Xiao-Song Yang

This paper investigates the stability and Turing instability of a migration-influenced two-species model. Based on the spectral analysis, we first consider how the chemotaxis affect the stability and instability of the linearized problem. Then, taking advantage of the ellipticity of the equation, using higher-order energy estimate, and imposing higher regularity constraints on the initial values, we further study the stability and instability for the nonlinear system in any dimensional space. Our theory not only eliminates the dimension limitation of the original theory (n ≤ 3), but also supplements the deficiency of the theory in the case of parabolic-elliptic coupling. Finally, we derive the amplitude equation by weakly nonlinear analysis, and verify the theoretical results by numerical simulation, revealing the transition between supercritical and subcritical bifurcation of competitive asymmetric regulation, and observing nonlinear phenomena such as hysteresis loop and bistable state.

In this paper, we revisit the well-known perturbed Duffing system and investigate its chaotic dynamics by means of numerical Runge--Kutta method based on topological horseshoe theory. Precisely, we investigate chaos through the topological horseshoes associated with the first, second, and third return maps, obtained by varying the amplitude of an external force term while keeping all other parameters fixed. Our new finding demonstrates that, when the force amplitude exceeds a certain value, the topological (Smale) horseshoe degenerates into a pseudo-horseshoe, while chaotic invariant set persists. This phenomenon indicates that the lower bound of the topological entropy decreases as the force amplitude increases, thereby enriching the dynamics in the perturbed Duffing system. Furthermore, we identify a critical value of the force amplitude governing the attractivity of the chaotic invariant set. For amplitudes slightly below this value, the basin of attraction of the chaotic invariant set progressively shrinks as the amplitude increases. In contrast, for larger amplitudes, both Lyapunov exponents become negative while the topological horseshoe persists, suggesting that the chaotic invariant set loses attractivity as the amplitude grows.

In this paper we present a mathematical framework on linking of embeddings of compact topological spaces into Euclidean spaces and separability of linked embeddings under a specific class of dimension reduction maps. As applications of the established theory, we provide some fascinating insights into classification and approximation problems in deep learning theory in the setting of deep neural networks.

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.

Systems with saturation nonlinearities are frequently encountered in engineering problems. In this paper, the dynamics of the three-dimensional saturated system [ ] is studied. We find rich dynamics in this system through numerical simulations, including chaotic attractors and transient chaos. We conduct both theoretical analysis and numerical analysis of this system. Numerical results suggest that there may exist a heteroclinic network among periodic orbits, which gives rise to chaos. Furthermore, based on the theory of topological horseshoes and by numerical methods, we find that for certain parameters there exist topological horseshoes under the fourth Poincaré return map in this system, thereby demonstrating the existence of chaos. Our study reveals that a simple saturation system can exhibit chaos, deepening our understanding of the influence of saturation on system dynamics.

This paper presents a Riemannian metric-based model to solve the optimal path planning problem on two-dimensional smooth submanifolds in high-dimensional space. Our model is based on constructing a new Riemannian metric on a two-dimensional projection plane, which is induced by the high-dimensional Euclidean metric on two-dimensional smooth submanifold and reflects the environmental information of the robot. The optimal path planning problem in high-dimensional space is therefore transformed into a geometric problem on the two-dimensional plane with new Riemannian metric. Based on the new Riemannian metric, we proposed an incremental algorithm RRT*-R on the projection plane. The experimental results show that the proposed algorithm is suitable for scenarios with uneven fields in multiple dimensions. The proposed algorithm can help the robot to effectively avoid areas with drastic changes in height, ground resistance and other environmental factors. More importantly, the RRT*-R algorithm shows better smoothness and optimization properties compared with the original RRT* algorithm using Euclidean distance in high-dimensional workspace. The length of the entire path by RRT*-R is a good approximation of the theoretical minimum geodesic distance on projection plane.

The Dijkstra algorithm is a classic path planning method, which operates in a discrete graph space to determine the shortest path from a specified source point to a target node or all other nodes based on non-negative edge weights. Numerous studies have focused on the Dijkstra algorithm due to its potential application. However, its application in surface path planning for mobile robots remains largely unexplored. In this letter, a surface optimal path planning algorithm called RM-Dijkstra is proposed, which is based on Riemannian metric model. By constructing a new Riemannian metric on the 2D projection plane, the surface optimal path planning problem is therefore transformed into a geometric problem on the 2D plane with new Riemannian metric. Induced by the standard Euclidean metric on surface, the constructed new metric reflects environmental information of the robot and ensures that the projection map is an isometric immersion. By conducting a series of simulation tests, the experimental results demonstrate that the RM-Dijkstra algorithm not only effectively solves the optimal path planning problem on surfaces, but also outperforms traditional path planning algorithms in terms of path accuracy and smoothness, particularly in complex scenarios.

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.