Chaotic systems and Chua’s
Circuit
by
Dao Tran
Missouri State University
KME Alpha Chapter
Presented at KME Regional Meeting,
Emporia State University, KANSAS
Outline
► Linear
systems
► Nonlinear systems
 Local behavior
 Global behavior
► Chaos
and Chua’s Circuit
 Bifurcation
 Periodic orbits
 Strange attractors
Motivation/Application
Secure Communication
S(t)
Information
signal
Transmitter
(Chaotic)
y(t)
S’(t)
Receiver
Transmitted
signal
Retrieved
signal
Transmitter
Chaos generator
(Chua’s circuit)
Message signal
Vc(t)
Buffer
+
Inverter
r(t)
Motivation/Application
Receiver
r(t)
Chaos generator
(Chua’s circuit)
Buffer
Vc(t)
s’(t)
-
What is Chaotic System?
► Phenomenon
that occurs widely in
dynamical systems
► Considered to be complex and no simple
analysis
► Study of chaos can be used in real-world
applications: secure communication,
medical field, fractal theory, electrical
circuits, etc.
What is Chua’s Circuit?
► Autonomous
circuit consisting two
capacitors, inductor, resistor, and nonlinear
resistor.
► Exhibits a variety of chaotic phenomena
exhibited by more complex circuits, which
makes it popular.
► Readily constructed at low cost using
standard electronic components
Linear systems
► Linear
System of D.E
.

 X  AX
n
;
X

R


 X (0)  X 0
► General
solution
X (t )  e At X 0
► The
solution is explicitly known for any t.
Linear systems (cont.)
► Stability
► Equilibrium
.
X  AX
points
► If
Re(λ)<0 => Stable
► If
Re(λ)>0 => Unstable
Linear systems (cont.)
► Stability
►Stability
of linear systems is determined by
eigenvalues of matrix A.
► Invariant
Sets
►(Generalized)
eigenvectors corresponding to
eigenvalues λ with negative, zero, or positive real
part form the stable, center, and unstable subspaces,
respectively.
Linear Systems (cont.)
Consider the linear RLC circuit
Applying KCL law and choosing V2 and IL as state variables
,we obtain the differential equation:
1
 dI 3


V2

L
 dt
, where G  1 / R
 dV
 2  1 I 3  G V2

C2
C2
 dt
Linear System (cont.)
With the fixed values of R, L, and C, using MATLAB, we
obtained the solution
Nonlinear systems
 .
X  F ( X ); X  R n

 F  R n  R n
► Even
for F smooth and bounded for all t є R, the
solution X (t) may become unpredictable or
unbounded after some finite time t.
► We
divide the study of nonlinear systems into local
and global behavior.
Local Behavior
►
Idea: use linear systems theory to study nonlinear systems,
at least locally, around some special sets, a technique
known as linearization.
►
In this work, we consider:
►Linearization
around equilibrium points.
►Linearization
around periodic orbits.
Local Behavior (cont.)
► Linearization
around equilibrium points
►Equilibrium
point is hyperbolic if no eigenvalues of
the Jacobian at the equilibrium point has zero real
part.
►Hartman-Grobman
Theorem: nonlinear system has
equivalent structure as linearized system, with
A=DF(x0), around hyperbolic equilibrium points.
Local behavior (cont.)
Linear system
Non-linear system
Local behavior (cont.)
► Linearization
►A
around periodic orbits
periodic solution satisfies
.

X  F( X )
, with A  DF ( X )


 X (t   )  X (t )
► Find
periodic orbit by solving the BVP
.

X  F(X )


 X (0)  X ( )
► Determine
the Jacobian matrix A(t) = DF(δ)
Local behavior (cont.)
►
The fundamental matrix of a linear system is the solution of
.

  A(t )


(0)  I
►
If the periodic orbit has period t, then we define the
monodromy matrix as (t )
►
Stability
► If
► If
|µ|<1, stability
|µ|>1, unstability
monodromy matrix has exactly one eigenvalue with |µ|=1,
then the periodic orbit is called hyperbolic
► If
Local behavior (cont.)
► Consider
the nonlinear system
.
x  x  y  x 3  xy2
.
y  x  y  x2 y  y3
.
z  z
► This
system has periodic orbit (cos t, sin t, 0), of
period 2
Local behavior (cont.)
1  3x 2  y 2  1  2 xy 0 


2
2
J F ( x)   1  2 xy
1 x  3y 0 


0
0



► Linearization
about the periodic orbit is the linear system x  Ax
where A is Jacobian evaluated at the periodic orbit, namely:
 2 cos 2 t

