Chaos in Electronic Circuits
K. THAMILMARAN
29.11.2012
Centre for Nonlinear Dynamics
School of Physics, Bharathidasan University
Tiruchirapalli-620 024
Aim

To introduce chaos as a dynamical behavior admitted by completely
deterministic nonlinear systems.

To give an idea of the equilibrium conditions like fixed points and limit
cycles and the routes or transition to chaos admitted by nonlinear
systems.

To model nonlinear systems using simple electronic circuits and
demonstrate their dynamics in real time for a wide range of control
parameters, with the limited facilities available in an undergraduate
electronics lab.
Plan of Talk

Classification of Dynamical systems

What is Chaos? How does it arise?

What is the characteristic (or) signature of chaos?

Transition to chaos – Routes

How to study or explore chaos?

Study of chaos using nonlinear circuits

Demo – circuit simulators

Conclusion
Dynamical systems
Dynamical Systems
Linear Dynamical Systems
Non - Linear Dynamical Systems
 Have linear forces acting on them
 Have nonlinear forces acting on them
 are modeled by linear ODE’s
 are modeled by nonlinear ODE’s
 Obey superposition principle
 Don’t obey superposition principle
 Frequency is independent of
 Frequency is not independent
the amplitude at all times
of the amplitude.
Dynamical Systems…
Linear Harmonic Oscillator – a linear dynamical system
x  02 x  f sin (t )
Corresponding two first
order equations
x y
y  02 x  f sin (t )
Dynamical systems…
Duffing Oscillator – a nonlinear dynamical system
x   x  02 x   x3  f sin (t )
Corresponding two first
order equations
x y
y   y  02 x   x3  f sin(t )
2
3
 2
2
 0    A  4  A3    A 2  f 2
Dynamical systems…
Dynamical systems whether linear or nonlinear are classified as
Dissipative systems: whose time evolution leads to contraction of
volume/area in phase space eventually resulting a bounded chaotic
attractor
Conservative or Hamiltonian systems: here chaotic orbits tend to visit
all parts of a subspace of the phase space uniformly, thereby conserving
volume in phase-space
Dynamical systems…
Phase space : N–dimensional geometrical space spanned
by the dynamical variables of the system.
Used to study time evolution behavior of
dynamical systems.
Dynamical systems…
1  2  
1  2  
1  2  
1  2  
1    i  ,
2    i
1 , 2  i
Various fixed points
1    i  ,
2    i 
1  
2  
Dynamical systems…
Fixed points limit cycles and strange attractors - equilibrium
states of dynamical systems
Fixed points:
are points in phase space to which
trajectories converge or diverge as
time progresses
limit cycles :
are bounded periodic motion of
forced damped or undamped
two dimensional systems
strange attractors :
characteristic behaviour of
systems when nonlinearity is
present
Dynamical systems…
Limit cycle attractor
Dynamical systems…
Quasi-periodic Motion
Dynamical systems…
strange attractors
What is Chaos?
 Chaos is the phenomenon of appearance of apparently random type
motion exhibited by deterministic nonlinear dynamical systems
whose time history shows a high sensitive dependence on initial
conditions
 Chaos is “deterministic – randomness”
 deterministic - because it arises from intrinsic causes and not from
external factors
 randomness - because of its unpredictable behavior
How does Chaos arise ?…
 Chaos is ubiquitous and arises as a result of nonlinearity present in
the dynamical systems
 It is observed in atmosphere, in turbulent sea, in rising columns of
cigarette smoke, variations of wild life population, oscillations of
heart and brain, fluctuations of stock market, etc.
 Most of the natural systems are nonlinear and therefore study of chaos
helps in understanding natural systems
Characteristics of Chaos…
Chaos is characterized by
 extreme sensitivity to infinitesimal changes in initial conditions
 a band distribution of FFT Spectrum (power spectrum)
 positive values for Lyapunov exponents
 fractal dimension
Characteristics of Chaos…
Insensitivity to initial conditions –
very high sensitive dependence on
observed when the absents of
initial conditions – observed
nonlinearity
when nonlinearity is present
Characteristics of Chaos…
Power spectrum for a simple
sinusoidal wave
Power spectrum of a chaotic
attractor
Lyapunov Exponent
For a 3-dim system we have 3 exponents:
 (t )   0 e t
Largest Lyapunov Exponent:
 1 ||  (t ) || 

