1. Chaotic systems and Chua’s Circuitby
Dao Tran
Missouri State University
KME Alpha Chapter
Presented at KME Regional Meeting,
Emporia State University, KANSAS

2. Outline Linear systems Nonlinear systems Chaos and Chua’s Circuit
Local behavior
Global behavior
Chaos and Chua’s Circuit
Bifurcation
Periodic orbits
Strange attractors

3. Motivation/Application
Secure Communication
S(t)
Transmitter
(Chaotic)
y(t)
Receiver
S’(t)
Information signal
Transmitted signal
Retrieved signal
Transmitter
Vc(t)
Chaos generator
(Chua’s circuit)
r(t) +
Buffer
Inverter
Message signal

4. Motivation/Application
Receiver
r(t)
Vc(t)
Chaos generator
(Chua’s circuit)
s’(t)
Buffer -

5. 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.

6. 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

7. Linear systems Linear System of D.E General solution
The solution is explicitly known for any t.

8. Linear systems (cont.) Stability Equilibrium points
If Re(λ)<0 => Stable
If Re(λ)>0 => Unstable

9. Linear systems (cont.) Stability Invariant Sets
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.

10. Linear Systems (cont.) Consider the linear RLC circuit
Applying KCL law and choosing V2 and IL as state variables
,we obtain the differential equation:

11. Linear System (cont.)
With the fixed values of R, L, and C, using MATLAB, we obtained the solution

12. Nonlinear systems
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.

13. 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.

14. 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.

15. Local behavior (cont.)
Linear system
Non-linear system

16. Local behavior (cont.) Linearization around periodic orbits
A periodic solution satisfies
Find periodic orbit by solving the BVP
Determine the Jacobian matrix A(t) = DF(δ)

17. Local behavior (cont.)
The fundamental matrix of a linear system is the solution of
If the periodic orbit has period t, then we define the monodromy matrix as
Stability
If |µ|<1, stability
If |µ|>1, unstability
If monodromy matrix has exactly one eigenvalue with |µ|=1, then the periodic orbit is called hyperbolic

18. Local behavior (cont.) Consider the nonlinear system
This system has periodic orbit (cos t, sin t, 0), of period

19. Local behavior (cont.)
Linearization about the periodic orbit is the linear system
where A is Jacobian evaluated at the periodic orbit, namely:

20. Local behavior (cont.)
The corresponding linear system has a fundamental matrix:
We evaluate at to get monodromy matrix. For α=1/2, MATLAB
gives eigenvalues 0,1 and , 1.0.

21. 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.

22. 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).

23. Chua’s Circuit (cont.)
If we let X1 = V1, X2 = V2 and X3 = I3, Chua's circuit is

24. Chua’s Circuit (cont.)
If we let X1 = V1, X2 = V2 and X3 = I3, the Chua's circuit is
The Jacobian matrix is
where

25. Chua’s circuit (cont.)
At (0,0,0) we have
Eigenvalues are

26. Bifurcation Bifurcation diagram starting value α = -1 (AUTO 2000)
Plot shows norm of the solution ||x|| versus parameter α.

27. Periodic orbits
Following Hopf bifurcation, two periodic orbits appear. The first with period (for α = ) and the second with period (for α= )
1st periodic orbit
2nd periodic orbit

28. Periodic orbit (cont.)

29. 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= ). However, we obtain a different periodic orbit, which seems to “encircle” the previous one.

30. Strange attractors

31. Strange attractors (cont.)

32. 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.

33. 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

34. 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
[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
[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