A(t )  1  sin 2t

0

.
 1  sin 2t 0 

2
 2 sin t 0 
0
 
Local behavior (cont.)
►
The corresponding linear system has a fundamental matrix:
e 2t cos t

 (t )   e  2t sin t
 0

►
We evaluate
(t )
at
2
 sin t 0 

cos t
0
0
et 
to get monodromy matrix. For α=1/2, MATLAB
gives eigenvalues 0,1 and 4.8105, 1.0.
Global Behavior
► Study
is more complex
► One investigates phenomena such as
heteroclinic and homoclinic trajectories,
bifurcations, and chaos.
► we focus in chaos, but this is closely related
to the other concepts and phenomena
mentioned above.
Chaos and Chua’s Circuit
Main goal is to give brief introduction to underlying ideas behind the
notion of chaos, by studying the system that models Chua’s circuit.
Chua’s circuit consists of two capacitors C1, C2, one inductor L, one
resistor R, and one non-linear resistor (Chua’s diode).
Chua’s Circuit (cont.)
If we let X1 = V1, X2 = V2 and X3 = I3, Chua's circuit is
 .
 X 1   [ X 2  h ( x )]
 .
 X 2  X1  X 2  X 3
 .
 X 3   X 2

where  14.3
h( x ) 
x 
2

2
3
x
7
14
 x 1 
arctan( 10 x )
x 1 
Chua’s Circuit (cont.)
If we let X1 = V1, X2 = V2 and X3 = I3, the Chua's
circuit is
 X 1   [ X 2  h( x )]

 X 2  X1  X 2  X 3
 X   X
3
2

where  14.3
h( x ) 
x 
2

2
3
x
7
14
h ' ( x1 )  a1 
x 1 
arctan( 10 x )
The Jacobian matrix is
where
 x 1 
 h ' ( x1 ) 

J ( x)   1
1
 0


1

1
0 


1
1
(a0  a1 ) 

2
2

1  100( x  1) 1  100( x  1) 
10
Chua’s circuit (cont.)
► At
(0,0,0) we have
 0  0  0 
  
J F  0   1  1 1 
 0  0   0 
  

► Eigenvalues
are
 1  1  4(    )

2
Bifurcation
Bifurcation diagram starting value α = -1 (AUTO 2000)
Plot shows norm of the solution ||x|| versus parameter α.
Periodic orbits
Following Hopf bifurcation, two periodic orbits appear. The first with
period 2.2835 (for α =8.19613) and the second with period 19.3835 (for
α=11.07941)
1st periodic orbit
2nd periodic orbit
Periodic orbit (cont.)
Sensitivity to initial data
To show that this dynamical system is sensitive to small changes in the
data (one sign of the presence of chaos), we solve the system again for
α=8.196 (not=8.196013). However, we obtain a different periodic orbit,
which seems to “encircle” the previous one.
Strange attractors
Strange attractors (cont.)
Strange attractor (cont.)
Finally, we compute another strange attractor solution to Chua’s circuit,
which is known in literature as double-scroll attractor. This type of
attractor has been mistaken for experimental noise, but they are now
commonly found in digital filter and synchronization circuits.
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Conclusions
► Chua’s
circuit is simple and has a rich variety of
phenomena:
► Equilibrium
points, periodic orbits
► Bifurcations and chaos
► Signs
of chaos:
► Sensitivity
to initial data
► Strange attractors
► Unpredictability
► Chaos
can be understood with elementary
knowledge of linear algebra and differential
equations
References
[1] W. E. Boyce, R.C. DiPrima, Elementary Differential Equations, seventh
edition, John Wiley & Sons, Inc. (2003)
[2] L. Dieci and J. Rebaza, “ Point to point and point to periodic
connections”, BIT, Numerical Mathematics. To appear, 2004.
[3] E. Doedel, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Sandstede, and
X. Wang. AUTO 2000: Continuation and bifurcation software for ordinary
differential equations. (2000). ftp://ftp.cs.concordia.ca.
[4] J. Hale and H. Kocak, Dynamics and Bifurcations, third edition, Springer
Verlag (1996).
[5] M. P. Kennedy, “Three steps to chaos, I: Evolution”, IEEE Transactions
on circuits and Systems, Vol. 40, No 10 (1993) pp. 640-656.
[6] M. P. Kennedy, “Three steps to chaos, II: A Chua’s circuit primer”, IEEE
Transactions on circuits and Systems, Vol. 40, No 10 (1993) pp. 657-674.
[7] Lawrence Perko, Differential Equations and Dynamical Systems.
Springer-Verlag, New York. (1991).
[8] L. Torres and L. Aguirre, “ Inductorless Chua’s circuit”, Electronic letters,
Vol. 36, No 23 (2000) pp. 1915-1916.