Simple Chaotic Systems and Circuits
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Simple Chaotic Systems and Circuits
J. C. SprottDepartment of PhysicsUniversity of Wisconsin - Madison
Presented at
University of CataniaIn 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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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)dx/dt = –y – z
dy/dt = x
dz/dt = Ay – Ay2 – Bz
6 terms, 1 quadratic nonlinearity, 2 parameters
“Probably the simplest strange attractor of a 3-D ODE”(1998)
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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: xVelocity: = dx/dtAcceleration: = d2x/dt2
Jerk: = d3x/dt3
x
x
x
)( x,x,xJx
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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:
b x xxxx –a
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Sprott (1997)dx/dt = y
dy/dt = z
dz/dt = –az + y2 – x
5 terms, 1 quadratic nonlinearity, 1 parameter
“Simplest Dissipative Chaotic Flow”
xxxax 2
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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:
xxxax 2
1–xxbxxax
1–2xxbxax
1–xxcxxbxax 2 ...
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Weaker Nonlinearitydx/dt = y
dy/dt = z
dz/dt = –az + |y|b – x
Seek path in a-b space that gives chaos as b 1.
xxxax b
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Regions of Chaos
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Linz and Sprott (1999)dx/dt = ydy/dt = zdz/dt = –az – y + |x| – 1
6 terms, 1 abs nonlinearity, 2 parameters (but one =1)
1 xxxax
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General Formdx/dt = ydy/dt = zdz/dt = – az – y + G(x)
G(x) = ±(b|x| – c)
G(x) = ±b(x2/c – c)
G(x) = –b max(x,0) + c
G(x) = ±(bx – c sgn(x))etc….
)(xGxxax
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Universal Chaos Approximator?
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Operational Amplifiers
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First Jerk Circuit
1 xxxax 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
CBA xxxx 15 components
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Chaos Circuit
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Third Jerk Circuit
)sgn(xxxxx A11 components
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Simpler Jerk Circuit
)- sgn( xxxxx CBA 9 components
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Inductor Jerk Circuit
)- sgn( xxxxx CBA 7 components
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Delay Lline Oscillator
xxx - sgn6 components
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References
http://sprott.physics.wisc.edu/
lectures/cktchaos/ (this talk)
http://sprott.physics.wisc.edu/
chaos/abschaos.htm