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Douglas D. Novaes

Publications and source records attributed to Douglas D. Novaes.

At least 19 recordsLinked to original sources
arXiv2026

Hopf-like bifurcations induced by hysteresis and time-delay near monodromic tangential singularities

We investigate planar piecewise-analytic vector fields, focusing on the merged focus and other monodromic tangential singularities when hysteresis or time-delay is incorporated into the switching condition. If the singularity is an asymptotically stable solution of the system with an instantaneous switch, then the introduction of hysteresis or time-delay causes an attracting limit cycle to be formed locally. We derive asymptotic expressions for the size and period of the limit cycle, allowing any degrees of tangency and any order for the smallest non-zero Lyapunov coefficient. We find that the growth rate of the limit cycle differs for hysteresis and time-delay, and differs to that of the related pseudo-Hopf bifurcation.

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arXiv2026

Infinite-Piecewise Expanding Maps: Chaos, Ergodicity and Invariant-Set Complexity

In this paper, we study a class of one-dimensional piecewise maps defined by infinitely many smooth expanding branches. This class arises naturally in the context of non-smooth dynamical systems and includes the first-return maps locally defined near sliding Shilnikov connections. By means of the theory of conformal iterated function systems (CIFS), we investigate several dynamical properties of these maps as well as the topological complexity of their invariant sets. In particular, we show that the dynamics restricted to the invariant set is topologically conjugate to the shift on $\mathbb{N}^{\mathbb{N}}$. We also establish the existence of a unique conformal measure that is invariant and ergodic under the map.

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arXiv2026

Weakly Normally Hyperbolic Invariant Tori: Persistence and an Averaging Principle

We prove a persistence theorem for attracting weakly normally hyperbolic invariant tori under small time-periodic perturbations. The theorem extends recent continuation results for weakly normally hyperbolic limit cycles to invariant tori of arbitrary dimension, providing a general analytical framework for their detection. As an application, we establish an averaging principle showing that attracting normally hyperbolic invariant $d$-tori of the averaged system give rise to attracting normally hyperbolic invariant $(d+1)$-tori of the original non-autonomous system. We further introduce a polynomiality-preserving construction that simultaneously lifts the dimensions of the phase space and the attracting normally hyperbolic invariant tori, yielding recursive lower bounds for the maximal number of codimension-$1$ normally hyperbolic invariant tori of polynomial vector fields of a given degree, thereby extending the counting aspect of Hilbert's sixteenth problem to higher-dimensional invariant tori. In particular, we prove that this number grows at least polynomially with the degree and becomes unbounded for invariant tori of higher codimension.

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arXiv2026

Growth estimate for the number of crossing limit cycles in planar piecewise polynomial vector fields

Motivated by the classical Hilbert's Sixteenth Problem, we extend some main developments obtained for Hilbert's number in the polynomial setting to the piecewise polynomial context. Specifically, we study the growth of the maximum number of crossing limit cycles in planar piecewise polynomial vector fields of degree $n$, denoted by $H_c(n)$. The best previously known general lower bound is $H_c(n)\geq 2n - 1$. In this work, we show that $H_c(n)$ grows at least as fast as $n^2/4.$ Furthermore, we prove that $H_c(n)$ is strictly increasing whenever it is finite, and that in such cases this maximum can be realized by piecewise polynomial systems whose crossing limit cycles are all hyperbolic. Finally, for the more restrictive class of piecewise polynomial Hamiltonian vector fields, we adapt the recursive construction of Christopher and Lloyd to demonstrate that the corresponding maximal number of crossing limit cycles, denoted by $\widehat{H}_c(n)$, grows at least as fast as $n\log n/(2\log 2)$, thereby improving previously established linear growth estimate.

