Bifurcation and Chaos
This chapter considers only deterministic dynamical systems. A deterministic dynamical system may have a continuous or discrete state space and a continuoustime or discrete-time dynamic. Discrete-time deterministic dynamical systems occur in electrical engineering as models of switched-capacitor and digital filters, sampled phase-locked loops, and sigma–delta modulators. Discrete-time dynamical systems arise when analyzing the stability of steady-state solutions of continuoustime systems. The chapter examines in detail steady-state behaviors, stability, structural stability, and bifurcations. The most complicated form of steady-state behavior is called quasiperiodicity. The chapter discusses three types of local bifurcation: the Hopf bifurcation, the saddle-node bifurcation, and the period-doubling bifurcation. These bifurcations are called local because they may be understood by linearizing the system close to an equilibrium point or limit cycle. Each of the three local bifurcations may give rise to a distinct route to chaos, and all three have been reported in electronic circuits.