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A four-dimensional parameter model that captures the essence of spring-block propagation with rate- and state- friction laws was stated. In order to establish the existence of waves phenomena for all velocity-dependent critical value of stiffness and stick-slip amplitude for stiffness variations, we derived the modulation equation for our model. Conditions under which the amplitude of the modulation wave equation produces many small solutions near the instability threshold for finitely large enough time scales were established.

MODULATION WAVE EQUATION FOR SPRING-BLOCK ORBITS WITH NONLINEAR RATE AND STATE FRICTION