Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 236

Fig. 4.2. Diagram of the inverted pendulum

4.1.5 Example of Nonlinear Oscillator: The Van Der Pol Oscillator

Stable oscillations in uncontrolled operating mode are another example of ad-
verse oscillations in physical devices. They arise very frequently from nonlin-
earities. A typical example is provided by the following Van der Pol differential
equation:

d 2 x
d t 2 −

2 zω 0 d x

d t + ω 0 x +3 kx 2 d x d t = u.

The parameter z is the damping rate of the system and ω 0 is the eigenfre-
quency of the oscillator. The state representation is 2-dimensional, i.e.,

x 1
x 2

= + 0

u

.

x 2

d
d t ω 0 x 1 − 3 kx 1 x 2

2 zω 0 x 2 −

Note that the system is linear with respect to the control. The dynamics
of the uncontrolled system ( u = 0) in the 2-dimensional sate space features
a limit cycle as an attractor. That means that, whatever the initial state,
the state trajectory winds around a specific periodic trajectory when times is
going on. This phenomenon is illustrated on Fig. 4.3:

4.1.6 Markov Chain as a Model for Discrete-Time Dynamical
Systems with Noise

Let us now return to discrete-time dynamical systems. Consider first the fol-
lowing very simple dynamical system: the random walk on a triangle. The
state space has three elements a,b and c . The dynamics is defined by the
following evolution function f :

f ( a )= b, f ( b )= c, f ( c )= a.