Chapter 3: Bifurcations

1. Chapter 3: Bifurcations • Dependence on Parameters is what makes 1-D systems interesting • Fixed Points can be created or destroyed, or the stability of the system itself can changed • These qualitative changes in stability are called Bifurcations • Bifurcation Points are the parameter values at which bifurcations occur

2. 3.1: Saddle-Node Bifurcations • Characterized by two fixed points moving towards each other, colliding, and being mutually annihilated as a parameter is varied. • Other ways of depicting saddle-node bifurcations • Stack of Vector fields • Limit of a continuous stack of vector fields • Bifurcation Diagram • Treat parameter as an independent variable and plot along the horizontal

3. Normal Forms • All Saddle-Node Bifurcations can be represented by x' = r – x^2 or x' = r + x^2 • Prototypical • Anything with this Algebraic Form has a Saddle-Node Bifurcation • Graphically, some function f(x) must have two roots near one another to have a saddle-node bifurcation

4. 3.2: Transcritical Bifurcations • These are situations where a fixed point must exist for all values of a parameter and can never be destroyed • i.e. In logistic population growth models there is a fixed point at 0 population, regardless of growth rate • Normal Form: x' = rx - x^2

5. 3.3: Laser Threshold Example • Consider a solid-state laser • Atoms are excited out of a ground state • When excitement is weak, we have a lamp • When excitement is strong, we have a laser • Model • Dynamic Variable is the number of photons in the laser field, n(t) • Rate of Change is represented by n' = gain - loss

6. 3.3: Continued • N' = gain – loss = GnN – kn • G is a gain coefficient, G > 0 • n(t) is the number of photons • N(t) is the number of atoms • k is a rate constant, k > 0 • As photons are emitted, N decreases. • N(t) = N(0) – αn • Where α > 0 and is the rate that atoms unexcite • N' = Gn(N(0) – α n) - kn

7. 3.4: Pitchfork Bifurcation • Common in Physical problems that have symmetry • Supercritical Bifurcations • Normal Form: x' = rx – x^3 • Invariant under the change of variables x = -x • Subcritical Bifurcations • Normal Form: x' = rx+x^3 • Where the cubic was stabilizing above, its destabilizing here

8. 3.4 Continued • Blow Up • x(t) can reach infinity in finite time if r > 0 is not opposed by the cubic term • In real physical systems, the cubic is usually opposed by a higher order term • X^5 is the first stabilized term that ensures symmetry • X'=rx + x^3 - x^5

9. 3.5: Overdamped Bead on a Rotating Hoop Example • What is the motion of the bead? • Acted on by centrifugal and gravitational forces • The whole system is immersed in molasses • Newton's law for the bead • Mrϕ'' = -bϕ' – mgsinϕ + mrω^2sinϕcosϕ • This is a second order equation however • Ignore second order term • Bϕ' = mgsinϕ((rω^2/g)cosϕ-1) • There are always fixed points at sinϕ=0 • Also fixed points at (rω^2/g) > 1

10. Dimensional Analysis and Scaling • When is it appropriate to drop a second order term? • Exploration through Dimensionless Forms • Allows us to define what small is (<< 1) • Reduces the number of parameters • There is a problem with this • Second order systems require two initial conditions • First order systems require only one • Questions of Validity

11. Phase Plane Analysis • A first order system is a vector field • A second order system can thus be regarded as a vector field on a phase plane • In this example, a graph of angle versus velocity • We want to see how they move about a trajectory • And what these trajectories actually look like

12. 3.6: Imperfect Bifurcations and Catastrophes • An imperfection can lead to a slight difference between the left and right • X' = h + rx – x^3 • If h != 0, symmetry is broken, thus h is the imperfection parameter • Cusp Point • Point where two bifurcations meet • Stability Diagrams • Cusp Catastrophe • Bifurcation Surface folding over itself in spaces • A discontinuous drop from an upper surface to a lower surface

13. Bead on a Tilted Wire • When the wire is horizontal, there is perfect symmetry • If the spring is in tension, the equilibrium point can remain • If the spring is compressed, the equilibrium becomes unstable • When the wire is not horizontal • Catastrophic change can occur in the direction of the tilt if it is too steep

14. 3.7: Insect Outbreak • Example of Catastrophe Bifurcation • N' = RN(1-(N/K))-p(N) • There is a catastrophic point where predation cannot keep population down, and the spruce budworms spread rapidly.