scholarly journals Orbital dynamics on invariant sets of contact Hamiltonian systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qihuai Liu ◽  
Pedro J. Torres
Keyword(s):  
Hamiltonian System ◽  
Equilibrium Points ◽  
Differential Invariant ◽  
Domain Of Attraction ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Zero Energy ◽  
Contact Phase ◽  
Hamiltonian Flows ◽  
Geometric Conditions

<p style='text-indent:20px;'>In this paper, we shall give new insights on dynamics of contact Hamiltonian flows, which are gaining importance in several branches of physics as they model a dissipative behaviour. We divide the contact phase space into three parts, which are corresponding to three differential invariant sets <inline-formula><tex-math id="M1">\begin{document}$ \Omega_\pm, \Omega_0 $\end{document}</tex-math></inline-formula>. On the invariant sets <inline-formula><tex-math id="M2">\begin{document}$ \Omega_\pm $\end{document}</tex-math></inline-formula>, under some geometric conditions, the contact Hamiltonian system is equivalent to a Hamiltonian system via the Hölder transformation. The invariant set <inline-formula><tex-math id="M3">\begin{document}$ \Omega_0 $\end{document}</tex-math></inline-formula> may be composed of several equilibrium points and heteroclinic orbits connecting them, on which contact Hamiltonian system is conservative. Moreover, we have shown that, under general conditions, the zero energy level domain is a domain of attraction. In some cases, such a domain of attraction does not have nontrivial periodic orbits. Some interesting examples are presented.</p>

scholarly journals Global Dynamics of the Hastings-Powell System

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Luis N. Coria
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
State Variables ◽  
System Parameters ◽  
First Order ◽  
Extremum Conditions

This paper studies the problem of bounding a domain that contains all compact invariant sets of the Hastings-Powell system. The results were obtained using the first-order extremum conditions and the iterative theorem to a biologically meaningful model. As a result, we calculate the bounds given by a tetrahedron with excisions, described by several inequalities of the state variables and system parameters. Therefore, a region is identified where all the system dynamics are located, that is, its compact invariant sets: equilibrium points, periodic-homoclinic-heteroclinic orbits, and chaotic attractors. It was also possible to formulate a nonexistence condition of the compact invariant sets. Additionally, numerical simulations provide examples of the calculated boundaries for the chaotic attractors or periodic orbits. The results provide insights regarding the global dynamics of the system.

Dynamical Analysis of Raychaudhuri Equations Based on the Localization Method of Compact Invariant Sets

2014 ◽  
Vol 24 (11) ◽  
pp. 1450136 ◽  
Author(s):  
Alexander P. Krishchenko ◽  
Konstantin E. Starkov
Keyword(s):  
Equilibrium Points ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Dynamical Analysis ◽  
Two Dimensional ◽  
Main Attention ◽  
Raychaudhuri Equations ◽  
Omega Limit Set

In this paper, we examine the localization problem of compact invariant sets of Raychaudhuri equations with nonzero parameters. The main attention is attracted to the localization of periodic/homoclinic orbits and homoclinic cycles: we prove that there are neither periodic/homoclinic orbits nor homoclinic cycles; we find heteroclinic orbits connecting distinct equilibrium points. We describe some unbounded domain such that nonescaping to infinity positive semitrajectories which are contained in this domain have the omega-limit set located in the boundary of this domain. We find a locus of other types of compact invariant sets respecting three-dimensional and two-dimensional invariant planes. Besides, we describe the phase portrait of the system obtained from the Raychaudhuri equations by the restriction on the two-dimensional invariant plane.

More Dynamical Properties Revealed from a 3D Lorenz-like System

2014 ◽  
Vol 24 (10) ◽  
pp. 1450133 ◽  
Author(s):  
Haijun Wang ◽  
Xianyi Li
Keyword(s):  
Numerical Simulations ◽  
Local Stability ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Mathematical Work ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Properties ◽  
Theoretical Results

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means — both "locally" and "globally" and both "finitely" and "infinitely" — to comprehensively explore a given system.

