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

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

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

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

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

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) – α n ● Where α > 0 and is the rate that atoms unexcite ● N' = Gn(N(0) – α n) - kn

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

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

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 ● Mr ϕ '' = -b ϕ ' – mgsin ϕ + mrω^2sin ϕ cos ϕ – This is a second order equation however ● Ignore second order term ● B ϕ ' = mgsin ϕ ((rω^2/g)cos ϕ -1) – There are always fixed points at sin ϕ =0 – Also fixed points at (rω^2/g) > 1

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

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

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

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

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.