Bifurcations Part 2 Transcritical Bifurcation Laser Model Example

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Transcritical bifurcations occur where a fixed point changes its stability type, or a pair of equilibrium points exchange their stability type. A model of a laser is given as an example. From online course 'Nonlinear Dynamics and Chaos'. • ► Jump to laser model 5:45 • ► Next, the pitchfork bifurcation • Bifurcations Part 3- Pitchfork Bifurc... • ► For a bifurcations introduction, see part 1 • • Bifurcations Part 1, Saddle-Node Bifu... • • ► Bifurcation topics • saddle-node bifurcation • Bifurcations Part 1, Saddle-Node Bifu... • transcritical bifurcation • Bifurcations Part 2- Transcritical Bi... • pitchfork bifurcation • Bifurcations Part 3- Pitchfork Bifurc... • robustness of bifurcations • Bifurcations Part 4- Robustness of Bi... • ► Online course playlist • https://is.gd/NonlinearDynamics • ► New topics posted on a regular basis. • Subscribe https://is.gd/RossLabSubscribe • ► Dr. Shane Ross, Virginia Tech professor (Caltech PhD) • http://chaotician.com • ► In-depth mathematical discussion of bifurcations • Bifurcation Theory: Saddle-Node, Hopf... • ► Course lecture notes (PDF) • https://is.gd/NonlinearDynamicsNotes • ► Chapters • 0:00 Bifurcation review, the saddle-node • 0:49 Transcritical bifurcation introduction • 5:45 Model laser, simple 1-D model • 9:30 Transcritical bifurcation in laser model • ► Reference: Steven Strogatz, Nonlinear Dynamics and Chaos , Chapter 3: Bifurcations • • • bifurcation bifurcations trans-critical laser model exchange of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field One-Dimensional 1-dimensional Functions • #NonlinearDynamics #Bifurcations #transcritical #laser #exchangebifurcation #exchangeofstability #transcriticalbifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #Poincare #Strogatz #graphicalmethod #FixedPoints #EquilibriumPoints #Stability #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #OneDimensional #Functions

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