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In this paper we research global dynamics and bifurcations of planar piecewise smooth quadratic quasi--homogeneous but non-homogeneous polynomial differential systems. We present sufficient and necessary conditions for the existence of a…
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…
For a class of polynomial non-autonomous differential equations of degree n, we use phase plane analysis to show that each equation in this class has n periodic solutions. The result implies that certain rigid two-dimensional systems have…
We analyze a pair of delay-coupled FitzHugh-Nagumo oscillators exhibiting in-out intermittency as a part of the generating mechanism of extreme events. We study in detail the characteristics of in-out intermittency and identify the…
For periodic linear control systems with bounded control range, an autonomized system is introduced by adding the phase to the state of the system. Here a unique control set (i.e., a maximal set of approximate controllability) with nonvoid…
Piecewise smooth hybrid systems, involving continuous and discrete variables, are suitable models for describing the multiscale regulatory machinery of the biological cells. In hybrid models, the discrete variables can switch on and off…
Hybrid dynamical systems have proven to be a powerful modeling abstraction, yet fundamental questions regarding the dynamical properties of these systems remain. In this paper, we develop a novel class of relaxations which we use to recover…
There exists a variety of physically interesting situations described by continuous maps that are nondifferentiable on some surface in phase space. Such systems exhibit novel types of bifurcations in which multiple coexisting attractors can…
We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value so-called event collisions occur. This paper classifies…
A. Gasull shared a list of 33 open problems in low dimensional dynamical systems in his work in 2021. The second part of Problem 3 is about whether the limit cycle of a quasi-homogeneous system $ \dot{x}=y,\; \dot{y}=-x^3+\alpha x^2y+y^3 $…
In this paper, we investigate a class of port-Hamiltonian systems with singular vector fields. We show that, under suitable conditions, their interconnection with passive systems ensures convergence to a prescribed non-equilibrium steady…
We consider time-invariant nonlinear $n$-dimensional strongly $2$-cooperative systems, that is, systems that map the set of vectors with up to weak sign variation to its interior. Strongly $2$-cooperative systems enjoy a strong…
In this paper we consider a class of piecewise affine Hamiltonian vector fields whose orbits are piecewise straight lines. We give a first classification result of such systems and show that the orbit-structure of the flow of such a…
We prove that the maximal number of limit cycles which bifurcate from an open period annulus under a given multi-parameter analytic deformation of a given analytic vector field is the same as in an appropriate one-parameter analytic…
A 3D nonlinear chaotic system, called the T system, is analyzed in this paper. Horseshoe chaos is investigated via the heteroclinic Shilnikov method constructing a heteroclinic connections between the saddle equilibrium points of the…
In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general…
We consider autonomous Lagrangian systems with two degrees of freedom, having an hyperbolic equilibrium of saddle-saddle type (that is the eingenvalues of the linearized system about the equilibrium are $\pm \lambda_1, \pm \lambda_2 $,…
Singular and sectional hyperbolic sets are the objects of the extension of the classical Smale Hyperbolic Theory to flows having invariant sets with singularities accumulated by regular orbits within the set. It is by now well-known that…
The first order condition of the constrained minimization problem leads to a saddle point problem. A multigrid method using a multiplicative Schwarz smoother for saddle point problems can thus be interpreted as a successive subspace…
We summarize recent developments of the semiclassical description of shell effects in finite fermion systems with explicit inclusion of spin degrees of freedom, in particluar in the presence of spin-orbit interactions. We present a new…