Chaos, Bifurcations and Fractals Around Us

The following sections are included:

• Contents

When we observe evolution in time of various phenomena in the macroscopic world that surrounds us, we often use the terms "chaos", or "chaotic", meaning that the changes in time are without pattern and out of control, and hence are unpredictable. The most frustrating phenomena are those, which concern long-term weather forecasting. We can never be sure about the change of weather patterns. The temperature, barometric pressure, wind direction, amount of precipitation and other important factors come as a surprise contradicting predictions made a few days ago. Sometimes we are caught in a storm, sometimes in a heat-wave…

At the International Symposium "The Impact of Chaos on Science and Society" organized by the United Nations University and the University of Tokyo in 1991, Prof. Yoshisuke Ueda presented a paper entitled Strange Attractors and Origin of Chaos…

The following sections are included:

• Equation of motion, linear and weakly nonlinear oscillations

• Method of Poincaré map

• Stable and unstable periodic solutions

• Bifurcation diagrams

• Basins of attraction of coexisting attractors

• Global homoclinic bifurcation

• Persistent chaotic motion — chaotic attractor

• Cantor set — an example of a fractal geometric object

The following sections are included:

• Physical and mathematical model of the system

• The single potential well motion

• Melnikov criterion

• Fractal boundaries of basins of attraction and transient chaos in the region of principal resonance

• Oscillating chaos and unpredictability of the final state after destruction of the resonant attractor

• Boundary crisis of the oscillating chaotic attractor

• Persistent cross-well chaos

• Lyapunov exponents

• Intermittent transition to chaos

• Large Orbit and the boundary crisis of the cross-well chaotic attractor

• Various types of attractors of the two-well potential system

The two simple mechanical models, i.e. pendulum and two-well potential system, introduced the Readers to fundamental theoretical concepts and essential chaotic phenomena that arise in nonlinear, dissipative oscillators, driven by periodic force. In both systems, the principal resonance curves were bowed to the left, towards the lower frequency, that is, their nonlinearity had a softening property. One may ask: what about oscillators with a hardening type of restoring force characteristics? Indeed, the oscillators with hardening elastic nonlinearity may also exhibit chaotic motion. Recall that "Ueda's strange attractors" were first found in the Duffing system with this type of nonlinearity…

The following sections are included:

• Bibliography

• Index