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arXiv2026

Detecting Limit Tori in Non-Smooth Systems: An Analytic Approach with Applications to 3D Piecewise Linear Systems

This work investigates a class of non-autonomous $T$-periodic piecewise smooth differential systems and their associated time-$T$ maps. Our main result provides an analytical approach for detecting, within this class of piecewise differential systems, isolated invariant tori associated with normally hyperbolic invariant closed curves of the time-$T$ map. To achieve this, we derive sufficient conditions under which smooth near-identity maps undergo a Neimark--Sacker bifurcation. As an application of our main result, we present a family of 3D piecewise linear differential systems exhibiting attracting and repelling isolated invariant tori which, moreover, persist under small perturbations. To the best of our knowledge, this family provides the first examples in which limit tori are analytically detected in piecewise linear systems.

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arXiv2026

Averaging theory and catastrophes

When a dynamical system is subject to a periodic perturbation, the averaging method can be applied to obtain an autonomous leading order "guiding system", placing the time dependence at higher orders. Recent research focused on investigating invariant structures in non-autonomous differential systems arising from hyperbolic structures in the guiding system, such as periodic orbits and invariant tori. Complementarily, the effect that bifurcations in the guiding system have on the original non-autonomous one has also been recently explored, albeit less frequently. This paper extends this study by providing a broader description of the dynamics that can emerge from non-hyperbolic structures of the guiding system. Specifically, we prove here that $\mathcal{K}$-universal bifurcations in the guiding system `persist' in the original non-autonomous one, while non-versal bifurcations, such as the transcritical and pitchfork, do not. We illustrate the results on examples of a fold, a transcritical, a pitchfork, and a saddle-focus.

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arXiv2025

Weak-Coppel problem for a class of Riccati differential equations

We study the $T$-periodic solutions of the real Riccati differential equation $x' = x^2 + γ(t),$ where $x=x(t)$ and $γ$ is a $T$-periodic function. Our goal is to define a real-valued discriminant $Δ_γ$ that determines whether the equation admits two, one, or no $T$-periodic solutions, in analogy with the classical discriminant of the quadratic algebraic equations. This problem is closely related to a question posed by Coppel concerning the characterization of bifurcation curves for planar quadratic differential systems. Although our result does not cover all periodic Riccati differential equations, many of them can be transformed into this particular form.

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arXiv2025

Bounding the number of limit cycles in piecewise linear differential systems: methodology and worked examples

In 2012, Huan and Yang introduced the first piecewise linear differential system with two zones separated by a straight line having at least three limit cycles, serving as a counterexample to the Han-Zhang conjecture that said that such systems have no more than two limit cycles. Over the past decade, extensive research has been conducted to explore periodic solutions in piecewise linear differential systems. However, the question of whether the Huan-Yang example indeed has exactly three limit cycles has remained unresolved, primarily due to the lack of techniques for bounding the number of limit cycles in these systems. Based on the authors' recent results, this paper presents a methodology for bounding the number of limit cycles in piecewise linear systems. This methodology conclusively establishes that the Huan-Yang example has exactly three limit cycles. Our methodology has a broader applicability and constitutes a powerful tool for analyzing and bounding the number of limit cycles in any explicit example of piecewise linear differential systems. We extend our analysis to several recent examples of significance in the literature and show that they also exhibit exactly three limit cycles. Finally, we present an algebraic criterion for bounding the number of crossing limit cycles in the focus-focus case.

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arXiv2025

Normally hyperbolic limit tori near monodromic singularities in 3D polynomial vector fields

We investigate the maximal number $N_h(m)$ of normally hyperbolic limit tori in three-dimensional polynomial vector fields of degree $m$, which extends the classical notion of Hilbert numbers to higher dimensions. Using recent developments in averaging theory, we show the existence of families of vector fields near monodromic singularities, including both Hopf-zero and nilpotent-zero cases, that exhibit multiple nested normally hyperbolic limit tori. This approach allows us to establish improved lower bounds: $N_h(2) \geq 3$, $N_h(3) \geq 5$, $N_h(4) \geq 7$, and $N_h(5) \geq 13$, which are currently the best available in the literature. Furthermore, these bounds are extended using the strict monotonicity of the function $N_h(m)$ and a recursive construction inspired by the Christopher-Lloyd method, leading to new estimates for higher degrees which improves all the previously known results.

