Hamilton-Poisson Realizations of the Integrable Deformations of the Rikitake System

Integrable deformations of an integrable case of the Rikitake system are constructed by modifying its constants of motions. Hamilton-Poisson realizations of these integrable deformations are given. Considering two concrete deformation functions, a Hamilton-Poisson approach of the obtained system is presented. More precisely, the stability of the equilibrium points and the existence of the periodic orbits are proved. Furthermore, the image of the energy-Casimir mapping is determined and its connections with the dynamical elements of the considered system are pointed out

Integrable Deformations and Dynamical Properties of Systems with Constant Population

In this paper we consider systems of three autonomous first-order differential equations x˙=f(x),x=(x,y,z),f=(f1,f2,f3) such that x(t)+y(t)+z(t) is constant for all t. We present some Hamilton–Poisson formulations and integrable deformations. We also analyze the case of Kolmogorov systems. We study from some standard and nonstandard Poisson geometry points of view the three-dimensional Lotka–Volterra system with constant population

Dynamical Properties, Deformations, and Chaos in a Class of Inversion Invariant Jerk Equations

In this paper, we consider a class of jerk equations which are invariant to inversion. We discuss the stability and some bifurcations of the considered equation. In addition, we construct integrable deformations in order to stabilize some equilibrium points. Finally, we introduce a piecewise chaotic system which belongs to the considered class of jerk equations

A mathematical model of infectious disease transmission

In this paper we consider a three-dimensional nonlinear system which models the dynamics of a population during an epidemic disease. The considered model is a SIS-type system in which a recovered individual automatically becomes a susceptible one. We take into account the births and deaths, and we also consider that susceptible individuals are divided into two groups: non-vaccinated and vaccinated. In addition, we assume a medical scenario in which vaccinated people take a special measure to quarantine their newborns. We study the stability of the considered system. Numerical simulations point out the behavior of the considered population

On a Family of Hamilton–Poisson Jerk Systems

In this paper, we construct a family of Hamilton–Poisson jerk systems. We show that such a system has infinitely many Hamilton–Poisson realizations. In addition, we discuss the stability and we prove the existence of periodic orbits around nonlinearly stable equilibrium points. Particularly, we deduce conditions for the existence of homoclinic and heteroclinic orbits. We apply the obtained results to a family of anharmonic oscillators

Stabilization of the

In this paper, some deformations of the T system are constructed. In order to stabilize the chaotic behavior of the T system, we particularize these deformations obtaining some external linear control inputs. In each case, we prove that the controlled system is asymptotically stable

Stability and Energy-Casimir Mapping for Integrable Deformations of the Kermack-McKendrick System

Integrable deformations of a Hamilton-Poisson system can be obtained altering its constants of motion. These deformations are integrable systems that can have various dynamical properties. In this paper, we give integrable deformations of the Kermack-McKendrick model for epidemics, and we analyze a particular integrable deformation. More precisely, we point out two Poisson structures that lead to infinitely many Hamilton-Poisson realizations of the considered system. Furthermore, we study the stability of the equilibrium points, we give the image of the energy-Casimir mapping, and we point out some of its properties

Symmetries and Properties of the Energy-Casimir Mapping in the Ball-Plate Problem

In this paper a system derived by an optimal control problem for the ball-plate dynamics is considered. Symplectic and Lagrangian realizations are given and some symmetries are studied. The image of the energy-Casimir mapping is described and some connections with the dynamics of the considered system are presented