In this chapter, we investigate the bifurcation phenomena observed in oscillatory circuits. The stability and bifurcation phenomena in autonomous systems are introduced by focusing on the equilibrium point and the fixed point. The characteristics and conditions of the saddle-node bifurcation, Hopf bifurcation, and pitchfork bifurcation are discussed for the equilibrium point. Likewise, the characteristics and conditions of the saddle-node bifurcation, period-doubling bifurcation, Neimark-Sacker bifurcation, and pitchfork bifurcation are introduced for the fixed point. The method for computing the bifurcation points of the equilibrium point and the periodic points is also introduced, and an example of an application is presented.

Chapter Contents:

• 2.1 Introduction
• 2.2 Analysis of bifurcations of autonomous systems
• 2.2.1 Stability of equilibrium point
• 2.2.2 Bifurcation at equilibrium point
• 2.2.2.1 Saddle-node bifurcation
• 2.2.2.2 Hopf bifurcation
• 2.2.2.3 Pitchfork bifurcation
• 2.2.3 Stability of fixed point
• 2.2.4 Bifurcation of fixed point
• 2.2.4.1 Saddle-node bifurcation
• 2.2.4.2 Period-doubling bifurcation
• 2.2.4.3 Neimark-Sacker bifurcation
• 2.2.4.4 Pitchfork bifurcation
• 2.3 Example of bifurcation analysis applied to an autonomous system
• 2.3.1 Single BVP oscillator
• 2.3.2 Coupled BVP oscillators
• 2.4 Conclusions
• Acknowledgments
• References

Inspec keywords: bifurcation; oscillators

Other keywords: fixed point; bifurcation analysis; autonomous systems; Hopf bifurcation; Neimark-Sacker bifurcation; pitchfork bifurcation; saddle-node bifurcation; oscillatory circuit; equilibrium point; period-doubling bifurcation

Subjects: Oscillators; Chaotic behaviour in circuits