Jacobi Stability of Hamiltonian Systems

Abolghasem, Hossein

doi:10.12732/ijpam.v87i1.11

Cited by 34 publications

(22 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…44,45 In other words, the Jacobi unstable trajectories of a dynamical system behave chaotically, because it is impossible to distinguish the trajectories that are very close at the initial moment after a finite time interval. 43 The Jacobi stability of some typical dynamical systems (Lorenz system, 46 Rössler system, 47 Chen system, 48 Chua circuit system, 49 Navier-Stokes system, 50 Rikitake system, 51 and others (see, e.g.,other studies [52][53][54][55] as well as their references) has been studied. They qualitatively described the chaotic evolution of the dynamical systems by analyzing the dynamics of the deviation vector.…”

Section: Figurementioning

confidence: 99%

New insights into a chaotic system with only a Lyapunov stable equilibrium

Chen

Liu

Wei

et al. 2020

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper gives some new insights into a chaotic system. The considered system has only one Lyapunov stable equilibrium and positive Lyapunov exponent in some certain parameter range, which means there coexists chaotic attractors and Lyapunov stable equibirium in system. First, from the perspective of analyzing the global structure of the system, based on the Poincaré compactification technique, the complete description of its dynamical behavior on the sphere at infinity is presented. The obtaining results show that its global dynamical behavior is very complex. Second, from a geometric perspective, the Jacobi stability of the system is investigated, including the unique equilibrium and a periodic orbit. Interestingly, we find that the unique Lyapunov stable equilibrium is always Jacobi unstable, a Lyapunov stable periodic orbit falls into both Jacobi stable and Jacobi unstable regions. It is shown that we might witness chaotic behavior of the system before the trajectories enter a neighborhood of the equilibrium point. It is hoped that the investigation of this paper can contribute to better understand and study the complex chaotic system with stable equilibria.

show abstract

Section: Figurementioning

confidence: 99%

New insights into a chaotic system with only a Lyapunov stable equilibrium

Chen

Liu

Wei

et al. 2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Jacobi stability is a natural generalization of the stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting [12]. The Jacobi stability conditions of non-degenerate equilibrium points are the same as the known conditions for Lyapunov stability [1,2,3]. When the potential surface of Lyapunov stability can be defined, the deviation curvature related to Jacobi stability corresponds to the Willmore energy density of the potential surface [30,28,29].…”

Section: Stability Analysis In Kcc Theorymentioning

confidence: 99%

“…This approach allows the evolution of a dynamic system to be described in geometric terms, by considering it as a geodesic in a Finsler space (e.g., [23,12,7]). From the geodesic equation, we can obtain the covariant form of the variational equations for the deviation of the whole trajectory to nearby ones, so the KCC theory represents a powerful mathematical method for the analysis of dynamic systems (e.g., [1,3,8,12,15]).…”

Section: Introductionmentioning

confidence: 99%

Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory

Yamasaki

Yajima

2016

Journal of Dynamical Systems and Geometric Theories

View full text Add to dashboard Cite

We considered the differential geometric structure of non-equilibrium dynamics in non-linear interactions, such as competition and predation, based on Kosambi-Cartan-Chern (KCC) theory. The stability of a geodesic flow on a Finslerian manifold is characterized by the deviation curvature (the second invariant in the dynamical system). According to KCC theory, the value of the deviation curvature is constant around the equilibrium point. However, in the non-equilibrium region, not only the value but also the sign of the deviation curvature depend on time. Next, we reapplied KCC theory to the dynamics of the deviation curvature and determined the hierarchical structure of the geometric stability. The dynamics of the deviation curvature in the nonequilibrium region is accompanied by a complex periodic (node) pattern in the predation (competition) system.

show abstract

“…Later, this system was named the Yang-Chen system [39,29]. On the one hand, when (a, b, c) = (35,3,35), system has one positive Lyapunov Exponent λ L1 = 1.0742, which implies that system (1) yields a chaotic attractor, as shown in Fig. 1.…”

mentioning

confidence: 99%

“…Nowadays, Jacobi stability analysis has become a useful tool in the study of the complexity of typical chaotic systems. This type of geometric analysis concerning many chaotic systems, such as Lorenz system, Chen system, Rössler system, Hamiltonian system, modified Chua circuit system, Rikitake system, Navier-Stokes system, Rabinovich-Fabrikant system, an unusual Lorenz-like chaotic system, can be found in the recent literature [18,24,3,19,47,20,50,26,21,16]. Research has shown that Jacobi instability can be used to interpret the nature of chaos.…”

mentioning

confidence: 99%

Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system

Liu¹,

Huang²,

Wei³

2021

DCDS-B

View full text Add to dashboard Cite

The present work is devoted to giving new insights into a chaotic system with two stable node-foci, which is named Yang-Chen system. Firstly, based on the global view of the influence of equilibrium point on the complexity of the system, the dynamic behavior of the system at infinity is analyzed. Secondly, the Jacobi stability of the trajectories for the system is discussed from the viewpoint of Kosambi-Cartan-Chern theory (KCC-theory). The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The obtained results show that in the sense of Jacobi stability, all equilibrium points of the system, including those of the two linear stable node-foci, are Jacobi unstable. These studies show that one might witness chaotic behavior of the system trajectories before they enter in a neighborhood of equilibrium point or periodic orbit. There exists a sort of stability artifact that cannot be found without using the powerful method of Jacobi stability analysis.

show abstract

Jacobi Stability of Hamiltonian Systems

Cited by 34 publications

References 10 publications

New insights into a chaotic system with only a Lyapunov stable equilibrium

New insights into a chaotic system with only a Lyapunov stable equilibrium

Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory

Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system

Contact Info

Product

Resources

About