arXiv subjects

Renato Huzak

Publications and source records attributed to Renato Huzak.

17 recordsLinked to original sources

arXiv2026

Cyclicity of sliding cycles in regularizations of piecewise linear two-folds

We study limit cycles produced by Sotomayor-Teixeira regularizations of piecewise linear vector fields with generic two-fold singularities. We focus on the visible-visible and visible-invisible cases where sliding cycles occur, treating cycles from both sides of the switching manifold in a unified way. In further details, by relating the cyclicity of sliding cycles and zeros of the slow divergence integral, we prove that the cyclicity of compact families of sliding cycles is bounded by two when such integral does not vanish identically. In contrast to previous works, we do not perform a case-by-base study, but instead relate zeros of slow divergence integrals to crossing limit cycles of a suitably defined auxiliary piecewise linear system. For visible folds, we provide necessary and sufficient conditions that assure the existence of a unique simple zero of the slow divergence integral, which implies the existence of two limit cycles. We also show that, when a hyperbolic singularity of the sliding vector field lies at the boundary of the sliding segment, then the cyclicity is bounded by one.

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arXiv2026

An unbounded number of canard limit cycles in linear regularizations of piecewise linear systems

The purpose of this paper is to study the number of limit cycles of canard type in linear regularizations of piecewise linear systems with non-monotonic transition functions. Using the notion of slow divergence integral and elementary breaking mechanisms, we construct systems with an arbitrary finite number of hyperbolic limit cycles. The Hopf breaking mechanism deals with transition functions with precisely one critical point in the interval $(-1,1)$. On the other hand, the jump breaking mechanism produces any number of limit cycles using transition functions with precisely three critical points in $(-1,1)$.

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arXiv2025

On entry-exit formulas for degenerate turning point problems in planar slow-fast systems

In this paper, we study degenerate entry-exit problems associated with planar slow-fast systems having an invariant line $\{(x,y)\,:\,y=0\}$ with a turning point at $x=0$. The degeneracy stems from the fact that the slow flow has a saddle-node of even order $2n$, $n\in \mathbb N$, at the turning point, i.e. $x' = -x^{2n}(1+o(1))$ for $ε=0$. We are motivated by the appearance of such turning point problems (for $n=1$) in the graphics $(I_2^1)$ and $(I_4^1)$, through a nilpotent saddle-node singularity at infinity, in the Dumortier-Roussarie-Rousseau program (for solving the finiteness part of Hilbert's 16th problem for quadratic polynomial systems). Our results show, under additional hypothesis, that in the case $n=1$ there is a well-defined entry-exit relation for $ε\rightarrow 0$. The associated Dulac map is smooth w.r.t. $(ε,ε\log ε^{-1})$. On the other hand for the cases $n\ge 2$, we show that the entry-exit relation requires additional control parameters. Our approach follows the one used by De Maesschalck, P. and Schecter, S. (JDE 2016) for a different type of degenerate entry-exit problem. In particular, we apply blow-up {after} having first performed a singular coordinate transformation of $y$. The degeneracy at $x=0$ requires an additional blow-up. We finally apply the result for $n=1$ to a normal form for the unfolding of the \NEW{relevant} graphics in the Dumortier-Roussarie-Rousseau program. Here we also demonstrate that the singular transformation of $y$ due to De Maesschalck, P. and Schecter, S. (JDE 2016) has practical significance in numerical computations.

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arXiv2025

Fractal analysis of slow-fast and regular systems: A survey of recent results and future perspectives

We survey recent developments in fractal analysis of regular and slow-fast dynamical systems using Minkowski dimension. Our focus is on spiral trajectories near monodromic limit periodic sets in regular systems and entry-exit sequences in slow-fast systems with degenerate singularities. For regular systems, we recall connections between Minkowski dimension and cyclicity. Key results include fractal classifications of weak foci, degenerate foci, and polycycles, where dimensional relationships predict limit cycle birth. For slow-fast systems, we survey the coordinate-free fractal methodology analyzing slow-fast Hopf points and canard cycles through slow divergence integrals and entry-exit sequences. The Minkowski dimension takes discrete values yielding upper bounds for the number of limit cycles without normal form transformations. The fractal approach provides computational advantages, works with original coordinates, and reveals geometric structures underlying bifurcation phenomena. Applications span neuroscience, chemistry, population dynamics, and climate modeling. We also discuss extensions to piecewise smooth and three-dimensional systems.