 t ||  0 || 
  lim ln 
t 
Chaos: λ1 > 0
λ2 = 0 , where | λ3| > | λ1|=> λ1 + λ2 + λ3 <0
λ3 < 0
Periodic Torus : λ1 = 0
λ2 = 0
λ3 < 0
Periodic Cycle : λ1 = 0
λ2 < 0
λ3 < 0
 λ < 0 the orbit attracts to a stable fixed point or stable periodic orbit.
 λ = 0 the orbit is a neutral fixed point (or an eventually fixed point).
 λ > 0 the orbit is unstable and chaotic.
Routes to Chaos
 Period doubling route
 Quasi-periodic route
 Intermittency route
Bifurcations are sudden qualitative changes in the system behavior as
the control parameters are varied
Routes to Chaos…
n 1  n
  lim
x  
n  2   n 1
  4.669 201 609....
Period doubling Bifurcations
Routes to Chaos…
Period doubling route to chaos
Routes to Chaos…
Experimentally observed Period doubling route
Routes to Chaos…
control parameter a
Routes to Chaos…
Quasiperiodic route to chaos
Routes to Chaos…
Intermittent bursts and transitions to chaos
Routes to Chaos…
Experimental result for Intermittent bursts and transitions to chaos
Rare routes to chaos
Period -3 doubling cascade
How to study chaos?
Chaos in dynamical systems can be studied by
mathematical modeling and solving the resultant equations of motion by
 Analytical methods
 Numerical computations
Experimental study by
 Hardware experiments
 Circuit simulations (PSpice, Multisim)
How to study chaos?...
 Analytical techniques like Fourier transforms, Laplace transforms,
Green’s Function method, etc., are available for linear ODE’s only
 Entirely new analytical tools for solving nonlinear ODE’s have to
be developed for each system separately
 Are often cumbersome and require rigorous knowledge of
mathematics
How to study chaos?...
Numerical computations
 require simple algorithms
 are easy to implement using any high level language
However
 they require enormous computing power and
 enormous time to unravel the full dynamics
Therefore real time hardware experiments are devised in the laboratory
Study of Chaos Using Electronic Circuits
 A circuit may be considered as a dynamical system if it contains
energy storing elements such as capacitors and/or capacitors and
inductors and linear or nonlinear elements
 Using these real time observation and measurements are easily
made for a wide range of parameters
 Various behaviors like bifurcation and chaos can be observed using
oscilloscopes, spectrum analyzers, etc.
Study of Chaos Using Electronic Circuits…
 Curve tracers and Poincaré section circuits help in easy analysis
 Circuit simulators like PSpice provide an "experimental comfort" and
help in exploring and confirming strange an unexpected behaviors
 Advances in IC technology help in building cheap models that truly
reflect the properties of any physical systems
Examples are Chua’s circuit, canonical Chua’s circuit and MLC circuit
CHAOS IN NONLINEAR ELECTRONIC CIRCUITS
Minimum requirements for realizing chaos in circuits
 One nonlinear element
 One locally active resistor
 Three energy storage elements for autonomous systems
 Two energy storage elements + forcing for nonautonomous
systems
Study of chaos using nonlinear circuits…
(a) A two-terminal linear resistor,
(b) its (v −i) characteristic curve,
(c) a nonlinear resistor and
(d) its (v − i) characteristic curve
v(t) = Ri(t);
Linear and Nonlinear Resistor
fR(v,i)=0
Study of chaos using nonlinear circuits…
(a) A two-terminal linear capacitor,
(b) its (v − q) characteristic curve,
(c) a nonlinear capacitor and
(d) its (v − q) characteristic curve
dv
iC ,
dt
Linear and Nonlinear Capacitor
fc  q, v   0
Study of chaos using nonlinear circuits…
(a) A two-terminal linear inductor,
(b) its (φ−i) characteristic curve,
(c) a nonlinear inductor and
(d) its (φ − i) characteristic curve
di
  L ,
dt
Linear and Nonlinear Inductor
f L ( , i)  0
The Fourth Element
Study of chaos using nonlinear circuits…
LC circuit
d 2v  1 

   0,
2
dt
 LC 
v  0   v0 ,
dv
= (i L0 )/C.
dt t=0
Study of Chaos Using Electronic Circuits…
Numerical time waveform of v (t) and phase portrait in the (v − iL) plane for LC circuit
Study of Chaos Using Electronic Circuits…
Study of Chaos Using Electronic Circuits…
Forced LCR circuit
d 2v  R  dv  1 
  
