Introduction to Chaos and Nonlinear Dynamics


The word "chaos" written in the Japanese characters

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Thank you for about 100,000 accesses from 17 Sept. 1997 to June 2005.

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Chaos in Android Smartphone

We offer some applications for android smartphone.

Fractal Mandelbrot Set for Android
Fractal Basin Boundary for Android
Dissipative and High-dimensional System Riddled Basin for Android

Interactive dynamic simulations

We present some simulators that demonstrate the well-known chaotic systems.
These interactive simulators will help you to understand the complex properties of nature.

Dissipative System Time Series of Logistic Map
Bifurcation Diagram of Logistic Map
Attractor of Hénon Map
Lorenz Attractor
Rössler Attractor
Attractor of Duffing Equation
Parametric Pendulum
Various Pendulums
Fractal Mandelbrot Set
Julia Set
Fractal Basin Boundary
Dissipative and High-dimensional System Riddled Basin
Blowout Bifurcation and On-off Intermittency
Coupled Map Lattice and Globally Coupled Map
Conservative System Double Pendulum
Extensible Pendulum
Standard Map
Harper Map

Brain Dynamics

The theories of chaos and nonlinear dynamics are applied to many fields such as sociology, economics, and biology.
The one of the most active field is the brain science.

Here we present some simulators which introduce some researches to understand our complex brain.

Single Neuron FitzHugh-Nagumo Equation
Synchrony Analysis of Synchronization in Pulse Neural Networks with Phase Response Function
Chaos in a Pulse Neural Network: Analysis of Synchronization with the Fokker-Planck Equation
Stochastic Synchrony of Chaos in a Pulse Neural Network with Electrical Synapses
Rewiring-induced Synchronization and Chaos in Pulse-coupled Neural Networks
High-dimensional Dynamics Chaotic Pattern Transitions in Pulse Neural Networks
Sensory System Stochastic Resonance
Others Array-Enhanced Coherence Resonance

A Popular Pictorial Introduction to Chaos

Chaos Movies

Chaos Animations

I made some animations of periodically varying chaotic attractor.

Chaos Gallery

A French mathematician, Henri Poincaré (1854-1912) proved that there is no analytical solution of the dynamical equations governing the three planets system.
He created an original method to understand such systems, and discovered a very complicated dynamics, namely, chaos. He said,

"It is so complicated that I cannot even draw the figure."

Fortunately we are living in the computer age, so we can see chaos in a computer display.

Quotations about Chaos (Translation from Japanese)

About the authors

Takashi Kanamaru J. Michael T. Thompson