In this paper, the dynamics of spring-block models are studied. A brief overview of the history of spring-block models relating to earthquakes is presented, along with the development of friction laws. Several mathematical topics relating to dynamical systems are also discussed. We consider two spring-block models; one with Dieterich-Ruina rate and state dependent friction and another with a modified Dieterich-Ruina style friction. For each system, the qualitative behavior and numerical solutions are presented. In the first case, we find that the system undergoes a Hopf bifurcation from a stationary solution to a periodic orbit, and eventually transitions to chaos. In the latter case, we find that a stationary solution exists along with the conditions for a Hopf bifurcation to occur. We develop the mathematical framework to compute periodic and chaotic behavior for the system. Future work will be to develop more efficient algorithms to perform the actual computations.


Brian Martensen

First Committee Member

Ruijun Zhao

Second Committee Member

Bryce Hoppie

Date of Degree




Document Type



Master of Science (MS)


Mathematics and Statistics


Science, Engineering and Technology

Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License



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