Chapter 4
Nonlinear Dynamics and Bifurcations in
Autonomous RLC Circuits
In this brief chapter we discuss some fundamental dynamic phenomena that can
be observed in nonlinear circuits containing time-invariant resistors, inductors,
capacitors, and dc sources (autonomous RLC circuits). In Chap. 6 we will study
analogous dynamic phenomena for nonlinear circuits containing also memristors.
While in first-order autonomous circuits any bounded solution converges to an
equilibrium point (EP), second-order circuits with locally active nonlinear resistors
can display nonvanishing oscillations, as negative resistance oscillators belonging to
the class of Van der Pol oscillators. More complex dynamics, as chaotic dynamics,
can be observed in third-order autonomous circuits, the most famous example being
Chua’s oscillator [1].
We then consider bifurcations due to changing parameters in autonomous circuits
briefly recalling some basic types of local bifurcations of EPs, as the saddlenode and Hopf bifurcation, and local bifurcations of cycles, as period-doubling
bifurcations, leading to the birth of complex attractors.
The theme of bifurcations is especially of interest. In fact, later in the book,
we will discuss a different type of bifurcations that can be observed for structural
reasons in circuits containing memristors. Those bifurcations will be named bifurcations without parameters since they are caused by changing the initial conditions
and are observable even in a circuit where parameters are held fixed (see Chap. 6).
The treatment on bifurcations in this chapter is basically descriptive. The
interested reader may further investigate these aspects in the classical books [2–4].
4.1 First-Order Circuits
Consider an autonomous first-order circuit made of one linear capacitor (or one
linear inductor) connected to a nonlinear resistor. The resistor may be replaced,
© Springer Nature Switzerland AG 2021
F. Corinto et al., Nonlinear Circuits and Systems with Memristors,
https://doi.org/10.1007/978-3-030-55651-8_4
131
Nonlinear Dynamics and Bifurcations in
Autonomous RLC Circuits
In this brief chapter we discuss some fundamental dynamic phenomena that can
be observed in nonlinear circuits containing time-invariant resistors, inductors,
capacitors, and dc sources (autonomous RLC circuits). In Chap. 6 we will study
analogous dynamic phenomena for nonlinear circuits containing also memristors.
While in first-order autonomous circuits any bounded solution converges to an
equilibrium point (EP), second-order circuits with locally active nonlinear resistors
can display nonvanishing oscillations, as negative resistance oscillators belonging to
the class of Van der Pol oscillators. More complex dynamics, as chaotic dynamics,
can be observed in third-order autonomous circuits, the most famous example being
Chua’s oscillator [1].
We then consider bifurcations due to changing parameters in autonomous circuits
briefly recalling some basic types of local bifurcations of EPs, as the saddlenode and Hopf bifurcation, and local bifurcations of cycles, as period-doubling
bifurcations, leading to the birth of complex attractors.
The theme of bifurcations is especially of interest. In fact, later in the book,
we will discuss a different type of bifurcations that can be observed for structural
reasons in circuits containing memristors. Those bifurcations will be named bifurcations without parameters since they are caused by changing the initial conditions
and are observable even in a circuit where parameters are held fixed (see Chap. 6).
The treatment on bifurcations in this chapter is basically descriptive. The
interested reader may further investigate these aspects in the classical books [2–4].
4.1 First-Order Circuits
Consider an autonomous first-order circuit made of one linear capacitor (or one
linear inductor) connected to a nonlinear resistor. The resistor may be replaced,
© Springer Nature Switzerland AG 2021
F. Corinto et al., Nonlinear Circuits and Systems with Memristors,
https://doi.org/10.1007/978-3-030-55651-8_4
131