Bifurcation, chaos, and their control in a wheelset model

Abstract

In this paper, we present an improved wheelset motion model with two degrees of freedom and study the dynamic behaviors of the system including the symmetry, the existence and uniqueness of the solution, continuous dependence on initial conditions, and Hopf bifurcation. The dynamic characteristics of the wheelset motion system under a nonholonomic constraint are investigated. These results generalize and improve some known results about the wheelset motion system. Meanwhile, based on multiple equilibrium analysis, calculation of Lyapunov exponents and Poincaré section, the chaotic behaviors of the wheelset system are discussed, which indicates that there are more complex dynamic behaviors in the railway wheelset system with higher order terms of Taylor series of trigonometric functions. This paper has also realized the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control. The results show that the chaotic wheelset system and bifurcation wheelset system are all well controlled, whether by controlling the yaw angle and the lateral displacement or only by controlling the yaw angle. Numerical simulations are carried out to further verify theoretical analyses.

Bifurcation, geometric constraint, chaos, and its control in a railway wheelset system

“…For the above two cases, the behaviors of the state (𝑦, p, 𝜑, q) of ( 11) with time are shown in Figures 21 and 22, respectively. It indicates that the controlled chaotic system (11) is asymptotically stable at zero-equilibrium point by selecting appropriate feedback coefficients.…”

“…Therefore, we assume k 𝑗 = 10 ( 𝑗 = 1, 2, 3, 4), (11) has only one equilibrium point (0, 0, 0, 0), and the corresponding eigenvalues are (0, 0, 0, 0) is asymptotically stable. Similarly, for the parameters selected in (8) and k 𝑗 = 20 ( 𝑗 = 1, 2, 3, 4), (11) has only one equilibrium point (0, 0, 0, 0), and the corresponding eigenvalues are (0, 0, 0, 0) is asymptotically stable. For the above two cases, the behaviors of the state (𝑦, p, 𝜑, q) of ( 11) with time are shown in Figures 21 and 22, respectively.…”

“…For the parameters selected in (7) and k 𝑗 = k ( 𝑗 = 1, 2, 3, 4), zero-equilibrium point is asymptotically stable when k > 2.73749. Therefore, we assume k 𝑗 = 10 ( 𝑗 = 1, 2, 3, 4), (11) has only one equilibrium point (0, 0, 0, 0), and the corresponding eigenvalues are (0, 0, 0, 0) is asymptotically stable. Similarly, for the parameters selected in (8) and k 𝑗 = 20 ( 𝑗 = 1, 2, 3, 4), (11) has only one equilibrium point (0, 0, 0, 0), and the corresponding eigenvalues are (0, 0, 0, 0) is asymptotically stable.…”

“…Miao et al 10 investigated the generalized Hopf bifurcation of a four‐dimension non‐smooth railway wheelset system. Li et al 11 investigated the bifurcation and chaotic behaviors of a wheelset system and discussed the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control method. Meanwhile, the dynamical behaviors with velocity constraint were also studied.…”

Local Dynamics of a New Four-Dimensional Quadratic Autonomous System

“…Efran and Manuel [12] addressed the problem of a robust tracking, surveillance and landing of a mobile ground target by Hopf bifurcation. Li et al [13] studied an improved wheelset motion model with two degrees of freedom through Hopf bifurcation method. By analyzing the existence of Hopf bifurcation, Wang et al [14] studied a delayed diffusive predator-prey model with predator interference or foraging facilitation.…”