The number of limit cycles for regularized piecewise polynomial systems is unbounded

Huzak, Renato; Kristiansen, Kristian

doi:10.1016/j.jde.2022.09.028

Cited by 5 publications

(3 citation statements)

References 66 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is another case of 'infinitely many sinks' arising in piecewise smooth systems, although the mechanism is totally different from the examples for the border collision normal form [13]. See also recent work [9], which shows that the number of limit cycles for regularized piecewise polynomial systems is unbounded.…”

Section: Discussionmentioning

confidence: 88%

Uncountably Many Cases of Filippov’s Sewed Focus

et al. 2023

View full text Add to dashboard Cite

The sewed focus is one of the singularities of planar piecewise smooth dynamical systems. Defined by Filippov in his book (Differential Equations with Discontinuous Righthand Sides, Kluwer, 1988), it consists of two invisible tangencies either side of the switching manifold. In the case of analytic focus-like behaviour, Filippov showed that the approach to the singularity is in infinite time. For the case of non-analytic focus-like behaviour, we show that the approach to the singularity can be in finite time. For the non-analytic sewed centre-focus, we show that there are uncountably many different topological types of local dynamics, including cases with infinitely many stable periodic orbits, and show how to create systems with periodic orbits intersecting any bounded symmetric closed set.

show abstract

Section: Discussionmentioning

confidence: 88%

Uncountably Many Cases of Filippov’s Sewed Focus

et al. 2023

View full text Add to dashboard Cite

show abstract

“…because we have ( 19), (20), ( 21), ( 22) and we can reject multiplicative constants, such as (−1) p . The first integral of the original system (3) appears by undoing the changes of variables (11) and (15). So…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…While in this paper, only one of the systems is Hamiltonian, otherwise the piecewise differential system cannot have limit cycles. Some authors are interested in knowing if the limit cycles of a discontinuous piecewise differential system persist when the piecewise differential is regularized (see [10]), or how many limit cycles can have such regularized piecewise differential systems (see [11]). For the discontinuous piecewise differential systems studied here, having one algebraic limit cycle, if they are regularized using the regularization of Sotomayor and Teixeira [12], the limit cycle persists but it is no longer algebraic.…”

Section: Introductionmentioning

confidence: 99%

Limit Cycles of Polynomially Integrable Piecewise Differential Systems

García

Llibre

Río

et al. 2023

Axioms

View full text Add to dashboard Cite

In this paper, we study how many algebraic limit cycles have the discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides. We assume that at least one of the systems is Hamiltonian. Under this assumption, piecewise differential systems have no more than one limit cycle. This study characterizes linear differential systems with polynomial first integrals.

show abstract

Sliding Cycles of Regularized Piecewise Linear Visible–Invisible Twofolds

Huzak,

Kristiansen

2024

Qual. Theory Dyn. Syst.

View full text Add to dashboard Cite

The goal of this paper is to study the number of sliding limit cycles of regularized piecewise linear visible–invisible twofolds 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.

show abstract

The number of limit cycles for regularized piecewise polynomial systems is unbounded

Cited by 5 publications

References 66 publications

Uncountably Many Cases of Filippov’s Sewed Focus

Uncountably Many Cases of Filippov’s Sewed Focus

Limit Cycles of Polynomially Integrable Piecewise Differential Systems

Sliding Cycles of Regularized Piecewise Linear Visible–Invisible Twofolds

Contact Info

Product

Resources

About