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arXiv2025

Normal Hyperbolicity in Secondary Hopf Bifurcations

We combine results available in the literature to prove that the torus emerging in a secondary Hopf bifurcation is normally hyperbolic. This result is then applied to establish sufficient conditions for the bifurcation of normally hyperbolic invariant tori in the extended phase space of systems with small time-periodic perturbations via an application of the averaging method.

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arXiv2025

Strict increase in the number of normally hyperbolic limit tori in 3D polynomial vector fields

The second part of Hilbert's 16th problem concerns determining the maximum number $H(m)$ of limit cycles that a planar polynomial vector field of degree $m$ can exhibit. A natural extension to the three-dimensional space is to study the maximum number $N(m)$ of limit tori that can occur in spatial polynomial vector fields of degree $m$. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number $N_h(m)$, if finite, increases strictly with $m$. More precisely, we prove that $N_h(m+1) \geqslant N_h(m) + 1$. Our proof relies on the torus bifurcation phenomenon observed in spatial vector fields near Hopf-Zero equilibria. While conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of a torus bifurcation assuming only that the linear part of the unperturbed vector field is in Jordan normal form. This approach circumvents the need for intricate computations involving higher-order normal forms.

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arXiv2024

On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets

The concept of sliding Shilnikov connection has been recently introduced and represents an important notion in Filippov systems, because its existence implies chaotic behavior on an invariant subset of the system. The investigation of its properties has just begun, and understanding the topology and complexity of its invariant set is of interest. In this paper, we conduct a local analysis on the first return map associated to a Shilnikov sliding connection, which reveals a conformal iterated function system (CIFS) structure. By using the theory of CIFS, we estimate the Hausdorff dimension of the local invariant set of the first return map, showing, in particular, that it is strictly greater than $0$ and strictly less than $1$, and its one-dimensional Lebesgue measure is 0. Moreover, we prove that the closure of the local invariant set is a Cantor set and has the same Hausdorff dimension and Lebesgue measure of the original invariant set. Furthermore, it is given by the invariant set adjoined with the set of all pre-images of the regular-fold point.

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arXiv2024

A note on a recent attempt to solve the second part of Hilbert's 16th Problem

For a given natural number $n$, the second part of Hilbert's 16th Problem asks whether there exists a finite upper bound for the maximum number of limit cycles that planar polynomial vector fields of degree $n$ can have. This maximum number of limit cycle, denoted by $H(n)$, is called the $n$th Hilbert number. It is well-established that $H(n)$ grows asymptotically as fast as $n^2 \log n$. A direct consequence of this growth estimation is that $H(n)$ cannot be bounded from above by any quadratic polynomial function of $n$. Recently, the authors of the paper [Exploring limit cycles of differential equations through information geometry unveils the solution to Hilbert's 16th problem. Entropy, 26(9), 2024] affirmed to have solved the second part of Hilbert's 16th Problem by claiming that $H(n) = 2(n - 1)(4(n - 1) - 2)$. Since this expression is quadratic in $n$, it contradicts the established asymptotic behavior and, therefore, cannot hold. In this note, we further explore this issue by discussing some counterexamples.

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arXiv2024

Unveiling the Cyclicity of Monodromic Tangential Singularities: Insights Beyond the Pseudo-Hopf Bifurcation

The cyclicity problem, crucial in analyzing planar vector fields, consists in estimating the number of limit cycles emanating from monodromic singularities. Traditionally, this estimation relies on Lyapunov coefficients. However, in nonsmooth systems, besides the limit cycles bifurcating by varying the Lyapunov coefficients, monodromic singularities on the switching curve can always be split apart yielding, under suitable conditions, a sliding region and an additional limit cycle surrounding it. This bifurcation phenomenon, known as pseudo-Hopf bifurcation, has enhanced lower-bound cyclicity estimations for monodromic singularities in Filippov systems. In this study, we push beyond the pseudo-Hopf bifurcation, demonstrating that the destruction of $(2k,2k)$-monodromic tangential singularities yields at least $k$ limit cycles surrounding sliding segments. This new bifurcation phenomenon expands our understanding of limit cycle bifurcations in nonsmooth systems and, in addition to the theoretical significance, has practical relevance in various applied models involving switches and abrupt processes.

