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New walking dynamics in the simplest passive bipedal walking model. (English) Zbl 1254.70020
Summary: We revisit the simplest passive walking model by M. Garcia et al. “The simplest walking model: stability, complexity, and scaling”, J. Biomech. Eng. Trans. ASME 120, 281–288 (1998)] which displays chaos through period doubling from a stable period-1 gait. By carefully numerical studies, two new gaits with period-3 and -4 are found, whose stability is verified by estimates of eigenvalues of the corresponding Jacobian matrices. A surprising phenomenon uncovered here is that they both lead to higher periodic cycles and chaos via period doubling. To study the three different types of chaotic gaits rigorously, the existence of horseshoes is verified and estimates of the topological entropies are made by computer-assisted proofs in terms of topological horseshoe theory.

MSC:
70E17 Motion of a rigid body with a fixed point
34C23 Bifurcation theory for ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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