introducing undergraduate electrical engineering students to chaotic dynamics: computer simulations...
TRANSCRIPT
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.