Formal First Integrals of General Dynamical Systems

Advances in Mathematical Physics ◽

10.1155/2016/1036089 ◽

2016 ◽

Vol 2016 ◽

pp. 1-9

Author(s):

Jia Jiao ◽

Wenlei Li ◽

Qingjian Zhou

Keyword(s):

Dynamical Systems ◽

Nonlinear Systems ◽

Invariant Manifolds ◽

First Integrals ◽

Higher Dimensional ◽

Complete Study ◽

Planar Systems

The goal of this paper is trying to make a complete study on the integrability for general analytic nonlinear systems by first integrals. We will firstly give an exhaustive discussion on analytic planar systems. Then a class of higher dimensional systems with invariant manifolds will be considered; we will develop several criteria for existence of formal integrals and give some applications to illustrate our results at last.

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On the Nonlinear Observability of Polynomial Dynamical Systems

SYSTEM THEORY, CONTROL AND COMPUTING JOURNAL ◽

10.52846/stccj.2021.1.1.15 ◽

2021 ◽

Vol 1 (1) ◽

pp. 88-94

Author(s):

Daniel Gerbet ◽

Klaus Röbenack

Keyword(s):

Dynamical Systems ◽

Nonlinear Systems ◽

State Space ◽

Polynomial Dynamical Systems ◽

Higher Dimensional ◽

Dimensional State Space ◽

System Properties ◽

Controllability And Observability ◽

Important System ◽

Nonlinear Observability

Controllability and observability are important system properties in control theory. These properties cannot be easily checked for general nonlinear systems. This paper addresses the local and global observability as well as the decomposition with respect to observability of polynomial dynamical systems embedded in a higher-dimensional state-space. These criteria are applied on some example system.

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First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach

Symmetry ◽

10.3390/sym11101244 ◽

2019 ◽

Vol 11 (10) ◽

pp. 1244

Author(s):

Muhammad Umar Farooq ◽

Chaudry Masood Khalique ◽

Fazal M. Mahomed

Keyword(s):

Dynamical Systems ◽

Nonlinear Systems ◽

First Integrals ◽

Lagrangian Approach ◽

Noether Symmetry ◽

Symmetry Condition ◽

Two Dimensional ◽

Physical Systems ◽

First Order

The aim of the present work is to classify the Noether-like operators of two-dimensional physical systems whose dynamics is governed by a pair of Lane-Emden equations. Considering first-order Lagrangians for these systems, we construct corresponding first integrals. It is seen that for a number of forms of arbitrary functions appearing in the set of equations, the Noether-like operators also fulfill the classical Noether symmetry condition for the pairs of real Lagrangians and the generated first integrals are reminiscent of those we obtain from the complex Lagrangian approach. We also investigate the cases in which the underlying systems are reducible via quadrature. We derive some interesting results about the nonlinear systems under consideration and also find that the algebra of Noether-like operators is Abelian in a few cases.

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Exact Solutions and Bifurcations in Invariant Manifolds for a Nonic Derivative Nonlinear Schrödinger Equation

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501364 ◽

2016 ◽

Vol 26 (08) ◽

pp. 1650136 ◽

Cited By ~ 2

Author(s):

Jibin Li

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Bifurcation Theory ◽

Invariant Manifolds ◽

First Integrals ◽

Phase Portraits ◽

Planar Dynamical System ◽

Nonlinear Schrödinger ◽

Derivative Nonlinear Schrödinger Equation ◽

Dimensional Integral

Propagating modes in a class of nonic derivative nonlinear Schrödinger equations incorporating ninth order nonlinearity are investigated by the method of dynamical systems. Because the functions [Formula: see text] and [Formula: see text] in the solutions [Formula: see text], [Formula: see text] satisfy a four-dimensional integral system having two first integrals (i.e. the invariants of motion), a planar dynamical system for the squared wave amplitude [Formula: see text] can be derived in the invariant manifold of the four-dimensional integrable system. By using the bifurcation theory of dynamical systems, under different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions for this planar dynamical system can be given. Therefore, under some parameter conditions, solutions [Formula: see text] and [Formula: see text] can be exactly obtained. Thirty six exact explicit solutions of equation are derived.

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