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

2. Chapter # 2 Flows on the Line One-Dimensional or First-order Systems
Single equation can be a system
f(x,t) is two-dimensional

3. Geometric Way of Thinking

4. Geometric Way of Thinking

5. Geometric Way of Thinking

6. Fixed Points and Stability

8. Example 2.2.2

9. Example 2.2.3
dx/dt = x – cos(x)

10. Population Growth dN/dt = rN implies exponential growth
dN/dt = rN(1 – N/K)
N(t) = Population at time ‘t’
r = Growth Rate
K = Carrying Capacity

11. Linear Stability Analysis
Perturbation grows exponentially if
Perturbation decays if

12. Example 2.4.1

13. Example 2.4.2

14. Example 2.4.3

15. Example 2.4.3

16. Existence and Uniqueness
Consider the initial value problem
Suppose that f(x) and d/dt(f(x)) are continuous on an open interval R of the x-axis, and suppose that x0 is a point in R, then the initial value problem has a solution x(t) on some interval (τ, -τ) about t = 0, and the solution is unique.

17. Impossibility of Oscillations
Flow on a line – Not a circle
No periodic solutions to d/dt(x) = f(x)

18. Mechanical Analog: Overdamped Systems

19. Potentials Potential v(x)
Particles always move toward lower potential.
dV/dx = 0: Equilibrium Point

20. Example 2.7.1

21. Example 2.7.2

22. Solving Equations on Computer
Numerical Integration
MATLAB
See ODE, PDE solution

23. To do:
2.2.1 – 2.2.7
2.4.1 – 2.4.7
2.7.1 – 2.7.6