Chaotic-Duffing-Oscillator

About

This project aims to demonstrate a nonlinear (chaotic) dynamic system by an application of Duffing Osillator (Equation). This chaotic structural model poses a chaotic behavior for a set of defined parameters and initial conditions, and to approximate the dynamics of the system, a fourth-order Runge-Kutta method will be applied to solve the problem.

Abstract

The chaotic behavior in motion is not unusual to observe in practice in common nonlinear differential equations. Through close examination of nonlinear dynamics of a system governed by a Duffing’s Equation, one can observe how phase-space motion follows a strange attractor as harmonic excitations are applied to the system. A Duffing oscillator system of complexity as simple as a second-order ordinary differential equation exhibits bifurcation past some threshold value of excitation due to periodic doubling, bounded by five parameters and initial conditions. Through visual aids and graphical responses, the chaotic motion of a Duffing system (i.e. spring) is observed, and verified in simple visualization techniques in Python.

Tools/Packages

Visual Python, Matplotlib, Scipy, Numpy, FFMpeg

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Purpose

Analyze stability through measures including eigenvalues of Jacobian/Lyapunov exponent, strange attractor in phase-space diagram, Poincare Sections and bifurcation due to period doubling

Determine point of bifurcation in terms of ratio of excitationo amplitude to damping coefficient

Make graphs representing periodic, quasi-periodic, and chaotic behavior of the system

Compare behaviors in time-,space- and frequency- domains under different conditions for initial conditions and parameters of interest

Duffing Oscillator

Duffing Oscillator(Equation) is a (nonlinear) system that is often periodically forced and damped with some nonlinear elasticity associated. The governing equation is as follows: , where the parameters are damping coefficient, nonlinearity coefficient, linear stiffness, excitation amplitude, excitation angular frequency, in order.

Related Topics

Visualization of numerically approximated data

Solving second-order ordinary differential equations

Observing chaos through eigenvalues and time series graphs

Methods Used

Runge-Kutta Algorithm (4th order) - widely used numerical method for differential equations, by interpolating derivatives at half-way points and taking a weighted average with the step size into account

Harmonic Balance Method - most commonly used method in studying the steady-state frequency response in a nonlinear dynamical system (autonomous and non-autonomous). By assumed the response in terms of a Fourier series in the differential equation and separating the harmonic coefficients, the frequency amplitude and the unknown coefficients in the response relation of the system can be determined (assuming the response to be in form of a sinusoid).

Link to final paper:

• Final Paper