arXiv subjects

Paulo R. da Silva

Publications and source records attributed to Paulo R. da Silva.

6 recordsLinked to original sources

arXiv2026

Limit cycles in piecewise smooth systems with circular switching manifold

We study limit cycles in piecewise complex systems with switching manifold $\mathbb{S}^1$. Using Möbius transformations we establish an equivalence between circular and straight-line discontinuities that preserves periods, stability, and algebraic structure. For piecewise polynomial holomorphic systems we obtain lower bounds on the number of limit cycles via second-order averaging and, for low degrees, via Lyapunov quantities. For piecewise antiholomorphic systems we prove upper bounds: at most $3$ limit cycles in the linear case and $10$ in the quadratic case. We also prove a rigidity theorem: when both components admit classical holomorphic normal forms at the origin no crossing limit cycles exist. Finally, we construct explicit algebraic limit cycles in the circular context, providing, as far as we know the first such examples in the literature.

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arXiv2026

Piecewise Smooth Dynamical Systems Regularized by Convolution

We present a general regularization procedure for piecewise smooth vector fields whose discontinuity locus is a variety of normal crossings type. We show that such regularization can be smoothed through a finite sequence of blowings-up, thereby reducing the problem to study of the dynamics of a smooth vector field in a manifold with corners. The procedure will be illustrated in the cases of piecewise smooth vector fields on $\mathbb{R}^2$ with discontinuity locus $x=0$ or $xy=0$, and on $\mathbb{R}^3$ with discontinuity locus $xyz=0$. We will see that some unexpected dynamical phenomena may arise even in the case of piecewise constant vector fields.

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arXiv2026

Complex potentials and holomorphic differential equations

A complex potential is a holomorphic function $Ω:\mathbb{C} \to \mathbb{C}$ whose real and imaginary parts generate a pair of orthogonal foliations, representing the equipotential lines and the streamlines of $\dot{z} = \overline{Ω'(z)}$. In this work, we generalize the concept of potential to the broader class of dynamical systems of the form $\dot{z} = f(z)$, with $f:\mathbb{C} \to \mathbb{C}$ holomorphic. The resulting potential induces a rectification mapping providing a natural framework for the topological classification of phase portraits of planar polynomial vector fields. The existence of complex potentials serves as a powerful tool in addressing fundamental problems, such as the establishment of bounds for the number of limit cycles in piecewise-smooth systems, and the local configuration of curvature lines around umbilic points, among others.

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arXiv2025

Simultaneous bifurcation of limit cycles for Piecewise Holomorphic systems

Let $\dot{z}=f(z)$ be a holomorphic differential equation with center at $p$. In this paper we are concerned about studying the piecewise perturbation systems $\dot{z}=f(z)+εR^\pm(z,\overline{z}),$ where $R^\pm(z,\overline{z})$ are complex polynomials defined for $\pm\operatorname{Im}(z)> 0.$ We provide an integral expression, similar to an Abelian integral, for the period annulus of $p.$ The zeros of this integral control the bifurcating limit cycles from the periodic orbits of this annular region. This expression is given in terms of the conformal conjugation between $\dot{z}=f(z)$ and its linearization $\dot{z}=f'(p)z$ at $p$. We use this result to control the simultaneous bifurcation of limit cycles of the two annular periods of $\dot{z}={\rm i} (z^2-1)/2$, after both complex and holomorphic piecewise polynomial perturbations. In particular, as far as we know, we provide the first proof of the existence of non nested limit cycles for piecewise holomorphic systems.

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arXiv2023

On the number of limit cycles for piecewise polynomial holomorphic systems

In this paper we are concerned with determining lower bounds of the number of limit cycles for piecewise polynomial holomorphic systems with a straight line of discontinuity. We approach this problem with different points of view: study of the number of zeros of the first and second order averaging functions, or with the control of the limit cycles appearing from a monodromic equilibrium point via a degenerated Andronov-Hoph type bifurcation, adding at the very end the sliding effects. We also use the Poincaré-Miranda theorem for obtaining an explicit piecewise linear holomorphic systems with 3 limit cycles, result that improves the known examples in the literature that had a single limit cycle.

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arXiv2023

Holomorphic slow-fast systems

In this paper, we are concerned with studying the existence of invariant complex manifolds of two-dimensional holomorphic systems. From the geometric singular perturbation theory we know that if a slow-fast system has associated a normally hyperbolic compact critical manifold, then there exists a smooth locally invariant manifold. However, this smooth manifold does not necessarily have a complex structure. Here, we provide conditions to guarantee the existence of one-dimensional invariant complex manifolds. Consequently, this allows us to establish that the centers, foci, and nodes of the reduced problem are persistent by singular perturbation. The tools used by us are the usual techniques of Fenichel and Briot-Bouquet Theories.

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