Simple%20Chaotic%20Systems%20and%20Circuits

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Simple Chaotic Systems and Circuits

• J. C. Sprott
• Department of Physics
• University of Wisconsin - Madison
• Presented at
• University of Catania
• In Catania, Italy
• On July 15, 2014

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Outline

• Abbreviated History
• Chaotic Equations
• Chaotic Electrical Circuits

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Abbreviated History

• Poincaré (1892)
• Van der Pol (1927)
• Ueda (1961)
• Lorenz (1963)
• Knuth (1968)
• Rössler (1976)
• May (1976)

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Chaos in Logistic Map
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Mathematical Models of Dynamical Systems

• Logistic Equation (Map)
• xn1 Axn(1-xn)
• Newtons 2nd Law (ODE)
• md2x/dt2 F(x,dx/dt,t)
• Wave Equation (PDE)
• ?2x/?t2 c2?2x/?r2

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Poincaré-Bendixson Theorem(in 2-D flow)
Fixed Point
Limit Cycle
y
x
Trajectory cannot intersect itself (no chaos)
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Autonomous Systems

• d2x/dt2 Adx/dt x Bsin?t
• let y dx/dt
• and z ?t
• dx/dt y
• dy/dt x Ay Bsin(z)
• dz/dt ?

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Lorenz Equations (1963)

• dx/dt Ay Ax
• dy/dt xz Bx y
• dz/dt xy Cz
• 7 terms, 2 quadratic nonlinearities, 3 parameters

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Rössler Equations (1976)

• dx/dt y z
• dy/dt x Ay
• dz/dt B xz Cz
• 7 terms, 1 quadratic nonlinearity, 3 parameters

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Lorenz Quote (1993)

• One other study left me with mixed feelings.
  Otto Roessler of the University of Tübingen had
  formulated a system of three differential
  equations as a model of a chemical reaction. By
  this time a number of systems of differential
  equations with chaotic solutions had been
  discovered, but I felt I still had the
  distinction of having found the simplest.
  Roessler changed things by coming along with an
  even simpler one. His record still stands.

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Rössler Toroidal Model (1979)
Probably the simplest strange attractor of a 3-D
ODE
(1998)

• dx/dt y z
• dy/dt x
• dz/dt Ay Ay2 Bz
• 6 terms, 1 quadratic nonlinearity, 2 parameters

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Sprott (1994)

• 14 additional examples with 6 terms and 1
  quadratic nonlinearity
• 5 examples with 5 terms and 2 quadratic
  nonlinearities

J. C. Sprott, Phys. Rev. E 50, R647 (1994)
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Gottlieb (1996)

• What is the simplest jerk function that gives
  chaos?
• Displacement x
• Velocity dx/dt
• Acceleration d2x/dt2
• Jerk d3x/dt3

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Linz (1997)

• Lorenz and Rössler systems can be written in jerk
  form
• Jerk equations for these systems are not very
  simple
• Some of the systems found by Sprott have simple
  jerk forms

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Sprott (1997)
Simplest Dissipative Chaotic Flow

• dx/dt y
• dy/dt z
• dz/dt az y2 x
• 5 terms, 1 quadratic nonlinearity, 1 parameter

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Bifurcation Diagram
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Return Map
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Zhang and Heidel (1997)

• 3-D quadratic systems with fewer than 5 terms
  cannot be chaotic.
• They would have no adjustable parameters.

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Eichhorn, Linz and Hänggi (1998)

• Developed hierarchy of quadratic jerk equations
  with increasingly many terms

...
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Weaker Nonlinearity

• dx/dt y
• dy/dt z
• dz/dt az yb x
• Seek path in a-b space that gives chaos as b ? 1.

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Regions of Chaos
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Linz and Sprott (1999)

• dx/dt y
• dy/dt z
• dz/dt az y x 1
• 6 terms, 1 abs nonlinearity, 2 parameters (but
  one 1)

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General Form

• dx/dt y
• dy/dt z
• dz/dt az y G(x)
• G(x) (bx c)
• G(x) b(x2/c c)
• G(x) b max(x,0) c
• G(x) (bx c sgn(x))
• etc.

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Universal Chaos Approximator?
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Operational Amplifiers
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First Jerk Circuit
18 components
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Bifurcation Diagram for First Circuit
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Strange Attractor for First Circuit
Calculated
Measured
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Second Jerk Circuit
15 components
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Chaos Circuit
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Third Jerk Circuit
11 components
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Simpler Jerk Circuit
9 components
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Inductor Jerk Circuit
7 components
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Delay Lline Oscillator
6 components
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References

• http//sprott.physics.wisc.edu/
  lectures/cktchaos/ (this talk)
• http//sprott.physics.wisc.edu/chaos/abschaos.htm
• sprott_at_physics.wisc.edu