Liapunov Stability Versus Jacobi Stability

Abolghasem, Hossein

doi:10.1080/1726037x.2012.10698604

Cited by 26 publications

(40 citation statements)

References 12 publications

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“…In order to study of the geometrical properties associated to the dynamical system defined by Eqs. (19), on the base manifold M we introduce first the nonlinear connection N , with coefficients N i j , defined as [25]…”

Section: A the Geometry Of Arbitrary Dynamical Systemsmentioning

confidence: 99%

“…From a physical point of view, and in the framework of classical mechanics, the first KCC invariant ǫ i may be interpreted as an external force. As a next step in our study we vary the trajectories x i (t) of the system (19) into nearby ones according to the rules…”

Section: A the Geometry Of Arbitrary Dynamical Systemsmentioning

confidence: 99%

“…(24) into Eqs. (19), and by taking the limit η → 0, we obtain the so-called deviation, or Jacobi, equations as given by [10][11][12][13]…”

Section: A the Geometry Of Arbitrary Dynamical Systemsmentioning

confidence: 99%

“…In many scientific applications, including the study of the stability of physical, chemical, biological or engineering systems, the study of the behavior of the trajectories of the dynamical system given by Eqs. (19) in the vicinity of a point x i (t 0 ) is of extreme importance. In the following for simplicity we take the origin of the time coordinate at t 0 = 0.…”

Section: Jacobi Stability Of Dynamical Systemsmentioning

confidence: 99%

“…In the following for simplicity we take the origin of the time coordinate at t 0 = 0. We interpret the trajectories x i = x i (t) of the system of equations (19) as curves in the Euclidean space (R n , ., . ), where ., .…”

Section: Jacobi Stability Of Dynamical Systemsmentioning

confidence: 99%

See 4 more Smart Citations

Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

Harkó

Pantaragphong

Sabau

2016

Int. J. Geom. Methods Mod. Phys.

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The Kosambi-Cartan-Chern (KCC) theory represents a powerful mathematical method for the investigation of the properties of dynamical systems. The KCC theory introduces a geometric description of the time evolution of a dynamical system, with the solution curves of the dynamical system described by methods inspired by the theory of geodesics in a Finsler spaces. The evolution of a dynamical system is geometrized by introducing a non-linear connection, which allows the construction of the KCC covariant derivative, and of the deviation curvature tensor. In the KCC theory the properties of any dynamical system are described in terms of five geometrical invariants, with the second one giving the Jacobi stability of the system. Usually, the KCC theory is formulated by reducing the dynamical evolution equations to a set of second order differential equations. In the present paper we introduce and develop the KCC approach for dynamical systems described by systems of arbitrary n-dimensional first order differential equations. We investigate in detail the properties of the n-dimensional autonomous dynamical systems, as well as the relationship between the linear stability and the Jacobi stability. As a main result we find that only even-dimensional dynamical systems can exhibit both Jacobi stability and instability behaviors, while odd-dimensional dynamical systems are always Jacobi unstable, no matter their Lyapunov stability. As applications of the developed formalism we consider the geometrization and the study of the Jacobi stability of the complex dynamical networks, and of the Λ-Cold Dark Matter cosmological models, respectively.

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Section: A the Geometry Of Arbitrary Dynamical Systemsmentioning

confidence: 99%

Section: A the Geometry Of Arbitrary Dynamical Systemsmentioning

confidence: 99%

“…(24) into Eqs. (19), and by taking the limit η → 0, we obtain the so-called deviation, or Jacobi, equations as given by [10][11][12][13]…”

Section: A the Geometry Of Arbitrary Dynamical Systemsmentioning

confidence: 99%

Section: Jacobi Stability Of Dynamical Systemsmentioning

confidence: 99%

Section: Jacobi Stability Of Dynamical Systemsmentioning

confidence: 99%

See 3 more Smart Citations

Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

Harkó

Pantaragphong

Sabau

2016

Int. J. Geom. Methods Mod. Phys.

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

Chen

Liu

Wei

et al. 2020

Math Methods in App Sciences

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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.

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Qualitative geometric analysis of traveling wave solutions of the modified equal width Burgers equation

Liu

Chen

Huang

et al. 2022

Math Methods in App Sciences

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This paper devotes to the qualitative geometric analysis of the traveling wave solutions of MEW-Burgers wave equation. Firstly, MEW-Burgers equation is transformed into an equivalent planar dynamical system by using traveling wave transformation. Then the global structure of the planar system is presented, and solitary waves, kink waves (anti-kink waves), and periodic waves are found. Secondly, Jacobi stability for the planar system is studied based on KCC theory, and Jacobi stability of any point on the trajectory of system is dicussed. The dynamical behavior of the deviation vector near the equilibrium points is analyzed, and the numerical simulation is consistent with the theoretical analysis. Lyapunov stability and Jacobi stability of the equilibrium points are also compared and analyzed. The obtaining results show that Lyapunov stability of the equilibrium points of the system is not exactly consistent with Jacobi stability. Finally, the planar system with periodic disturbance is transformed into a six dimensional nonlinear system, and the periodic, quasi-periodic, and chaotic dynamical behaviors of the system are numerically simulated.

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Liapunov Stability Versus Jacobi Stability

Cited by 26 publications

References 12 publications

Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

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

Qualitative geometric analysis of traveling wave solutions of the modified equal width Burgers equation

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