# Why Chaos?

## Conditions for Chaos

Necessary conditions for a set of evolution equations (assumed to
be first order differential equations) to show chaos are:

- There must be at least three variables and equations
- There must be some nonlinearity

Kirchoff's equations for the circuit, involving equations for the
voltages *V*_{1} across *C*_{1}, and
*V*_{2} across *C*_{2}, and the current
*I* through the inductor
*L*satisfy
these requirements

- The requirement of 3 equations is given by the inclusion of
three reactive elements
*C*_{1},
*C*_{2}, and *L.*
- The nonlinearity in the circuit comes form the piecewise
linear effective negative resistance
*g(V) *produced by the op-amp, diodes and associated
resistors. Thus the* only nonlinearity in the effective circuit
is the kink in the I-V characteristics g(V).*

## Oscillations

The d.c. operating point *(I*_{0} ,V_{0} ) is
the intersection of the "load line" of slope *-1/R* with
*g(V)*. There are actually two possibilities, at positive and
negative values. For suitable circuit values these stationary
solutions are unstable to growing oscillations about the dc point.
Since *g(V)* is locally linear, the amplitude of the
oscillations will continue to grow, until the voltage somewhere in
the cycle reaches the kink in *g(V) *at *V*_{c}.*
*The d.c. resistance in the circuit then becomes positive over
part of the cycle, saturating (if we are lucky!) the growth.

*A typical observation will be oscillations where
the voltage V*_{1} oscillates about one of the two d.c.
operating points V_{0 } (or -V_{0 } ) with an
amplitude large enough to swing the voltage past the kink at
V_{c} (or - V_{c} ).

## Chaos

It is hard to predict the effect of the non-linearity: Does the
periodic orbit persist or does it break down to chaos? Which of the
routes to chaos occurs? What is the nature of the chaotic dynamics?

Observation on the physical circuit or the simulation shows that
the periodic orbit undergoes several
period doubling bifurcations (I
managed to see up to period 16 in the physical circuit), and then
becomes noisy. At first the noise is weak (as expected for the period
doubling route to chaos), perturbing the main oscillation about
*V*_{0} , but eventually, as parameters change to
increase the nonlinearity, the dynamics begins to switch randomly
back and forth between oscillations about *V*_{0 } and
then *-V*_{0 } . In this regime the chaotic dynamics is
quite reminiscent of Lorenz chaos. If we
rescale the variables the equations

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Last modified 18 August, 2009

Michael Cross