Chaotic systems and Chua

1
Chaotic systems and Chuas Circuit
by

• Dao Tran

Missouri State University KME Alpha Chapter
Presented at KME Regional Meeting, Emporia
State University, KANSAS
2
Outline

• Linear systems
• Nonlinear systems
• Local behavior
• Global behavior
• Chaos and Chuas Circuit
• Bifurcation
• Periodic orbits
• Strange attractors

3
Motivation/Application

• Secure Communication

S(t)
Transmitter (Chaotic)
Receiver
y(t)
S(t)
Information signal
Transmitted signal
Retrieved signal
Transmitter
Vc(t)
Chaos generator (Chuas circuit)
r(t)

Buffer
Inverter
Message signal
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Motivation/Application
Receiver
r(t)
Vc(t)
Chaos generator (Chuas circuit)
s(t)
Buffer
-
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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.

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What is Chuas 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

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Linear systems

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

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Linear systems (cont.)

• Stability
• Equilibrium points
• If Re(?)lt0 gt Stable
• If Re(?)gt0 gt Unstable

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Linear systems (cont.)

• Stability
• Stability of linear systems is determined by
  eigenvalues of matrix A.
• Invariant Sets
• (Generalized) eigenvectors corresponding to
  eigenvalues ? with negative, zero, or positive
  real part form the stable, center, and unstable
  subspaces, respectively.

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Linear Systems (cont.)
Consider the linear RLC circuit
Applying KCL law and choosing V2 and IL as state
variables ,we obtain the differential equation
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Linear System (cont.)
With the fixed values of R, L, and C, using
MATLAB, we obtained the solution
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Nonlinear systems

• Even for F smooth and bounded for all t ? 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.

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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.

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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
  ADF(x0), around hyperbolic equilibrium points.

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

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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 µlt1, stability
• If µgt1, unstability
• If monodromy matrix has exactly one eigenvalue
  with µ1, then the periodic orbit is called
  hyperbolic

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Local behavior (cont.)

• Consider the nonlinear system
• This system has periodic orbit (cos t, sin t, 0),
  of period

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Local behavior (cont.)

• Linearization about the periodic orbit is the
  linear system
• where A is Jacobian evaluated at the periodic
  orbit, namely

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Local behavior (cont.)

• The corresponding linear system has a fundamental
  matrix
• We evaluate at to get monodromy matrix.
  For a1/2, MATLAB
• gives eigenvalues 0,1 and 4.8105, 1.0.

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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.

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Chaos and Chuas Circuit

• Main goal is to give brief introduction to
  underlying ideas behind the notion of chaos, by
  studying the system that models Chuas circuit.
• Chuas circuit consists of two capacitors C1, C2,
  one inductor L, one resistor R, and one
  non-linear resistor (Chuas diode).

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

• At (0,0,0) we have
• Eigenvalues are

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Bifurcation

• Bifurcation diagram starting value a -1 (AUTO
  2000)

• Plot shows norm of the solution x versus
  parameter a.

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Periodic orbits

• Following Hopf bifurcation, two periodic orbits
  appear. The first with period 2.2835 (for a
  8.19613) and the second with period 19.3835 (for
  a11.07941)

1st periodic orbit
2nd periodic orbit
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Periodic orbit (cont.)
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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
a8.196 (not8.196013). However, we obtain a
different periodic orbit, which seems to
encircle the previous one.
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Strange attractors
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Strange attractors (cont.)
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Strange attractor (cont.)
Finally, we compute another strange attractor
solution to Chuas 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.
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Conclusions

• Chuas 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

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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
  Chuas 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
  Chuas circuit, Electronic letters, Vol. 36, No
  23 (2000) pp. 1915-1916.