Chaos Theory MS Electrical Engineering Department of Engineering GC University Lahore

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

Geometric Way of Thinking

Geometric Way of Thinking

Geometric Way of Thinking

Fixed Points and Stability

Example 2.2.2

Example 2.2.3 dx/dt = x – cos(x)

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

Linear Stability Analysis Perturbation grows exponentially if Perturbation decays if

Example 2.4.1

Example 2.4.2

Example 2.4.3

Example 2.4.3

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.

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

Mechanical Analog: Overdamped Systems

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

Example 2.7.1

Example 2.7.2

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

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