Symmetry in Nonlinear Dynamics and Chaos

Self Cite

The traditional theory of chaotic intermittency developed for return maps hypothesizes a uniform density of reinjected points from the chaotic zone to the laminar one. In the past few years, we have described how the reinjection probability density function (RPD) can be generalized as a power law function. Here, we introduce a broad and general analytical approach to determine the RPD function and other statistical variables, such as the characteristic relation traditionally utilized to characterize the chaotic intermittency type. The proposed theoretical methodology is simple to implement and includes previous studies as particular cases. It is compared with numerical data, the M function methodology, and the Perron–Frobenius technique, showing high accuracy between them.

Section: Introductionmentioning

confidence: 99%

Theoretical Evaluation of the Reinjection Probability Density Function in Chaotic Intermittency

Elaskar

Río

2023

Self Cite

On New Symmetric Fractional Discrete-Time Systems: Chaos, Complexity, and Control

Hammad,

Diabi,

Dababneh

et al. 2024

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the C0 algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map’s states in commensurate and incommensurate cases.

Asymmetry and Symmetry in New Three-Dimensional Chaotic Map with Commensurate and Incommensurate Fractional Orders

Al-Taani,

Abu Hammad,

Abudayah

et al. 2024

According to recent research, discrete-time fractional-order models have greater potential to investigate behaviors, and chaotic maps with fractional derivative values exhibit rich dynamics. This manuscript studies the dynamics of a new fractional chaotic map-based three functions. We analyze the behaviors in commensurate and incommensurate orders, revealing their impact on dynamics. Through the maximum Lyapunov exponent (LEmax), phase portraits, and bifurcation charts. In addition, we assess the complexity and confirm the chaotic features in the map using the approximation entropy ApEn and C0 complexity. Studies show that the commensurate and incommensurate derivative values influence the fractional chaotic map-based three functions, which exhibit a variety of dynamical behaviors, such as hidden attractors, asymmetry, and symmetry. Moreover, the new system’s stabilizing employing a 3D nonlinear controller is introduced. Finally, our study validates the research results using the simulation MATLAB R2024a.