Chapter 3: Bifurcations
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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
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These qualitative changes in stability are called
Bifurcations
Bifurcation Points are the parameter values at which
bifurcations occur
3.1: Saddle-Node Bifurcations
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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
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Stack of Vector fields
Limit of a continuous stack of vector fields
Bifurcation Diagram
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Treat parameter as an independent variable and plot along
the horizontal
Normal Forms
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All Saddle-Node Bifurcations can be represented
by x' = r – x^2 or x' = r + x^2
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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
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These are situations where a fixed point must exist
for all values of a parameter and can never be
destroyed
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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
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Consider a solid-state laser
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Atoms are excited out of a ground state
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When excitement is weak, we have a lamp
When excitement is strong, we have a laser
Model
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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
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N' = gain – loss = GnN – kn
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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.
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N(t) = N(0) – αn
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Where α > 0 and is the rate that atoms unexcite
N' = Gn(N(0) – α n) - kn
3.4: Pitchfork Bifurcation
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Common in Physical problems that have symmetry
Supercritical Bifurcations
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Normal Form: x' = rx – x^3
Invariant under the change of variables x = -x
Subcritical Bifurcations
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Normal Form: x' = rx+x^3
Where the cubic was stabilizing above, its destabilizing
here
3.4 Continued
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Blow Up
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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
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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
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What is the motion of the bead?
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Acted on by centrifugal and gravitational forces
The whole system is immersed in molasses
Newton's law for the bead
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This is a second order equation however
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Mrϕ'' = -bϕ' – mgsinϕ + mrω^2sinϕcosϕ
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
Dimensional Analysis and Scaling
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When is it appropriate to drop a second order
term?
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Exploration through Dimensionless Forms
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Allows us to define what small is (<< 1)
Reduces the number of parameters
There is a problem with this
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Second order systems require two initial conditions
First order systems require only one
Questions of Validity
Phase Plane Analysis
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A first order system is a vector field
A second order system can thus be regarded as a
vector field on a phase plane
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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
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An imperfection can lead to a slight difference
between the left and right
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X' = h + rx – x^3
If h != 0, symmetry is broken, thus h is the
imperfection parameter
Cusp Point
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Point where two bifurcations meet
Stability Diagrams
Cusp Catastrophe
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Bifurcation Surface folding over itself in spaces
A discontinuous drop from an upper surface to a lower
surface
Bead on a Tilted Wire
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When the wire is horizontal, there is perfect
symmetry
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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
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Catastrophic change can occur in the direction of the
tilt if it is too steep
3.7: Insect Outbreak
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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.