scholarly journals Global Phase Portraits for the Abel Quadratic Polynomial Differential Equations of the Second Kind With Z2-symmetries

2018 ◽  
Vol 61 (1) ◽  
pp. 149-165 ◽  
Author(s):  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Differential Equations ◽  
Normal Forms ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Polynomial Differential Equations ◽  
Poincaré Disk

AbstractWe provide normal forms and the global phase portraits on the Poincaré disk for all Abel quadratic polynomial diòerential equations of the second kind with -symmetries.

scholarly journals Bifurcation Diagrams and Global Phase Portraits for Some Hamiltonian Systems with Rational Potentials

2018 ◽  
Vol 28 (13) ◽  
pp. 1850168
Author(s):  
Ting Chen ◽  
Jaume Llibre
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Normal Forms ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Linear Change ◽  
Rational Potential ◽  
Change Of Variables ◽  
Poincaré Disk

In this paper, we study the global dynamical behavior of the Hamiltonian system [Formula: see text], [Formula: see text] with the rational potential Hamiltonian [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomials of degree 1 or 2. First we get the normal forms for these rational Hamiltonian systems by some linear change of variables. Then we classify all the global phase portraits of these systems in the Poincaré disk and provide their bifurcation diagrams.

scholarly journals Bifurcations of the Riccati Quadratic Polynomial Differential Systems

2021 ◽  
Vol 31 (06) ◽  
pp. 2150094
Author(s):  
Jaume Llibre ◽  
Bruno D. Lopes ◽  
Paulo R. da Silva
Keyword(s):  
Phase Portrait ◽  
Differential System ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Topological Equivalence ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Global Phase Portrait ◽  
Poincaré Disk

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of [Formula: see text] adding the circle [Formula: see text] of the infinity) modulo topological equivalence.

Global Phase Portraits for a Planar ℤ2-Equivariant Kukles Systems of Degree 3

2020 ◽  
Vol 30 (12) ◽  
pp. 2050164
Author(s):  
Fabio Scalco Dias ◽  
Ronisio Moises Ribeiro ◽  
Claudia Valls
Keyword(s):  
Normal Forms ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Poincaré Disk

We provide normal forms and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree [Formula: see text] with [Formula: see text]-equivariant symmetry. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

Global Phase Portraits for the Kukles Systems of Degree 3 with ℤ2-Reversible Symmetries

2021 ◽  
Vol 31 (06) ◽  
pp. 2150083
Author(s):  
Fabio Scalco Dias ◽  
Ronisio Moises Ribeiro ◽  
Claudia Valls
Keyword(s):  
Normal Forms ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Poincaré Disk

We provide the normal forms, the bifurcation diagrams and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree [Formula: see text] with [Formula: see text]-symmetries.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

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