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Optimal Oscillations and Chaos Generation in Biologically-Inspired Systems


Biological systems display a variety of complex dynamic behaviors, ranging from periodic orbits to chaos. Regular rhythmic behavior, for instance, is associated with locomotion, while chaotic behavior is observed in neural interactions. Both these cases can be mathematically expressed as the interaction of a collection of coupled bodies or oscillators that are actuated to behave with a desired pattern. In animal locomotion, this desired pattern is the periodic body motion (gait) that interacts with the environment to generate thrust for motion. By contrast, the observed behavior of a network of neurons is possibly chaotic and flexible. This research focuses on the design and analysis of these two types of behaviors in biologically-inspired systems.

A fundamental problem in animal locomotion is determining a gait that optimizes an essential performance while satisfying a desired velocity constraint. In this study, a functional model is developed for a general class of three dimensional locomotors with full (six) degrees of freedom, in addition to arbitrary finite degrees of freedom for body shape deformation. An optimal turning gait problem is then formulated for a periodic body movement that minimizes a quadratic cost function while achieving a steady turning motion with prescribed average linear and angular velocities. The problem is shown to reduce equivalently to two separate and simpler minimization problems that are both solvable for globally optimal solutions.

Optimal gait theory can also be utilized in order to determine analytical justifications for observed behavior in biological systems. In this study, a simple body-fluid fish model is developed, and steady swimming at various speeds is analyzed using optimal gait theory. The results show that the gait that minimizes bending moment over tail movements and stiffness matches data from observed swimming of saithe. Furthermore, muscle tension is reduced when undulation frequency matches the resonance frequency, which maximizes the ratio of tail-tip velocity to bending moment.

The final task is to design the interconnections in a network of Andronov-Hopf oscillators in order to generate desired chaotic behavior. Due to the structure of the oscillators, it is possible to generate chaos by using weak linear coupling to destabilize the phase difference between the oscillators. To this end, a set of sufficient conditions are determined to guarantee the instability of a desired periodic solution through phase destabilization. Subsequently, a condition is found to guarantee the absence of any stable harmonic orbit. Finally, additional properties are considered, where small variations in a parameter can lead to chaotic behavior. With additional research, these results can be expanded to the design of a chaotic neural controller to generate adaptive locomotion for a mechanical rectifier.

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