scholarly journals On the Integrable Deformations of the Maximally Superintegrable Systems

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1000
Author(s):  
Cristian Lăzureanu
Keyword(s):  
Differential Equations ◽  
Arbitrary System ◽  
Superintegrable System ◽  
First Order ◽  
Integrable Deformation ◽  
Superintegrable Systems ◽  
Constants Of Motion ◽  
Integrable Deformations ◽  
Autonomous Differential Equations ◽  
New System

In this paper, we present the integrable deformations method for a maximally superintegrable system. We alter the constants of motion, and using these new functions, we construct a new system which is an integrable deformation of the initial system. In this manner, new maximally superintegrable systems are obtained. We also consider the particular case of Hamiltonian mechanical systems. In addition, we use this method to construct some deformations of an arbitrary system of first-order autonomous differential equations.

scholarly journals Integrable Deformations and Dynamical Properties of Systems with Constant Population

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1378
Author(s):  
Cristian Lăzureanu
Keyword(s):  
Differential Equations ◽  
Three Dimensional ◽  
Poisson Geometry ◽  
Volterra System ◽  
First Order ◽  
Integrable Deformations ◽  
Dynamical Properties ◽  
Points Of View ◽  
First Order Differential Equations

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.

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

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Cristian Lăzureanu ◽  
Camelia Petrişor
Keyword(s):  
Integrable Systems ◽  
Equilibrium Points ◽  
Poisson System ◽  
Poisson Structures ◽  
Integrable Deformation ◽  
Constants Of Motion ◽  
Integrable Deformations ◽  
Dynamical Properties ◽  
The Stability ◽  
Mckendrick Model

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.

scholarly journals On the Limit Cycles for a Class of Perturbed Fifth-Order Autonomous Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Nabil Sellami ◽  
Romaissa Mellal ◽  
Bahri Belkacem Cherif ◽  
Sahar Ahmed Idris
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Limit Cycles ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Nonlinear Function ◽  
Averaging Theory ◽  
First Order ◽  
Autonomous Differential Equations ◽  
Fifth Order

We study the limit cycles of the fifth-order differential equation x ⋅ ⋅ ⋅ ⋅ ⋅ − e x ⃜ − d x ⃛ − c x ¨ − b x ˙ − a x = ε F x , x ˙ , x ¨ , x ⋯ , x ⃜ with a = λ μ δ , b = − λ μ + λ δ + μ δ , c = λ + μ + δ + λ μ δ , d = − 1 + λ μ + λ δ + μ δ , e = λ + μ + δ , where ε is a small enough real parameter, λ , μ , and δ are real parameters, and F ∈ C 2 is a nonlinear function. Using the averaging theory of first order, we provide sufficient conditions for the existence of limit cycles of this equation.

scholarly journals SUPERINTEGRABLE SYSTEMS, MULTI-HAMILTONIAN STRUCTURES AND NAMBU MECHANICS IN AN ARBITRARY DIMENSION

2004 ◽  
Vol 19 (03) ◽  
pp. 393-409 ◽  
Author(s):  
A. TEĞMEN ◽  
A. VERÇIN
Keyword(s):  
Arbitrary Dimension ◽  
Functional Independence ◽  
Poisson Structures ◽  
Algebraic Condition ◽  
Superintegrable System ◽  
Hamiltonian Structures ◽  
Superintegrable Systems ◽  
Constants Of Motion ◽  
Nambu Bracket ◽  
Moser System

A general algebraic condition for the functional independence of 2n-1 constants of motion of an n-dimensional maximal superintegrable Hamiltonian system has been proved for an arbitrary finite n. This makes it possible to construct, in a well-defined generic way, a normalized Nambu bracket which produces the correct Hamiltonian time evolution. Existence and explicit forms of pairwise compatible multi-Hamiltonian structures for any maximal superintegrable system have been established. The Calogero–Moser system, motion of a charged particle in a uniform perpendicular magnetic field and Smorodinsky–Winternitz potentials are considered as illustrative applications and their symmetry algebras as well as their Nambu formulations and alternative Poisson structures are presented.

ANALYSIS OF HYPERCHAOTIC COMPLEX LORENZ SYSTEMS

2008 ◽  
Vol 19 (10) ◽  
pp. 1477-1494 ◽  
Author(s):  
GAMAL M. MAHMOUD ◽  
MANSOUR E. AHMED ◽  
EMAD E. MAHMOUD
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Differential Equations ◽  
Bifurcation Analysis ◽  
State Feedback ◽  
State Feedback Control ◽  
Lyapunov Dimension ◽  
First Order ◽  
Autonomous Differential Equations

This paper introduces and analyzes new hyperchaotic complex Lorenz systems. These systems are 6-dimensional systems of real first order autonomous differential equations and their dynamics are very complicated and rich. In this study we extend the idea of adding state feedback control and introduce the complex periodic forces to generate hyperchaotic behaviors. The fractional Lyapunov dimension of the hyperchaotic attractors of these systems is calculated. Bifurcation analysis is used to demonstrate chaotic and hyperchaotic behaviors of our new systems. Dynamical systems where the main variables are complex appear in many important fields of physics and communications.

scholarly journals III. On the differential equations of dynamics

1863 ◽  
Vol 12 ◽  
pp. 481-481
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Original System ◽  
General Character ◽  
Partial Differential ◽  
First Order ◽  
Numerical Result ◽  
The Common ◽  
New System

It is shown in the general paper that if an integral of any one equation of the peculiar system of (partial differential) equations there discussed be found, then if a certain numerical result of subsequent and always possible operations prove odd , an integral of the entire system can be found by the solution of a single differential equation of the first order. It is shown in the paper now sent that, when the above numerical result is even , we can reduce the original system of partial differential equations into a new system, fewer in number by unity at least, and of the same general character, so as to admit of a repetition of the same procedure. Thus the common integral sought will finally be given either by the solution of a single differential equation of the first order, or by finding one integral of the single partial differential equation, which, in the most unfavourable case conceivable, will remain at last.

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