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arXiv2025

Canard cycles of non-linearly regularized piecewise smooth vector fields

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 non-linear regularization and blow-up. We give a simple criterion for upper bounds and the existence of limit cycles of canard type, expressed in terms of zeros of the slow divergence integral. Using the criterion we can construct a quadratic regularization of piecewise linear center such that for any integer $k>0$ it has at least $k+1$ limit cycles, for a suitably chosen monotonic transition function $φ_k:\mathbb{R}\rightarrow\mathbb{R}$. We prove a similar result for regularized invisible-invisible fold-fold singularities of type II$_2$. Canard cycles of dodging layer are also considered, and we prove that such limit cycles undergo a saddle-node bifurcation.

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arXiv2025

Cyclicity of sliding cycles with singularities of regularized piecewise smooth visible-invisible two-folds

In this paper we study the cyclicity of sliding cycles for regularized piecewise smooth visible-invisible two-folds, in the presence of singularities of the Filippov sliding vector field located away from two-folds. We obtain a slow-fast system after cylindrical blow-up and use a well-known connection between the divergence integral along orbits and transition maps for vector fields. Since properties of the divergence integral depend on the location and multiplicity of singularities, we divide the sliding cycles into different classes, which can then produce different types of cyclicity results. As an example, we apply our results to regularized piecewise linear systems.

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arXiv2024

Reading multiplicity in unfoldings from epsilon-neighborhoods of orbits

We consider generic 1-parameter unfoldings of parabolic vector fields. It is known that the box dimension of orbits of their time-one maps is discontinuous at the bifurcation value. Here, we expand asymptotically the Lebesgue measure of the epsilon-neighborhoods of orbits of the time-one maps in a Chebyshev scale, uniformly with respect to the bifurcation parameter. We use the so-called Ecalle-Roussarie-type compensators. We read from the expansion the number of hyperbolic points born in the unfolding of the parabolic point (i.e. the codimension of the bifurcation). We consider generic analytic 1-parameter unfoldings of saddle-node germs of analytic vector fields on the real line, their time-one maps and the Lebesgue measure of $\varepsilon$-neighborhoods of the orbits of these time-one maps.The box dimension of an orbit gives the asymptotics of the principal term of this Lebesgue measure and it is known that it is discontinuous at bifurcation parameters. In order to recover continuous dependence of the asymptotics on the parameter, here we expand asymptotically the Lebesgue measure of $\varepsilon$-neighborhoods of orbits of time-one maps in a Chebyshev system, uniformly with respect to the bifurcation parameter. We use Écalle-Roussarie-type compensators. We show how the number of fixed points of the time-one map born in the universal analytic unfolding of the parabolic point corresponds to the number of terms vanishing in this uniform expansion of the Lebesgue measure of $\varepsilon$-neighborhoods of orbits.

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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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arXiv2024

Ergodicity in planar slow-fast systems through slow relation functions

In this paper, we study ergodic properties of the slow relation function (or entry-exit function) in planar slow-fast systems. It is well known that zeros of the slow divergence integral associated with canard limit periodic sets give candidates for limit cycles. We present a new approach to detect the zeros of the slow divergence integral by studying the structure of the set of all probability measures invariant under the corresponding slow relation function. Using the slow relation function, we also show how to estimate (in terms of weak convergence) the transformation of families of probability measures that describe initial point distribution of canard orbits during the passage near a slow-fast Hopf point (or a more general turning point). We provide formulas to compute exit densities for given entry densities and the slow relation function. We apply our results to slow-fast Liénard equations.

