Related papers: Symmetric Limit Cycles in 3D Piecewise Linear Syst…
We analyse three codimension-two bifurcations occurring in nonsmooth systems, when a non-hyperbolic cycle (fold, flip, and Neimark-Sacker cases, both in continuous- and discrete-time) interacts with one of the discontinuity boundaries…
The already proved Lum-Chua's conjecture says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In this paper, we provide a new proof by using a novel…
In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line $x=0$. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector…
In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations $\dot{x}=y, \ \dot{y}=-x-\varepsilon \cdot (f(x)\cdot y +{\rm sgn}(y)\cdot g(x))$. Using the averaging method, we were able…
We study piecewise-smooth systems with three zones, $\dot{z} = f_i(z)$, $i = 1,2,3,$ whose discontinuity set $\Sigma$ consists either of a pair of parallel lines or a pair of circles tangent to each other internally or externally. Each…
Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are…
The study of the dynamics of a continuous observable and non-controllable three-dimensional symmetric piecewise linear system with three zones can be reduced to the study of the existence of limit cycles for the piecewise differential…
In this paper, we study crossing limit cycles of planar discontinuous piecewise differential systems separated by a nonregular switching line, where one subsystem is a linear differential center and the other belongs to one of six families…
A 3-dimensional reduction of the seld-dual Yang-Mills (SDYM) equation, named SDYM3, is examined from the point of view of its symmetry and integrability characteristics. By using a non-auto-Backlund transformation, this equation is…
In this paper we study the maximum number $N$ of limit cycles that can exhibit a planar piecewise linear differential system formed by two pieces separated by a straight line. More precisely, we prove that this maximum number satisfies…
This Letter outlines 20 geometric mechanisms by which limit cycles are created locally in two-dimensional piecewise-smooth systems of ODEs. These include boundary equilibrium bifurcations of hybrid systems, Filippov systems, and continuous…
We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking…
Piecewise smooth systems are intensively studied today in many application areas, such as economics, finance, engineering, biology, and ecology. In this work, we consider a class of one-dimensional piecewise linear discontinuous maps with a…
We consider families of planar polynomial vector fields of degree $n$ and study the cyclicity of a type of unbounded polycycle~$\Gamma$ called hemicycle. Compactified to the Poincar\'e disc,~$\Gamma$ consists of an affine straight line…
This paper presents a criterion that provides an easy sufficient condition for a collection of line integrals to have the Chebyshev property. The condition is based on the functions appearing in the line integrals. The criterion is used to…
We study both analytically and numerically a coupled system of spherically symmetric SU(2) Yang-Mills-dilaton equation in 3+1 Minkowski space-time. It has been found that the system admits a hidden scale invariance which becomes transparent…
In this paper, by using Picard-Fuchs equations and Chebyshev criterion, we study the bifurcate of limit cycles for quadratic Hamilton system $S^{(2)}$ and $S^{(3)}$: $\dot{x}= y+2axy+by^2$, $\dot{y}=-x+x^2-ay^2$ with $a\in(-\frac{1}{2},1)$,…
Continuing the investigation for the number of crossing limit cycles of nonsmooth Li\'enard systems in [Nonlinearity 21(2008), 2121-2142] for the case of a unique equilibrium, in this paper we consider the case of any number of equilibria.…
For piecewise-smooth differential systems, in this paper we focus on crossing limit cycles and sliding loops bifurcating from a grazing loop connecting one high multiplicity tangent point. For the low multiplicity cases considered in…
We study continuous symmetry reduction of dynamical systems by the method of slices (method of moving frames) and show that a `slice' defined by minimizing the distance to a single generic `template' intersects the group orbit of every…