Chaos in Neurons and Adaptive Control of Birkhoff-Shaw Strange Chaotic Attractor

Chaos is an important applied area in nonlinear dynamical systems and it is applicable to many real-world systems including the biological systems. Nerve membranes are known to exhibit their own nonlinear dynamics which generate and propagate action potentials. Such nonlinear dynamics in nerve membranes can produce chaos in neurons and related bifurcations. In 1952, A.L. Hodgkin and A.F. Huxley proposed a nonlinear dynamical system as a mathematical model of nerve membranes based on their electrophysiological experiments with squid giant atoms. Chaos in nerve membranes have been studied in the chaos literature both theoretically and experimentally. In this research work, we discuss the properties of the Birkhoff-Shaw chaotic attractor, which is a forced oscillator and this strange chaotic attractor exhibits the structure of beaks and wings, typically observed in chaotic neuronal models. We also derive new results for the adaptive control of the Birkhoff-Shaw chaotic attractor (1981).All the main results are proved using Lyapunov stability theory. Also, numerical simulations have been plotted using MATLAB to illustrate the main results for the BirkhoffShaw chaotic attractor.