dx/dt = sin(x)

Viewed as a flow on the line

Viewed as a Flow on the circle
π/2
- π/2
0
π/2
π
3π/2
2π
0 and 2π are not the same point
π
0
2π
3π/2
!
x
1
π/2
0.5
-10
-5
5
10
x
π
0
-0.5
2π
-1
3π/2
dθ/dt = ƒ(θ)

so a 1D system oscillates only when if can be viewed as
flow on a circle
only works if ƒ (θ) is a 2π periodic function
ƒ(θ)=ƒ (θ + 2π)


This ensures that that each point on the circle has a unique
velocity dθ/dt

so it doesn’t work for … d θ /dt = θ
1
Uniform Oscillators

Uniform Oscillator

velocity is doesn’t vary
with θ
!! = "



ω is the angular velocity
T = 2π / ω
T is the period
Nonuniform Oscillators

Nonuniform Oscillator

velocity varies with θ

example:
!! = " # a sin(! )
behavior depends on the
value of a relative to ω
ω
ω=1
slow
π/2
π
a=1.5
a=1.0
a=0.5
a
0, 2π
3π/2
what happens at a=w?
fast
2
Bifurcation at a = ω
a<ω
a=ω
π/2
π/2
π
0, 2π
3π/2
π
a>ω
π/2
0, 2π
3π/2
π
0, 2π
3π/2
saddle node … and its ghost for a<w but close to a=w
Basic properties of an excitable system
1.
The system contains a unique and global resting
state that it returns to when not being
stimulated
2.
The system contains a threshold value such that
when an input stimulates the system above that
threshold value the phase point travels a long
excursion through the phase space before
returning to the rest state.
Neurons are an example of Excitable Cells
3
Oscillations / Rhythms Occur in Nature

Circadian Rhythms (24 hours)


Biochemical Oscillations (1 – 20 min)





sleep wake cycles
metabolites oscillate
Neuronal Oscillations (ms – s)
Cardiac Rhythms (1 s)
Hormonal Oscillations (10 min - 24 hour)
Communication in Animals

firefly
4