1. Chaos Theory MS Electrical Engineering Department of Engineering
GC University Lahore

2. Course Contents Introduction Flows on the line Bifurcations
Flows on the circle
Linear Systems
Phase Planes
Limit Cycles
Lorenz Equation
One dimensional Maps
Fractals
Strange Attractors

3. Books
“NONLINEAR DYNAMICS AND CHAOS With Applications to Physics, Biology, Chemistry, and Engineering”, STEVEN H. STROGATZ
“CHAOS AND NONLINEAR DYNAMICS: An Introduction for Scientists and Engineers”, Robert C. Hilborn
“The illustrated Dictionary of NONLINEAR DYNAMICS AND CHAOS”, Tomasz Kapitaniak, Steven R. Bishop

4. Research Chaos, An Interdisciplinary Journal of Nonlinear Science
International Journal of Bifurcation and Chaos
Chaos, Solitons and Fractals

5. Chaos – Meaning Pronunciation:/keɪɒs/noun
complete disorder and confusion
Example: “snow caused chaos in the region”
Physics:
the property of a complex system whose behaviour is so unpredictable as to appear random, owing to great sensitivity to small changes in conditions.
the formless matter supposed to have existed before the creation of the universe.
Reference: Oxford Dictionary

6. Probabilistic vs Deterministic
Random
Deterministic

7. Static vs Dynamic Static Systems Dynamic Systems
Dynamics: Subject that deals with change
systems that evolve in time, settles down to equilibrium, keeps repeating in cycles, or does something more complicated

8. Brief History of Dynamics
17th Century – Newton solving Two-Body Problem (Sun & Earth) using Differential Equations and Law of Gravitation
Three-Body Problem (Sun, Moon and Earth) – N o explicit solution
Late 19th Century – Poincare – Qualitative Solution rather than Quantitative, Geometric Approach
Dynamics restricted to Nonlinear oscillators in radio, radar, phase-locked loops, and lasers
Lorenz's discovery in 1963 of chaotic motion on a strange attractor – Weather Model: Aperiodic, very sensitive to Initial Conditions

9. Types of Dynamical Systems
Differential equations: the evolution of systems in continuous time, whereas iterated maps arise in problems
Iterated maps (Difference equations): time is discrete

10. Differential Equations
LINEAR

11. Differential Equations
Exponential Growth of population of organisms

12. Differential Equations
NONLINEAR

13. Trajectory & Phase-Space
Draw the trajectories without actually solving the system

14. Non-autonomous Systems
Time Dependent Systems

15. Dimension of the Phase Space
n = 1: Growth, Decay or Equilibrium e.g. RC Circuit (Linear), Logistic Equation (Nonlinear)
n = 2: Oscillations e.g RLC Circuit (Linear), Pendulum (Nonlinear)
n >= 3: Three-Body Problem, Chaos & Fractals (Nonlinear)

16. Example:

17. Example (Continued)

18. Application Areas Mathematics Biology Computer science Economics
Engineering
Finance
Philosophy
Physics
Politics
Population dynamics
Psychology

19. Chaos in Electrical Circuits

20. Chaos in Electrical Circuits
Secure Communication