Introducing Undergraduate Electrical Engineering

Students to Chaotic Dynamics: Computer Simulations with Logistic Map and Buck Converter

Sajid Iqbal Ph.D student

Harbin Institute of Technology

Contents

• Determinism

• Nonlinear Dynamics: bifurcations and chaos

• Introducing nonlinear dynamics in Undergraduate

Electrical Engineering

• Simulation results

Logistic map

DC-DC buck converter

Laplace described determinism as, “If you give me the positions and momenta of all the particles in the Universe, I will predict all past and future.”

Bifurcation is a sudden qualitative change in the behavior of a dynamical system caused by the variation of its parameters.

“We collectively wish to apologize for having misled the general educated public by spreading ideas about the determinism of systems satisfying Newtons’ laws of motion that, after 1960, were proved to be incorrect.”

Sir James Lighthill collective apology on behalf of all scientists

Deterministic Chaos is an unstable aperiodic behavior in deterministic dynamical system, which shows sensitive dependence on initial conditions.

Edward Lorenz coined the term ‘Butterfly Effect’.

The advent of chaos theory shattered and obscured the well-regarded Newtonian vision.

The consequence of chaos is that complex behavior need not have complex causes.

The logistic map also known as the “Verhulst model” is given as Xn +1 = k * Xn (1-Xn) 0 < k < 4 Where parameter ‘k’ represents the population growth rate and ‘Xn’ is the variable at the nth iteration.

DC-DC Buck Converter

A time series plot is a display of data points that shows how values have changed over uniform time intervals.

A bifurcation diagram is a visual summary chart of the behaviors exhibited by a dynamical system, when some parameters are varied.

Logistic map simulation results The iterates settle down to a fixed value

Period-1 orbit

1n nv v

Logistic map simulation results (cont.)

The iterated solutions reappear every second value

Period-2 orbit

2n nv v

Period-4 orbit

4n nv v

Logistic map simulation results (cont.)

Chaotic orbit

Logistic map simulation results (cont.)

Bifurcation diagram for logistic map

Buck converter simulation results

Period-1 output waveform and attractor

Buck converter simulation results (cont.)

Period-2 output waveform and

Period-2 attractor

Buck converter simulation results (cont.)

Aperiodic output and

chaotic attractor

Buck converter simulation results (cont.)

Periodic output waveform and

periodic attractor

Such dynamical systems are excellent vehicles for explaining concepts of chaotic dynamics. They provide an easy-to-understand idea of this novel and productive way of thinking.

Don't curse the darkness, light a candle.