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Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin on February 24, 2014
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Page 1: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Multistability and Hidden Attractors

Clint Sprott

Department of Physics

University of Wisconsin - Madison

Presented to the

UW Math Club

in Madison, Wisconsin

on February 24, 2014

Page 2: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Types of Equilibria

Attractor Repellor

Page 3: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Dynamics near Attractor

Page 4: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Phase Space

v

x

v

x

Focus Node

A system with n physical dimensions has a 2n-dimensional phase space.

In a linear system, there can be only oneattractor, and it is a point in phase space.

Page 5: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Two-well Oscillator

Three equilibrium points

Example of bistabilityU = x4 – x2

x

U

Page 6: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Basins of Attraction

x’ = vv’ = x(1–x2) – 0.05v

Page 7: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Direction of Flow

x’ = vv’ = x(1–x2) – 0.05v

Page 8: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Saddle Point

Page 9: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

x’ = dx/dt = v

v’ = dv/dt = x(1–x2) – 0.05v

x’ = dx/dt = v = 0 (no velocity)

v’ = dv/dt = x(1–x2) – 0.05v = 0 (no acceleration)

Finding the Equilibria

Three equilibria:

v = 0, x = 0 (unstable)

v = 0, x = 1 (stable)

v = 0, x = –1 (stable)

Calculation of stabilityis almost as simple.

Page 10: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Tacoma Narrows Bridge

November 7, 1940Washington State

Two attractors!

Page 11: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Metastability

“Tipping Point” (Al Gore)

All stable equilibria are attractors,but not all attractors are equlibria.

Page 12: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Hopf Bifurcation

Page 13: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Limit Cycles

x’ = yy’ = zz’ = –2.3z + y2 – x

Page 14: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Millennium Bridge

June 10, 2000London

Limit cycle!

Page 16: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Period Doubling Chaos

x’ = yy’ = zz’ = –az + y2 – x

Page 17: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Strange Attractor Basin

x’ = yy’ = zz’ = –2.02z + y2 – x

Unboundedsolutions

Basinof strangeattractor

Page 18: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Lunch with Ron Chen

Page 19: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.
Page 20: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Tri-stability in Lorenz System

x’ = 10(y–x)y’ = 24.4x – y – xzz’ = xy – 8z/3

Page 21: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Three Coexisting Attractors

x’ = yz + 0.01y’ = x2 – yz’ = 1 – 4x

Page 22: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Three Basins

x’ = yz + 0.01y’ = x2 – yz’ = 1 – 4x

Page 23: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Main Collaborators

Sajad JafariAmirkabir University of Technology, TerhanIran

Chunbiao LiSoutheast University,Nanjing China

Page 24: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

23 Additional Examples

All 3-Dquadraticwith 1 stableequilibrium

Page 25: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Chaos with no Equilibria

17 cases3-Dquadratic

Page 26: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.
Page 27: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Chaos with Line Equilibrium

9 cases

Example:x’ = yy’ = yz – xz’ = –x(1–15y–z)

Page 28: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Basin of Line Equilibrium

x’ = yy’ = yz – xz’ = –x(1–15y–z)

(0, 0, z)

Page 29: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

System with 5 Attractors

x’ = y + yzy’ = yz – xzz’ = –0.55z – xy + 0.8

Page 30: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Chaos with Parallel Lines

x’ = x2 – y – y2

y’ = –xzz’ = 0.3x2 + xy

(0, 0, z)(0, −1, z)

Page 31: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Chaos with Perpendicular Lines

x’ = x(2 + z)y’ = x(x – 1)z’ = x(1 – 4y) – yz

(0, y, 0)(0, 0, z)

Page 32: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Chaos with Plane Equilibrium

(0, y, z)

x’ = xyy’ = xzz’ = x(1.54y2 – x – xz)

Page 33: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Chaos with Three Planes

f = xyz

x’ = f(−0.1y + yz)y’ = f(2z − y2 − z2)z’ = f(−0.2x2 + y2)

Page 34: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Chaos with Spherical Equilibrium

x' = 0.4fy y' = fxz z' = – f(z + x2 + 6yz)

f = 1 – x2 – y2 – z2

Page 35: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Hyperchaos with Line Equilibrium

x' = y – xz – yz + u y' = 4xz z' = y2 – 0.28z2

u' = –0.1y

Page 36: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

Summary

Systems with multiple attractors that were previously thought to be rare may be rather common.

Some of these attractors are “hidden” in the sense that they are not associated with any unstable equilibrium point.

Page 37: Multistability and Hidden Attractors Clint Sprott Department of Physics University of Wisconsin - Madison Presented to the UW Math Club in Madison, Wisconsin.

References

http://sprott.physics.wisc.edu/ lectures/multistab.pptx (this talk)

http://sprott.physics.wisc.edu/chaostsa/ (my chaos textbook)

[email protected] (contact me)


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