  Cellular migration is crucial for biological processes including embryonic
development, immune response, and wound healing. The myosin-clutch model is a
framework that describes how cells control migration through the interactions
between myosin, the clutch mechanism, and the substrate. This model is related
to how cells regulate adhesion, generate traction forces, and move on compliant
substrates. In this study, we present a five-dimensional nonlinear autonomous
system to investigate the influences of myosin, clutches, substrate, and
external load on the system's stability. Moreover, we analyze the effects of
various parameters on fixed points and explore the frequency and amplitude of
the limit cycle associated with oscillations. We discovered that the system
demonstrates oscillatory behavior when the velocity of the myosin motor is
relatively low, or when the ratio of the motor attachment rate to motor
detachment rate is relatively high. The external load shares a fraction of the
force exerted by myosin motors, thereby diminishing the force endured by the
clutches. Within a specific range, an increase in external load not only
diminishes and eventually eliminates the region lacking fixed points but also
decelerates clutch detachment, enhancing clutch protein adherence.