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arXiv2023

Sliding cycles of the regularized piecewise linear $VI_3$ two-fold

The goal of this paper is to study the number of sliding limit cycles of a regularized piecewise linear $VI_3$ two-fold using the notion of slow divergence integral. We focus on limit cycles produced by canard cycles located in the half-plane with an invisible fold point. We prove that the integral has at most $1$ zero counting multiplicity (when it is not identically zero). This will imply that the canard cycles can produce at most $2$ limit cycles. Moreover, we detect regions in the parameter space with $2$ limit cycles.

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arXiv2023

Slow divergence integral in regularized piecewise smooth systems

In this paper we define the notion of slow divergence integral along sliding segments in regularized planar piecewise smooth systems. The boundary of such segments may contain diverse tangency points. We show that the slow divergence integral is invariant under smooth equivalences. This is a natural generalization of the notion of slow divergence integral along normally hyperbolic portions of curve of singularities in smooth planar slow-fast systems. We give an interesting application of the integral in a model with visible-invisible two-fold of type $VI_3$. It is related to a connection between so-called Minkowski dimension of bounded and monotone "entry-exit" sequences and the number of sliding limit cycles produced by so-called canard cycles.

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arXiv2023

Fractal analysis of hyperbolic saddles with applications

In this paper we express the Minkowski dimension of spiral trajectories near hyperbolic saddles and semi-hyperbolic singularities in terms of the Minkowski dimension of intersections of such spirals with transversals near these singularities. We apply these results to hyperbolic saddle-loops and hyperbolic $2$-cycles to obtain upper bounds on the cyclicity of such limit periodic sets.

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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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arXiv2022

Slow-Fast Torus Knots

The goal of this paper is to study global dynamics of $C^\infty$-smooth slow-fast systems on the $2$-torus of class $C^\infty$ using geometric singular perturbation theory and the notion of slow divergence integral. Given any $m\in\mathbb{N}$ and two relatively prime integers $k$ and $l$, we show that there exists a slow-fast system $Y_ε$ on the $2$-torus that has a $2m$-link of type $(k,l)$, i.e. a (disjoint finite) union of $2m$ slow-fast limit cycles each of $(k,l)$-torus knot type, for all small $ε>0$. The $(k,l)$-torus knot turns around the $2$-torus $k$ times meridionally and $l$ times longitudinally. There are exactly $m$ repelling limit cycles and $m$ attracting limit cycles. Our analysis: a) proves the case of normally hyperbolic singular knots, and b) provides sufficient evidence to conjecture a similar result in some cases where the singular knots have regular nilpotent contact with the fast foliation.

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arXiv2021

The box dimension of degenerate spiral trajectories of a class of ordinary differential equations

In this paper we initiate the study of the box dimension of degenerate spiral trajectories of a class of ordinary differential equations. A class of singularities of focus type with two zero eigenvalues (nilpotent or more degenerate) has been studied. We find the box dimension of a polynomial degenerate focus of type $(n,n)$ by exploiting the well-known fractal results for $α$-power spirals. In the general $(m,n)$ case, we formulate a conjecture about the box dimension of a degenerate focus. Further, we reduce the fractal analysis of planar nilpotent contact points to the study of the box dimension of a slow-fast spiral generated by their "entry-exit" function. There exists a bijective correspondence between the box dimension of the slow-fast spiral and the codimension of contact points. We also construct a three-dimensional vector field that contains a degenerate spiral, called an elliptical power spiral, as a trajectory.

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arXiv2021

Jump-induced mixed-mode oscillations through piecewise-affine maps

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 differential equations of slow-fast type, and are typically related to the presence of so-called folded singularities and the corresponding canard trajectories in such systems. Here, we introduce a canonical family of three-dimensional slow-fast systems that exhibit MMOs which are induced by relaxation-type dynamics, and which are hence based on a "jump mechanism", rather than on a more standard canard mechanism. In particular, we establish a correspondence between that family and a class of associated one-dimensional piecewise affine maps (PAMs) which exhibit MMOs with the same signature. Finally, we give a preliminary classification of admissible mixed-mode signatures, and we illustrate our findings with numerical examples.

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