arXiv subjects

Claudio Buzzi

Publications and source records attributed to Claudio Buzzi.

8 recordsLinked to original sources

arXiv2026

Homoclinic Theorems for piecewise smooth vector fields

In this paper we provide extensions of the Birkhoff-Smale Homoclinic Theorem and the $\lambda$-Lemma for piecewise smooth vector fields free of sliding motion. We prove that if the vector field is $T$-periodic in its independent variable and has a transverse homoclinic point, then its time-$T$-map is conjugated with a Bernoulli Shift and thus the vector field is chaotic. The Bernoulli Shifts considered here may have any finite number of symbols, or countably infinite many.

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arXiv2024

On the cyclicity of hyperbolic polycycles

Let $X$ be a planar smooth vector field with a polycycle $\Gamma^n$ with $n$ sides and all its corners, that are at most $n$ singularities, being hyperbolic saddles. In this paper we study the cyclicity of $\Gamma^n$ in terms of the hyperbolicity ratios of these saddles, giving explicit conditions that ensure that it is at least $k,$ for any $k\leqslant n.$ Our result extends old results and also provides a more accurate proof of the known ones because we rely on some recent powerful works that study in more detail the regularity with respect to initial conditions and parameters of the Dulac map of hyperbolic saddles for families of vector fields. We also prove that when $X$ is polynomial there is a polynomial perturbation (in general with degree much higher that the one of $X$) that attains each of the obtained lower bounds for the cyclicities. Finally, we also study some related inverse problems and provide concrete examples of applications in the polynomial world.

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arXiv2023

On structural stability of Evolutionary Stables Strategies

The introduction of concepts of Game Theory and Ordinary Differential Equations into Biology gave birth to the field of Evolutionary Stable Strategies, with applications in Biology, Genetics, Politics, Economics and others. In special, the model composed by two players having two pure strategies each results in a planar polynomial vector field with an invariant octothorpe. In this paper we provide sufficient and necessary conditions for structural stability in this model.

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arXiv2022

Evolutionary Stable Strategies and Cubic Vector Fields

The introduction of concepts of Game Theory and Ordinary Differential Equations into Biology gave birth to the field of Evolutionary Stable Strategies, with applications in Biology, Genetics, Politics, Economics and others. In special, the model composed by two players having two pure strategies each results in a planar cubic vector field with an invariant octothorpe. Therefore, in this paper we study such class of vector fields, suggesting the notion of genericity and providing the global phase portraits of the generic systems with a singularity at the central region of the octothorpe.

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arXiv2021

Monodromy and Dulac's Problem for Piecewise analytical planar vector fields

Consider an analytical function $f:V\subset\mathbb R^2\rightarrow\mathbb R$ having $0$ as its regular value, a switching manifold $\Sigma=f^{-1}(0)$ and a piecewise analytical vector field $X=(X^+,X^-)$, i.e. $X^\pm$ are analytical vector fields defined on $\Sigma^\pm=\{p\in V: \pm f(p)>0\}$. We characterize when the vector field $X$ has a monodromic singular point in $\Sigma$, called $\Sigma$-monodromic singular point. Moreover, under certain conditions, we show that a $\Sigma$-monodromic singular point of $X$ has a neighborhood free of limit cycles.

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arXiv2014

Chaotic Planar Piecewise Smooth Vector Fields With Non Trivial Minimal Sets

In this paper some aspects on chaotic behavior and minimality in planar piecewise smooth vector fields theory are treated. The occurrence of non-deterministic chaos is observed and the concept of orientable minimality is introduced. It is also investigated some relations between minimality and orientable minimality and observed the existence of new kinds of non-trivial minimal sets in chaotic systems. The approach is geometrical and involve the ordinary techniques of non-smooth systems.

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arXiv2010

Canard Cycles and Poincaré Index of Non-Smooth Vector Fields on the Plane

This paper is concerned with closed orbits of non-smooth vector fields on the plane. For a subclass of non-smooth vector fields we provide necessary and sufficient conditions for the existence of canard kind solutions. By means of a regularization we prove that the canard cycles are singular orbits of singular perturbation problems which are limit periodic sets of a sequence of limit cycles. Moreover, we generalize the Poincaré Index for non-smooth vector fields.

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arXiv2010

Branching of periodic orbits in reversible hamiltonian systems

This paper deals with the dynamics of time-reversible Hamiltonian vector fields with 2 and 3 degrees of freedom around an elliptic equilibrium point in presence of symplectic involutions. The main results discuss the existence of one-parameter families of reversible periodic solutions terminating at the equilibrium. The main techniques used are Birkhoff and Belitskii normal forms combined with the Liapunov-Schmidt reduction.

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