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Related Experiment Video

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Fabrication and Testing of Microfluidic Optomechanical Oscillators

Fabrication and Testing of Microfluidic Optomechanical Oscillators

Published on: May 29, 2014

Exact folded-band chaotic oscillator.

Ned J Corron¹, Jonathan N Blakely

¹U. S. Army RDECOM, RDMR-WSS, Redstone Arsenal, Alabama 35898, USA. ned.corron@us.army.mil

Chaos (Woodbury, N.Y.)

|July 5, 2012

Summary

This study presents an exactly solvable chaotic oscillator with a folded-band structure, demonstrating provably chaotic behavior and an exact solution derived from its hybrid dynamics. The findings reveal a connection between the folded-band topology and symbol grammar.

Area of Science:

Dynamical Systems
Chaos Theory
Nonlinear Dynamics

Background:

Chaotic oscillators are fundamental in understanding complex systems.
Hybrid dynamical systems offer unique behaviors not found in purely continuous or discrete systems.
Folded-band dynamics, observed in systems like Rössler's oscillator, are crucial for complex chaotic behavior.

Purpose of the Study:

To introduce and analyze an exactly solvable chaotic oscillator.
To investigate the properties of its folded-band dynamics.
To establish a connection between the oscillator's dynamics and symbolic representations.

Main Methods:

The study employs a hybrid dynamical system combining a linear ordinary differential equation with a nonlinear switching condition.

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Exact solutions are derived using linear convolution of a basis pulse and a binary sequence.

Symbolic dynamics are obtained from the exact solution.

Main Results:

The bounded oscillations of the hybrid system are proven to be chaotic.
A one-dimensional piecewise-linear return map with segments of positive and negative slopes is generated from waveform maxima.
The continuous-time dynamics exhibit a folded-band topology.
An exact solution is formulated, enabling the derivation of equivalent symbolic dynamics.

Conclusions:

The developed oscillator is exactly solvable and exhibits chaotic behavior.
The folded-band topology is intrinsically linked to the symbol grammar of the system.
This work provides a novel model for studying chaotic phenomena in hybrid systems.