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Complex Behavior of Simple Systems
J. C. SprottDepartment of Physics
University of Wisconsin - Madison
Presented to
International Conference on Complex Systems
in Nashua, NH
on May 23, 2000
Lorenz Equations (1963)
dx/dt = y - x
dy/dt = -xz + rx - y
dz/dt = xy - bz
7 terms, 2 quadratic
nonlinearities, 3 parameters
Rössler Equations (1976)
dx/dt = -y - z
dy/dt = x + ay
dz/dt = b + xz - cz
7 terms, 1 quadratic
nonlinearity, 3 parameters
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.”
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)
Sprott (1994)
14 more examples with 6 terms and 1 quadratic nonlinearity
5 examples with 5 terms and 2 quadratic nonlinearities
Gottlieb (1996)
What is the simplest jerk function that gives chaos?
Displacement: x
Velocity: = dx/dt
Acceleration: = d2x/dt2
Jerk: = d3x/dt3
x
x
x
)( x,x,xJx
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
Bifurcation Diagram
Return Map
Fu and Heidel (1997)
Dissipative quadratic
systems with less than 5
terms cannot be chaotic.
They would have no
adjustable parameters.
Weaker Nonlinearity
dx/dt = y
dy/dt = z
dz/dt = -az + |y|b - x
Seek path in a-b space that
gives chaos as b 1.
xxxaxb
Regions of Chaos
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)
1 xxxax
General Formdx/dt = y
dy/dt = z
dz/dt = -az - y + G(x)
G(x) = ±(b|x| - c)
G(x) = -bmax(x,0) + c
G(x) = ±(bx - csgn(x))etc….
)(xGxxax
First Circuit
1 xxxax
Bifurcation Diagram for First Circuit
Second Circuit
Third Circuit
)sgn(xxxxax
Chaos Circuit