Junfeng Cheng

In this paper, we revisit the classical perturbed Duffing system and investigate its intricate dynamical behavior through the numerical method based on the topological horseshoe theory employing the Runge–Kutta method. Based on the classical Melnikov analysis, we explore the persistence of chaotic dynamics beyond the parameter regimes in which the Melnikov condition guarantees the existence of a transverse homoclinic intersection. Specifically, we examine the second return map and demonstrate the existence of a topological horseshoe at parameter values εγ=0.4, εδ=0.54, and ω=1. This provides numerical evidence to Smale horseshoe-type chaos in a regime where the Melnikov criterion is not satisfied. Furthermore, we provide a more rigorous treatment on the existence of the topological horseshoe in terms of crossing stability.

Coverage path planning is an algorithm that generates efficient robot trajectories to ensure complete coverage of a designated area. It is widely used in applications such as cleaning, surveillance, and exploration of unknown environments. The chaotic mobile robot, driven by a controller with chaotic dynamics, ensures full workspace coverage without a map or a global motion plan. In this paper, different variables of the Chen system and the Lorenz system are used to generate the chaotic motion of the robot. By comparing the trajectory coverage rate of the three variables [ ] in the Chen system under the same initial conditions and running time, it is found that the third variable [ ] of the Chen system performs the best in bounded workspace and is better than those in the Lorenz system. Furthermore, in order to illustrate the advantages of chaotic path planning, we consider several more complex robot workspaces: the case of a bounded workspace with obstacles and the case of various nonconvex bounded workspaces. The simulation results show that the robot trajectory corresponding to the [ ]-variable of the Chen system not only maintains the highest coverage rate but also can successfully pass through narrow channels in nonconvex areas for comprehensive retrieval. Finally, the relationship between the qualitative properties of the two chaotic systems and the coverage rate of the chaotic robot’s trajectory is discussed in detail.

Time-series analysis plays a crucial role in understanding the dynamics of real-world systems across various scientific and engineering disciplines. We in this paper propose a novel approach to identifying chaotic dynamics by a geometric method based on deep learning. Specifically, we construct a map from the observed time-series data and seek the existence of a topological horseshoe in the map, which indicates chaotic behavior. We demonstrate the effectiveness of our method by numerical experiments on the Hénon map, the Lorenz system, and the Duffing system. The results show that the topological horseshoe theory combined with deep neural works provides a valuable tool for detection of chaos in complex nonlinear systems from time series.

In this paper, we explore the rich dynamics in the Chen system using Runge–Kutta method and Deep Neural Networks (DNNs). Compared with the Lorenz system, we find that the first return map [ ] of the Chen system exhibits a more complex structure of continuous regions in the Poincaré section. Moreover, the existence of six crossing blocks with respect to the second return map implies that there is a closed invariant set [ ] in the Poincaré section such that [ ] is semi-conjugate to a 6-shift map, thus demonstrating remarkably chaos in the Chen system.

Deep Neural Networks (DNNs) have been successfully applied to investigations of numerical dynamics of finite-dimensional nonlinear systems such as ODEs instead of finding numerical solutions to ODEs via the traditional Runge–Kutta method and its variants. To show the advantages of DNNs, in this paper, we demonstrate that the DNNs are more efficient in finding topological horseshoes in chaotic dynamical systems.