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Multistability and Hidden AttractorsPowerPoint Presentation

Multistability and Hidden Attractors

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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

Phase Space

v

v

x

x

Focus

Node

A system with n physical dimensions

has a 2n-dimensional phase space.

In a linear system, there can be only one

attractor, and it is a point in phase space.

Finding the Equilibria

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)

Three equilibria:

v = 0, x = 0 (unstable)

v = 0, x = 1 (stable)

v = 0, x = –1 (stable)

Calculation of stability

is almost as simple.

Metastability

“Tipping Point” (Al Gore)

All stable equilibria are attractors,

but not all attractors are equlibria.

Hopf Bifurcation

Strange Attractor Basin

Basin

of strange

attractor

Unbounded

solutions

x’ = y

y’ = z

z’ = –2.02z + y2 – x

Main Collaborators

SajadJafari

Amirkabir University of Technology, Terhan

Iran

Chunbiao Li

Southeast University，Nanjing

China

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.

Summaryhttp://sprott.physics.wisc.edu/ lectures/multistab.pptx thought to be rare may be rather common. (this talk)

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

[email protected] (contact me)

References
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