Chaotic Dynamics in Electrical and Biological Systems

Prerequisites:

Math 240, ; Physics 150, or permission of instructor

Description:

The world surrounding us is nonlinear, yet most models of the real world that are dealt with in undergraduate engineering courses are predominantly linear. They are intentionally linearized to facilitate their analysis and understanding. Unfortunately, this frequently hides from view and general experience immensely interesting behavior that is rich in complexity and is only seen in nonlinear systems.

In the past two decades, considerable progress in the understanding and mathematical analysis of nonlinear dynamical systems has taken place with bifurcation and chaos in nonlinear dynamical deterministic systems being a paramount example. The recognition that determinism does not imply regular or predictable behavior is having a major impact on many facets of science, engineering, biology and mathematics. The fact that even very simple systems, describable by entirely deterministic physical laws, can exhibit behavior, seemingly random, and extremely rich in complexity, represents what amounts to a revolution in our view of the world around us.

The goal of this course, which can be offered as a science or engineering elective, is to introduce seniors to the mathematical tools and physical insight needed for modern understanding of nonlinear dynamics in electrical and biological circuits and systems. The pendulum, several nonlinear circuits including the Choa circuit, and several iterative maps of the interval onto itself such as the logistic and the circle map will be used to furnish a unified approach to the study and understanding of nonlinear dynamics, chaos, synchronicity, bifurcation, self-organization and fractals. Selected areas of application such as biological clocks, chaos and cardiac arrythmias, nerve cell dynamics, coupled lattice maps, attractor type neural networks will also be covered along with self-organization and other collective emergent phenomena in attractor-type neural networks and their applications.

Course Content:

I. Elements of Nonlinear Dynamics: (5 weeks)

Linear vs. nonlinear, the power of nonlinear dynamics, simple nonlinear dynamic models, stability considerations, the driven pendulum, phase-space, phase-space trajectories, dissipative vs. non-dissipative systems, attractors and basins of attraction, periodic and strange attractors, energy functions and the physics of computing with attractors; Chaos and when is a dynamical system apt to exhibit chaos, demonstration of chaotic oscillations in the double pendulum; Iterative maps, the logistic map, graphical (cobweb) iterations, bifurcation diagram; university of chaos, pitchfork bifurcation, routes to chaos and the Feigenbaum constant; measures of chaos, Lyapunov exponent, entropy, sensitivity of chaotic orbits to initial conditions, complexity and information related to the logistic map; Geometric characterization of attractors, fractal and other geometric dimensions of strange attractors; The circle-map, phase-locking and entrainment, rotation numbers, the Devil's Staircase diagram and the Arnold Tongue's diagram, driven nonlinear oscillators (limit-cycle and the Van der Pol oscillators).

II. Biological Clocks and Mechanisms of Neural Control: (5 weeks)

Theory of clocks, phase-resetting, modulated clocks, the nerve cell membrane, structure and function of the living neuron, the electrophysics of nerve fiber and active pulse propagation, the dendritic-tree as passive multibranched transmission line, the Hodgkin-Huxley model and its simplifications, the periodically driven biological neuron: experimental evidence of entrainment, (phase-locking) bifurcation, and chaos; Artificial models of the living neuron, mathematical models and hardware embodiments: The McColough-Pitts neuron, the sigmoidal neuron, the voltage-controlled oscillator neuron (VCON), phase-locking properties of the VCON, the limit-cycle oscillator neuron and the integrate-and-fire neuron, analytical and experimental tools for characterizing the various neuron models; The bifurcating neuron, the phase-transition map, bifurcation diagram, Lyapunov exponent, and entropy of the periodically driven bifurcating neuron, functional complexity and structural simplicity of the bifurcating neuron, computing with diverse attractors.

III. Population of Coupled Nonlinear Processing Elements and Dynamical Computation with Attractors: (4 weeks)

• Numerical study of coupled maps of the interval
• Attractor type neural networks, concepts of associative memory, learning, and optimization.
• Networks of coupled oscillators, the fire-fly machine, relation to higher-level brain function, (speculations on how the brain works).

Text:

• G.L. Baker and J.P. Gollub, Chaotic Dynamics - An Introduction, 2nd ed., Cambridge University Press, (1996).
• Supplementary course notes will be used to cover parts II and III.
• Several classroom and laboratory demonstrations will be used as aids in teaching the course and to generate ideas for senior design projects.

Reference Text:

• E. Beltrami, Mathematics for Dynamic Modeling, Academic Press, Orlando, Fla. (1987).
• S. Strogatz, Nonlinear Dynamics and Chaos, Addison Wesley, (1994).

NB: Code of Academic Integrity

Using or attempting to use unauthorized assistance, material, lab results, or solutions (in part or whole) is a violation of the Code of Academic Integrity and will result in a zero grade for the course.