complex periodic behavior of multi-scale systems has become increasingly popular. Krupa et al. \cite{krupa2} provided a way to obtain relaxation oscillations in slow-fast systems through singular Hopf bifurcations…
We consider double canard cycles including two canards in singularly perturbed planar systems with two canard points. Previous work studied the complex oscillations including relaxation oscillations and canard cycles in singularly perturbed…