arXiv subjects

Ansfried Janssens

Publications and source records attributed to Ansfried Janssens.

3 recordsLinked to original sources

arXiv2024

Fractal analysis of canard cycles and slow-fast Hopf points in piecewise smooth Liénard equations

The main goal of this paper is to give a complete fractal analysis of piecewise smooth (PWS) slow-fast Liénard equations. For the analysis, we use the notion of Minkowski dimension of one-dimensional orbits generated by slow relation functions. More precisely, we find all possible values for the Minkowski dimension near PWS slow-fast Hopf points and near bounded balanced crossing canard cycles. We study fractal properties of the unbounded canard cycles using PWS classical Liénard equations. We also show how the trivial Minkowski dimension implies the non-existence of limit cycles of crossing type close to Hopf points. This is not true for crossing limit cycles produced by bounded balanced canard cycles, i.e. we find a system undergoing a saddle-node bifurcation of crossing limit cycles and a system without limit cycles (in both cases, the Minkowski dimension is trivial). We also connect the Minkowski dimension with upper bounds for the number of limit cycles produced by bounded canard cycles.

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arXiv2023

Minkowski dimension and slow-fast polynomial Liénard equations near infinity

In planar slow-fast systems, fractal analysis of (bounded) sequences in $\mathbb R$ has proved important for detection of the first non-zero Lyapunov quantity in singular Hopf bifurcations, determination of the maximum number of limit cycles produced by slow-fast cycles, defined in the finite plane, etc. One uses the notion of Minkowski dimension of sequences generated by slow relation function. Following a similar approach, together with Poincaré--Lyapunov compactification, in this paper we focus on a fractal analysis near infinity of the slow-fast generalized Liénard equations $\dot x=y-\sum_{k=0}^{n+1} B_kx^k,\ \dot y=-ε\sum_{k=0}^{m}A_kx^k$. We extend the definition of the Minkowski dimension to unbounded sequences. This helps us better understand the fractal nature of slow-fast cycles that are detected inside the slow-fast Liénard equations and contain a part at infinity.

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arXiv2022

Fractal codimension of nilpotent contact points in two-dimensional slow-fast systems

In this paper we introduce the notion of fractal codimension of a nilpotent contact point $p$, for $λ=λ_0$, in smooth planar slow$-$fast systems $X_{ε,λ}$ when the contact order $n_{λ_0}(p)$ of $p$ is even, the singularity order $s_{λ_0}(p)$ of $p$ is odd and $p$ has finite slow divergence, i.e., $s_{λ_0}(p)\leq 2(n_{λ_0}(p)-1)$. The fractal codimension of $p$ is a generalization of the traditional codimension of a slow-fast Hopf point of Liénard type, introduced in (Dumortier and Roussarie (2009)), and it is intrinsically defined, i.e., it can be directly computed without the need to first bring the system into its normal form. The intrinsic nature of the notion of fractal codimension stems from the Minkowski dimension of fractal sequences of points, defined near $p$ using the so$-$called entry$-$exit relation, and slow divergence integral. We apply our method to a slow$-$fast Hopf point and read its degeneracy (i.e., the first nonzero Lyapunov quantity) as well as the number of limit cycles near such a Hopf point directly from its fractal codimension. We demonstrate our results numerically on some interesting examples by using a simple formula for computation of the fractal codimension. We demonstrate our results numerically on some interesting examples by using a simple formula for computation of the fractal codimension.

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