Transcript Poincare Map

Poincare Map
Oscillator Motion

Harmonic motion has both a
mathematical and geometric
description.
• Equations of motion
• Phase portrait
Plane pendulum
q  2 E 1 
2
q
sin 2
E
2
 02
l 12
T  2 ( ) (1  )
g
16
q
E>2

The motion is characterized
by a natural period.
E=2
E<2
q
Convergence

q
The damped driven oscillator
has both transient and
steady-state behavior.
• Transient dies out
• Converges to steady state
q
q  2q  02 q  f cos t

2 

f cos t  arctan 2
2 
0   

q  ae t cos 02   2 t   
2
02   2  4 2 2




Equivalent Circuit

L
vin
C
• Inductance as mass
• Resistance as damping
• Capacitance as inverse
spring constant
v
R
q
C
dq
vR  Ri  R
dt
vout  vC 
vL   L
Oscillators can be simulated
by RLC circuits.
d 2v R dv 1
2


v


V0 sin t
2
dt
L dt LC
2
di
d q
 L 2
dt
dt
vin  vL  vC  vR  0

R
2L
02 
1
LC
Negative Resistance

Devices can exhibit negative
resistance.
•
•

Negative slope current vs.
voltage
Examples: tunnel diode,
vacuum tube
These were described by Van
der Pol.
R. V. Jones, Harvard University


d 2v d
3
2
2


v


v


v


V0 sin t
0
2
dt
dt
Relaxation Oscillator

The Van der Pol oscillator
shows slow charge build up
followed by a sudden
discharge.


y   1  y 2 y  y  0
• Self sustaining without a
driving force

The phase portraits show
convergence to a steady
state.
• Defines a limit cycle.
Wolfram Mathworld
Stroboscope Effect

q
E>2
• Exact period maps to a
point.
E=2
E<2
The values of the motion
may be sampled with each
period.
q

The point depends on the
starting point for the system.
• Same energy, different point
on E curve.

This is a Poincare map
Damping Portrait

Damped simple harmonic
motion has a well-defined
period.

The phase portrait is a spiral.

The Poincare map is a
sequence of points
converging on the origin.
E  exp[ 2 (
4mk

2

Damped harmonic mq
  kq  q
motion
k
2  2
T  2 ( 
)
2
m 4m
1
q
q
1
2
 1) ]
Undamped
curves
Energetic Pendulum


A driven double pendulum
exhibits chaotic behavior.

The Poincare map consists
of points and orbits.
l
m
l
f
m
pf
f
• Orbits correspond to
different energies
• Motion stays on an orbit
• Fixed points are non-chaotic