Complex Dynamical Behaviors in a Spring-Block Model with Periodic Perturbation

Cun Chen,Jingli Ren,Xueping Li

doi:10.1155/2019/5253496

Cun Chen, Jingli Ren + Show 1 more

Open Access

https://doi.org/10.1155/2019/5253496

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Abstract

A generalization of a Burridge-Knopoff spring-block model is investigated to illustrate the dynamics of transform faults. The model can undergo Hopf bifurcation and fold bifurcation of limit cycles. Considering the cyclical nature of the spring stiffness, the model with periodic perturbation is further explored via a continuation technique and numerical bifurcation analysis. It is shown that the periodic perturbation induces abundant dynamics, the existence, the switch, and the coexistence of multiple attractors including periodic solutions with various periods, quasiperiodic solutions, chaotic solutions through torus destruction, or cascade of period doublings. Throughout the results obtained, one can see that the system manifests complex dynamical behaviors such as chaos, self-organized criticality, and the transition of dynamical behaviors when it comes to periodic perturbations. Even very small variation of a parameter can result in radical changes of the dynamics, which provides a new insight into the fault dynamics.

Highlights

The system contains a series of spring-blocks manifests complicated nonlinear dynamic behavior which can be related to the phenomenon in earthquake
After traveling wave transformation and nondimensionalize, we introduce the perturbation with time dependent, which will present that even a small oscillation in the spring stiffness can sufficiently change the original behavior, leading to the burst of chaos or other complex dynamical behaviors
To further investigate the influence of a periodic perturbation on the spring-block system, we present the numerical simulation of the discrete system (2)

Summary

Introduction

The system contains a series of spring-blocks manifests complicated nonlinear dynamic behavior which can be related to the phenomenon in earthquake. Afterwards, in 1989, based on the Burridge-Knopoff model, Carlson and Langer further presented another version of BK model for fault dynamics consisting of a uniform chain of blocks and springs pulled slowly across a rough surface [7] This model can give rise to events of various sizes, and the numerical evaluation of the distribution of earthquake magnitudes results in a power-law spectrum similar to the earthquake in nature. Time delay is introduced in one-block system transforming the system into infinite-dimensional, which makes it possible for the occurrence of deterministic chaos [11] When it comes to periodic parameter perturbations, Kosticet al. We explore a model with periodic perturbation which can give rise to much more complex dynamical behavior in the spring-block system

The Model and Bifurcations

The Model with Periodic Perturbation

Complex Dynamic Behaviors

Conclusion

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