Chaotic Dynamics in a Class of Delay Controlled Partial Difference Equations

Zhang, Xuanxuan; Liang, Wei; Zhang, Yongjun

doi:10.1007/s10883-023-09675-9

Cited by 1 publication

(2 citation statements)

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“…When the system (2) describes the geodesic equations in either Riemann or Finsler geometry, Equation ( 6) is the Jacobi field equation. The tensor P i j is the second KCC invariant, that is, the deviation curvature tensor of system (2). It serves as the essential quantity in both the KCC theory and the Jacobi stability analysis.…”

Section: Kcc Geometric Theorymentioning

confidence: 99%

“…One of the key contributors to the development of nonlinear dynamics is Edward Lorenz. In the 1960s, Lorenz made a ground-breaking discovery known as the Lorenz system, which exhibits the butterfly effect, attractor coexistence and intransitivity [1,2]. He demonstrated that even tiny changes in the initial conditions of a dynamic system could have large-scale effects on its long-term behavior.…”

Section: Introductionmentioning

confidence: 99%

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Jacobi Stability Analysis of Liu System: Detecting Chaos

Liu,

Zhang

2024

Mathematics

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By utilizing the Kosambi–Cartan–Chern (KCC) geometric theory, this paper is dedicated to providing novel insights into the Liu dynamical system, which stands out as one of the most distinctive and noteworthy nonlinear dynamical systems. Firstly, five important geometrical invariants of the system are obtained by associating the nonlinear connection with the Berwald connection. Secondly, in terms of the eigenvalues of the deviation curvature tensor, the Jacobi stability of the Liu dynamical system at fixed points is investigated, which indicates that three fixed points are Jacobi unstable. The Jacobi stability of the system is analyzed and compared with that of Lyapunov stability. Lastly, the dynamical behavior of components of the deviation vector is studied, which serves to geometrically delineate the chaotic behavior of the system near the origin. The onset of chaos for the Liu dynamical system is obtained. This work provides an analysis of the Jacobi stability of the Liu dynamical system, serving as a useful reference for future chaotic system research.

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Section: Kcc Geometric Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%