   F Sin ( t ).
2
dt  L  dt  LC 
Study of Chaos Using Electronic
Circuits…
Unforced damped oscillation (F=0, R/L > 0)
Numerical time waveform of v(t) and phase portrait in the (v − iL) plane of the series LCR
circuit
Study of Chaos Using Electronic
Circuits…
Forced damped oscillation (F>0, R/L > 0)
Numerical time waveform of v(t) and phase portrait in the (v − iL) plane.
Study of Chaos Using Electronic
Circuits…
Forced damped oscillation (F > 0, R/L > 0)
Experimental time waveform of v(t) and phase portrait in the (v − iL) plane.
Study of Chaos Using Electronic Circuits…
Chaotic Colpitts Oscillator
Study of chaos using nonlinear circuits…
Study of Chaos Using Electronic Circuits…
Study of Chaos Using Electronic Circuits…
Oscillator VS with an active RC load composite.
V S = 10V/3kHz, R1 = 1kΩ, C1 = 4.7nF, R2 =994kΩ, C2 = 1.1nF, Q1 = 2N2222A.
Study of Chaos Using Electronic Circuits…
limit cycle
chaotic attractor
V S : 10V/3kHz. y: V (2) = V (C2), 0.5V/div, x: V (5) = V (V S), 2.0V/div
Study of Chaos Using Electronic
Circuits…
Chaotic Wien's Bridge oscillator
Study of chaos using nonlinear circuits…
Clockwise from top left. Fixed point, period 2-T, 4-T, 8-T chaotic oscillations.
Study of chaos using nonlinear circuits…
Wein bridge oscillator
Wein bridge Chua’s oscillator
Study of chaos using nonlinear circuits…
Experimental results of Wein bridge oscillator
Fixed point
4T
1T
1 bc attractor
2T
2 bc attractor
Study of chaos using nonlinear circuits…
Duffing Equation
x   x   x   x  f sin  t
2
0
3
 is an ubiquitous nonlinear differential equation, which makes its presence
felt in many physical, engineering and even biological problems.
 Introduced by the Dutch physicist Duffing in 1918 to describe the
hardening spring effect observed in many mechanical problems
Study of chaos using nonlinear circuits…
Duffing equation
can be also thought of as the equation of
motion for a particle of unit mass in the
potential well
  x
1 2 2  4
 0 x  x
2
4
subjected to a viscous drag force of strength α and driven by an external
periodic signal of period T=2 / and strength f
This found to exhibit interesting dynamics like period doubling route to chaos
Study of chaos using nonlinear circuits…
Single - well potential
Double -well potential
Single-hump potential
Double -hump potential
Study of chaos using nonlinear circuits…
Analog simulation of Duffing oscillator
x   x  02 x   x3  f sin  t
Study of chaos using nonlinear circuits…
Analog simulation of Duffing oscillator equation – regular dynamics
1T
3T
2T
4T
Study of chaos using nonlinear circuits…
Analog simulation of Duffing oscillator – Chaotic dynamics
Single band chaos
Double band chaos
v-i characteristic of different
Nonlinearities
Circuit realization of Chua’s Diode
NR
g (v1 )  Gb v1 + 0.5 (Ga - Gb ) [|v1 +B p | - |v1  B p |]
Chua’s Diode – A Nonlinear Resistor (NR)
Chaos in Chua’s Oscillator
dv1 1
 (v2  v1 )  g (v1 )
dt R
dv1 1
C2
 (v1  v 2 )  iL
dt R
di
L L  v 2
dt
C1
g (v1 )  Gb v1 + 0.5 (Ga - Gb ) [|v1 +B p | - |v1  Bp |]
Chaos in Chua’s Oscillator…
Chaos in Chua’s Oscillator…
Single scroll Chaotic attractor
Phase Portrait
Time waveform
Power spectrum
Chaos in Chua’s Oscillator…
Double scroll Chaotic attractor
Phase Portrait
Time waveform
Power spectrum
Lorenz Oscillator
x   ( y  x)
y  rx  y  xz
z  xy  bz
Hindmarsh–Rose model
Rössler Oscillator
x  ( y  z )
y  x  ay
z  b  z ( x  c)
Logistic map
xn1  axn (1  xn )
Conclusion
We have seen
 How dynamical systems are classified and their behaviors
 What is chaos?, how it is generated?, its characteristics, etc
 How systems undergo transitions to chaos
Exploration of chaos in some simple nonlinear circuit like
transistor circuits, Colpitts, Wien Bridge, Duffing, Chua and MLC
oscillators
Bifurcation and Chaos are observed by Multisim Simulation