Mathematical Modeling of the λ Switch

2010 ◽  
pp. 573-603
Author(s):  
Dmitriy Laschov ◽  
Michael Margaliot
Keyword(s):  
Mathematical Modeling ◽  
Differential Equation ◽  
Mathematical Model ◽  
Gene Regulation ◽  
Equilibrium Points ◽  
Domain Of Attraction ◽  
Verbal Description ◽  
Quantitative Understanding ◽  
Living Organisms ◽  
Scientific Challenge

Gene regulation plays a central role in the development and functioning of living organisms. Developing a deeper qualitative and quantitative understanding of gene regulation is an important scientific challenge. The Lambda switch is commonly used as a paradigm of gene regulation. Verbal descriptions of the structure and functioning of the Lambda switch have appeared in biological textbooks. We apply fuzzy modeling to transform one such verbal description into a well-defined mathematical model. The resulting model is a piecewise-quadratic, second-order differential equation. It demonstrates functional fidelity with known results while being simple enough to allow a rather detailed analysis. Properties such as the number, location, and domain of attraction of equilibrium points can be studied analytically. Furthermore, the model provides a rigorous explanation for the so-called stability puzzle of the Lambda switch.

scholarly journals Orbital Structure of the Two Fixed Centres Problem

1999 ◽  
Vol 172 ◽  
pp. 463-464
Author(s):  
A. Cordero ◽  
J. Martínez Alfaro ◽  
P. Vindel
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Artificial Satellite ◽  
Equilibrium Points ◽  
Integrable Hamiltonian System ◽  
Three Dimensional ◽  
First Integral ◽  
Material Point ◽  
Dimensional Sphere

The set of orbits of the Two Fixed Centres problem has been known for a long time (Chartier, 1902, 1907; Pars, 1965), since it is an integrable Hamiltonian system.We consider a plane that contains the fixed masses. Denote by φ the angle denned by this plane and the one that contains also the third body. The momentum pφ is a first integral of the system and when pφ is different from zero, the manifold generated by the generalized coordinates and momenta are two copies of the three-dimensional sphere S3. If pφ = 0, that is to say when the planet crosses the line joining both suns, the motion is restricted to a planar one. All the equilibrium points appears in this case and therefore the phase spaces are more complex. We restrict our attention to this case which has two degrees of freedom.It is again a Bott-integrable Hamiltonian system. The set of periodic orbits of this systems can be studied from a subset of them, the Non-Singular Morse-Smale type orbits (see Casasayas, 1992). It is proved in Campos (1997) that a small perturbation of a Bott-integrable Hamiltonian system transforms it into a Non-Singular Morse-Smale system. The NMS periodic orbits belong to both the NMS system and the Hamiltonian one. Moreover, The NMS p.o. can be continued to nearly Hamiltonian systems. For instance, in our case to the Restricted Three Body Problem and in the study of the motion of a material point moving inside the gravitational field generated by two stars. This approximation is also useful when the motion of an artificial satellite around a spheroidal body is considered.

scholarly journals Author’s reply to: Comments on “Asymptotically stable equilibrium points in new chaotic systems”

Nova Scientia ◽  
2017 ◽  
Vol 9 (19) ◽  
pp. 906-909
Author(s):  
K. Casas-García ◽  
L. A. Quezada-Téllez ◽  
S. Carrillo-Moreno ◽  
J. J. Flores-Godoy ◽  
Guillermo Fernández-Anaya
Keyword(s):  
Dynamic Systems ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Heteroclinic Orbits ◽  
Stable Equilibrium Point ◽  
Asymptotically Stable ◽  
Linear Quadratic ◽  
The Monte Carlo Method

Since theorem 1 of (Elhadj and Sprott, 2012) is incorrect, some of the systems found in the article (Casas-García et al. 2016) may have homoclinic or heteroclinic orbits and may seem chaos in the Shilnikov sense. However, the fundamental contribution of our paper was to find ten simple, three-dimensional dynamic systems with non-linear quadratic terms that have an asymptotically stable equilibrium point and are chaotic, which was achieved. These were obtained using the Monte Carlo method applied specifically for the search of these systems.