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arXiv2024

A version of Hilbert's 16th Problem for 3D polynomial vector fields: Counting isolated invariant tori

Hilbert's 16th Problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree $m$, has been one of the most important driving forces for new developments in the qualitative theory of vector fields. Increasing the dimension, one cannot expect the existence of a finite upper bound for the number of limit cycles of, for instance, $3$D polynomial vector fields of a given degree $m$. Here, as an extension of such a problem in the $3$D space, we investigate the number of isolated invariant tori in $3$D polynomial vector fields. In this context, given a natural number $m$, we denote by $N(m)$ the upper bound for the number of isolated invariant tori of $3$D polynomial vector fields of degree $m$. Based on a recently developed averaging method for detecting invariant tori, our first main result provides a mechanism for constructing $3$D differential vector fields with a number $H$ of normally hyperbolic invariant tori from a given planar differential vector field with $H$ hyperbolic limit cycles. The strength of our mechanism in studying the number $N(m)$ lies in the fact that the constructed $3$D differential vector field is polynomial provided that the given planar differential vector field is polynomial. Accordingly, our second main result establishes a lower bound for $N(m)$ in terms of lower bounds for the number of hyperbolic limit cycles of planar polynomial vector fields of degree $[m/2]-1$. Based on this last result, we apply a methodology due to Christopher & Lloyd to show that $N(m)$ grows as fast as $m^3/128$. Finally, the above-mentioned problem is also formulated for higher dimensional polynomial vector fields.

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arXiv2024

Poincaré-Hopf Theorem for Filippov vector fields on 2-dimensional compact manifolds

The Poincaré-Hopf Theorem relates the Euler characteristic of a 2-dimensional compact manifold to the local behavior of smooth vector fields defined on it. However, despite the importance of Filippov vector fields, concerning both their theoretical and applied aspects, until now, it was not known whether this theorem extends to Filippov vector fields. In this paper, we demonstrate that the Poincaré-Hopf Theorem applies to Filippov vector fields defined on 2-dimensional compact manifolds with smooth switching manifolds. As a result, we establish a variant of the Hairy Ball Theorem, asserting that "any Filippov vector field on a sphere with smooth switching manifolds must have at least one singularity (in the Filippov sense) with positive index". This extension is achieved by introducing a new index definition that includes the singularities of Filippov vector fields, such as pseudo-equilibria and tangential singularities. Our work extends the classical index definition for singularities of smooth vector fields to encompass those of Filippov vector fields with smooth switching manifolds. This extension is based on an invariance property under a regularization process, allowing us to establish all classical index properties. We also compute the indices of all generic $Σ$-singularities and some codimension-1 $Σ$-singularities, including fold-fold tangential singularities, regular-cusp tangential singularities, and saddle-node pseudo-equilibria.

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arXiv2024

Limit cycles bifurcating from periodic integral manifold in non-smooth differential systems

This paper addresses the perturbation of higher-dimensional non-smooth autonomous differential systems characterized by two zones separated by a codimension-one manifold, with an integral manifold foliated by crossing periodic solutions. Our primary focus is on developing the Melnikov method to analyze the emergence of limit cycles originating from the periodic integral manifold. While previous studies have explored the Melnikov method for autonomous perturbations of non-smooth differential systems with a linear switching manifold and with a periodic integral manifold, either open or of codimension 1, our work extends to non-smooth differential systems with a non-linear switching manifold and more general periodic integral manifolds, where the persistence of periodic orbits is of interest. We illustrate our findings through several examples, highlighting the applicability and significance of our main result.

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