In current undergraduate electrical engineering, the emphasis on linear systems develops a way of thinking that dismisses nonlinear dynamics as spurious oscillations. The linear systems approach oversimplifies the dynamics of nonlinear systems. This talk helps students understand chaotic behavior using simulations of a continuous-time circuit and a discrete-time system. The undergraduate students and teachers can recognize the educational value of chaotic phenomena from them.

1. 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

3. Contents
• Determinism
• Nonlinear Dynamics: bifurcations and chaos
• Introducing nonlinear dynamics in Undergraduate
Electrical Engineering
• Simulation results
 Logistic map
 DC-DC buck converter

4. 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.”

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

6. “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

7. 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’.

8. 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.

9. 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.

10. DC-DC Buck Converter

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

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

13. Logistic map simulation results
The iterates settle down to a fixed value
Period-1 orbit
1n nv v 

14. Logistic map simulation results (cont.)
The iterated solutions reappear every
second value
Period-2 orbit
2n nv v 

15. Period-4 orbit
4n nv v 
Logistic map simulation results (cont.)

16. Chaotic orbit
Logistic map simulation results (cont.)

17. Bifurcation diagram for logistic map

18. Buck converter simulation results
Period-1 output
waveform and attractor

19. Buck converter simulation results (cont.)
Period-2 output waveform
and
Period-2 attractor

20. Buck converter simulation results (cont.)
Aperiodic output
and
chaotic attractor

21. Buck converter simulation results (cont.)
Periodic output waveform
and
periodic attractor

22. 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.