On the Synthesis of Brain-State-in-a-Box Neural Models with Application to Associative Memory

2000 ◽  
Vol 12 (2) ◽  
pp. 451-472 ◽  
Author(s):  
Fation Sevrani ◽  
Kennichi Abe
Keyword(s):  
Neural Networks ◽  
Associative Memory ◽  
Equilibrium Points ◽  
Domain Of Attraction ◽  
Neural Model ◽  
Brain State ◽  
Qualitative Properties ◽  
The Stability ◽  
The Brain ◽  
Insight Into

In this article we present techniques for designing associative memories to be implemented by a class of synchronous discrete-time neural networks based on a generalization of the brain-state-in-a-box neural model. First, we address the local qualitative properties and global qualitative aspects of the class of neural networks considered. Our approach to the stability analysis of the equilibrium points of the network gives insight into the extent of the domain of attraction for the patterns to be stored as asymptotically stable equilibrium points and is useful in the analysis of the retrieval performance of the network and also for design purposes. By making use of the analysis results as constraints, the design for associative memory is performed by solving a constraint optimization problem whereby each of the stored patterns is guaranteed a substantial domain of attraction. The performance of the designed network is illustrated by means of three specific examples.

Algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations via Hamiltonian approach

2015 ◽  
Vol 12 (03) ◽  
pp. 1550038
Author(s):  
Dianlou Du ◽  
Xiao Yang
Keyword(s):  
Hamiltonian System ◽  
Nonlinear Equations ◽  
Poisson Structure ◽  
Hamiltonian Approach ◽  
Nonlinear Schrödinger ◽  
Hamiltonian Flows ◽  
Jacobi Inversion ◽  
Liouville Integrable ◽  
Separated Variables

The algebraic-geometrical solutions of three (2 + 1)-dimensional equations (including mKP equation and coupled mKP equation) are discussed by Hamiltonian approach. First, the Poisson structure on CN × RN is introduced to give a Hamiltonian system associated with the derivative nonlinear Schrödinger (DNLS) hierarchy. The Hamiltonian system is proved to be Liouville integrable, accordingly the solutions of three (2 + 1)-dimensional nonlinear equations can be solved by three compatible Hamiltonian flows. Second, the canonical separated variables and Hamilton–Jacobi theory is used to definite action-angle variables for Hamiltonian flows. At last, by Riemann–Jacobi inversion, the algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations are obtained. Besides, the algebraic-geometrical solutions of the first two DNLS equations are also given.

LOCALIZING BOUNDS FOR COMPACT INVARIANT SETS OF NONLINEAR SYSTEMS POSSESSING FIRST INTEGRALS WITH APPLICATIONS TO HAMILTONIAN SYSTEMS

2010 ◽  
Vol 20 (05) ◽  
pp. 1477-1483 ◽  
Author(s):  
KONSTANTIN E. STARKOV
Keyword(s):  
Hamiltonian System ◽  
Nonlinear Systems ◽  
Hamiltonian Systems ◽  
Positive Definite ◽  
First Integrals ◽  
Invariant Sets ◽  
Yang Mills ◽  
First Order ◽  
Special Case ◽  
Extremum Conditions

In this paper, we study the localization problem of compact invariant sets of nonlinear systems possessing first integrals by using the first order extremum conditions and positive definite polynomials. In the case of natural polynomial Hamiltonian systems, our results include those in [Starkov, 2008] as a special case. This paper discusses the application to studies of the generalized Yang–Mills Hamiltonian system and the Hamiltonian system describing dynamics of hydrogenic atoms in external fields.

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