Bifurcations in the Lozi map

We study the presence in the Lozi map of a type of abrupt order-to-order and order-to-chaos transitions which are mediated by an attractor made of a continuum of neutrally stable limit cycles, all with the same period.Comment: 17 pages, 12 figure

Center boundaries for planar piecewise-smooth differential equations with two zones

Agraïments: The first author is partially supported by Procad-Capes88881.068462/2014-01 and FAPESP2013/24541-0. The second author is partially supported by program CAPES/PDSE grant number 7038/2014-03 and CAPES/DS program number 33004153071P0. The third author is supported by program CAPES/PNPD grant number 1271113.This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary. We prove that given a pair of systems that share a hyperbolic focus singularity p 0 , with the same orientation and opposite stability, and a ray Σ 0 with endpoint at the singularity p 0 , we can find a smooth manifold Ω such that Σ 0 ∪ p 0 ∪ Ω is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached

Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems

Canard-induced phenomena have been extensively studied in the last three decades, from both the mathematical and the application viewpoints. Canards in slow-fast systems with (at least) two slow variables, especially near folded-node singularities, give an essential generating mechanism for mixed-mode oscillations (MMOs) in the framework of smooth multiple timescale systems. There is a wealth of literature on such slow-fast dynamical systems and many models displaying canard-induced MMOs, particularly in neuroscience. In parallel, since the late 1990s several papers have shown that the canard phenomenon can be faithfully reproduced with piecewise-linear (PWL) systems in two dimensions, although very few results are available in the three-dimensional case. The present paper aims to bridge this gap by analyzing canonical PWL systems that display folded singularities, primary and secondary canards, with a similar control of the maximal winding number as in the smooth case. We also show that the singular phase portraits are compatible in both frameworks. Finally, we show using an example how to construct a (linear) global return and obtain robust PWL MMOs

On 3-parameter families of piecewise smooth vector fields in the plane

This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three-parameter families of a class of non$-$smooth vector fields are studied and the tridimensional bifurcation diagrams are exhibited. Our main results describe the unfolding of the so called fold-cusp singularity by means of the variation of 3 parameters.Comment: 33 Figures. arXiv admin note: text overlap with arXiv:1104.086

Maximum number of limit cycles for certain piecewise linear dynamical systems

Agraïments: The second author was partially supported by a FAPESP Grant 2012/10231-7. The third authors was partially supported by a FAPESP Grant 2012/18780-0. The three authors are also supported by a CAPES CSF-PVE Grant 88881.030454/2013-01 from the program CSF-PVE.This paper deals with the question of the determinacy of the maximum number of limit cycles of some classes of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line \Sigma. We restrict ourselves to the non-sliding limit cycles case, i.e., limit cycles that do not contain any sliding segment. Among all cases treated here, it is proved that the maximum number of limit cycles is at most 2 if one of the two linear differential systems of the discontinuous piecewise linear differential system has a focus in \Sigma, a center, or a weak saddle. We use the theory of Chebyshev systems for establishing sharp upper bounds for the number of limit cycles. Some normal forms are also provided for these systems

Uniqueness of limit cycles for sewing piecewise linear systems

Agraïments: The first author is partially supported by the FAPEG, by the CNPq grant numbers 475623/2013-4 and 306615/2012-6 and, by the CAPES grant numbers CSF/PVE-88881.030454/2013-01 and PROCAD-88881.068462/2014-01. The second author is supported by the UNAB 10-4E-378 grants.This paper proves the uniqueness of limit cycles for sewing planar piecewise linear systems with two zones separated by a straight line, \Sigma, and only one \Sigma-singularity of monodromic type. The proofs are based in an extension of Rolle's Theorem for dynamical systems on the plane

Smoothing of homoclinic-like connections to regular tangential singularities in Filippov systems

In this paper, we are concerned about smoothing of a class of $\Sigma$-polycycles of Filippov systems, namely homoclinic-like connections to regular-tangential singularities. Conditions are stablished in order to guarantee the existence of limit cycles bifurcating from such connections.Comment: arXiv admin note: text overlap with arXiv:2003.0954