Download Chaos and Determinism

Deterministic Chaos and the
Chao Circuit
Determinism and Randomness
• Classical physics is
deterministic!
– If you know where you
started you know where
you are going
• Randomness:
– Quantum randomness is
truly random and
unpredictable
– A lot of randomness is
actually complexity and
uncertainty
Deterministic Chaos
• If we have a system with
two things present
– A sensitivity to initial
conditions
– A non-linear response
• Then we can get a
system response that
appears random but is
actually chaotic
• “Chaotic” in this case
means complex and
unpredictable but not
truly random
Deterministic Chaos
• This combination can
produce a great deal of
complexity in the
response.
– Fractals -- geometric
chaos
– “Butterfly effect” -where we have
unpredictable transient
response due to a very
large sensitivity to small
disturbances.
Linear Dynamical Systems
• Well behaved
• Not chaotic
• x and y follow smooth
trajectories
• Generally solvable,
predictable and intuitive
“dot” indicates time derivative
RLC Driven Linear Oscillator
• Linear circuit
– VR = IR
– VL = L dI/dt
– I = C dVC/dt
• Excite with a step
– Oscillating response
– State-space (plot of v(t), i(t))
shows a spiral
Non-linear systems
• Lorentz system
– Simple non-linear system
of 3 variables
– Produces deterministic
chaos
– State-space plot shows
the movement
(trajectory) of x,y and z in
time
– The state-space
description shows two
“attractors” around which
the “system” orbits
Double Pendulum
• Another non-linear
system is the double
(jointed) pendulum
• Also produces chaos
• Trajectory is very
sensitive to the
initial starting point
Double Pendulum
Oscillator Circuits
• Circuits are used to create “self
starting” oscillators.
• Use a transistor to provide nonlinearity
• Design to oscillate at a particular
frequency.
• Your first oscillator will be your first
amplifier!
• Need to make sure it is not chaotic
The lab: Chua Circuit
• For your lab you will build and test a
non-linear circuit that can oscillate and
also produce chaos.
• Circuit uses diodes and a opamp to
produce a non-linear element (negative
resistance).
• By changing the value of R and R1 you
can change the behavior from
oscillation to chaos.
Non-linear elements
• Diode
– Simple semiconductor
device
– Exponential nonlinearity
• Opamp
–
–
–
–
Ideal amplifier
Very high gain
Complex circuit
Clamping at max/min
Other Chaotic Systems
• Weather
• Economics
• History
(cliodynamics)
• Etc.