Chaos Theory MS Electrical Engineering Department of Engineering GC University Lahore

Bifurcations Qualitative changes in the dynamics are called bifurcations, The parameter values at which they occur are called bifurcation points. Bifurcations are important scientifically-they provide models of transitions and instabilities as some control parameter is varied.

Saddle-Node Bifurcation Fixed-Points are created and destroyed

Saddle-Node Bifurcation

Saddle-Node Bifurcation

Bifurcation Diagram Fold Bifurcation, Turning-Point Bifurcation, Blue Sky Bifurcation

Saddle-Node Bifurcation

Saddle-Node Bifurcation

Transcritical Bifurcation At least one fixed-point always exists It may change its stability with change in parameter

Transcritical Bifurcation

Laser Threshold Solid-State Laser Each Atom is a tiny radiating antenna Below threshold – random phases After threshold – in-phase oscillations

Laser Threshold Number of photons – n(t) Gain is due to stimulated emission G > 0 – Gain coefficent N(t) – Number of excited atoms K>0 – rate constant that is reciprocal of typical lifetime of a photon in laser

Laser Threshold N decreases by emission of photons Thus

Laser Threshold

Pitchfork Bifurcation Supercritical Pitchfork Bifurcation Subcritical Pitchfork Bifurcation

Supercritical Pitchfork Bifurcation

Supercritical Pitchfork Bifurcation

Subcritical Pitchfork Bifurcation

Subcritical Pitchfork Bifurcation Stabilizing term

Overdamped Bead on a Rotating Hoop

Overdamped Bead on a Rotating Hoop

Imperfect Bifurcations and Catastrophe Imperfection parameter

Imperfect Bifurcations and Catastrophe

Imperfect Bifurcations and Catastrophe

Cusp Catastrophe

Cusp Catastrophe

Insect Outbreak spruce budworm vs balsam fir tree Budworms – Fivefold in a year (Characteristic Time Scale in Months) Trees – Replace their foliage in 7-10 years (Lifespan of 100-150 years without budworms)

Model

p(N)

Complete Model

Dimensionless Formulation Divide by B and let x = N/A

Analysis of Fixed Points Fixed Point at x = 0 Other Fixed Points:

Fixed Points Refuge vs Outbreak

Bifurcation Curves Condition for Saddle-Node Bifurcation is that the line r(1-x/k) intersects the curve x/(1+x2) tangentially. Thus,

Bifurcation Curves