Phase Transitions in Coupled
Nonlinear Oscillators
Tanya Leise
Amherst College
[email protected]
Materials available at
www.amherst.edu/~tleise
Single Finger Oscillation
Single Finger Oscillation
Bimanual Oscillations
Left hand
Right hand
Right hand
Phase portrait:
Left hand
Bimanual Oscillations
Left hand
Right hand
Left hand
Right hand
Right hand
Left hand
Bimanual Oscillations
• Increasing frequency:
Out-of-phase
In-phase
Transition
Basic Features
1. Only two stable states exist: in-phase and
out-of-phase.
2. As the frequency passes a critical value,
out-of-phase oscillation abruptly changes to
in-phase.
3. Beyond this critical frequency, only in-phase
motion is possible.
Developing a Model
• Goals:
– To develop a minimal model that can reproduce
these qualitative features
– To gain insight into underlying neuromuscular
system (how both flexibility and stability can be
achieved)
“Nature uses only the longest threads to weave her
pattern, so each small piece of the fabric reveals the
organization of the entire tapestry.”
R.P. Feynman
Developing a Model
• Control parameter:
– Frequency w of oscillation (1-6 Hz; divide
by 2 for radians/sec)
• Variables (describing oscillations):
– Relative phase f of fingers (0º or 180º)
– Amplitude r of finger motion (0-2 inches)
Differential equation models
x(t)  r cos(wt  f )  displacement of fingertip
Mass - spring with natural frequency
w  k /m :
..
mx  kx  0
Damped spring, always decays to zero :
..
.
mx  cx  kx  0
Forced spring :
..
.
mx  cx  kx  F0 cos(wt)
Nonlinear Oscillator
• Include nonlinear damping term(s) to
yield desired phase shifts as w increases
• Obtain “self-sustaining” oscillations if use
negative linear damping term
• Hybrid oscillator (Van der Pol/Rayleigh):
..
.
.3
2 .
x  x  x  x x  w 2 x  0
• Seek stable oscillatory solution of form
x(t)  r cos(wt  f )

Single Finger Oscillatory Solution
x(t)  r(t)coswt  f (t)
.
.
.
x(t)  r(t)w  f (t)sin wt  f (t)  r(t)coswt  f (t)
If nearly oscillatory,
velocity should be
x(t)  r(t)w sin wt  f (t).
.
For this to be true we must have
.
r(t)f (t)sin wt  f (t)  r(t)coswt  f (t)  0.
.
The acceleration is then
.
x(t)  r(t)sin wt  f (t)  r(t)w  f (t)coswt  f (t).
..
.
Single Finger Oscillatory Solution
x(t)  r(t)coswt  f (t) 

..
.
.3
2.
 x  x  x  x x  w 2 x  0
xÝ(t)  r(t)w sin wt  f (t)

.
.
r(t)f (t)sin wt  f (t)  r(t)coswt  f (t)  0
.
.
Solve for r and f :
r  r sin 2 (wt  f )  w 2 r 2 sin 2 (wt  f )  r 2 cos 2 (wt  f )
.
1
f   sin 2(wt  f )  w 2 r 2 sin 2 (wt  f )  r 2 cos 2 (wt  f )
2
.
Single Finger Oscillatory Solution
If amplitude and phase vary negligibly over a period
we can integrate over a period to approximate
.
r
r4  (3w 2   )r 2 
4w
A steady oscillation results if
.
and f  0.
.
.
r  0 and f  0 :

x(t)  2
cos(wt  f 0 ),
2
3w  
.
dr 2
which is stable since

 0.
dr
w
.
2
.w
r and f :
,
Coupled Nonlinear Oscillators
x L (t)  rL (t)cos(wt + f L (t)) = left fingertip displacement
x R (t)  rR (t)cos(wt + f R (t)) = right fingertip displacement
..
.
.3
..
.
.3
.
2
.
2
.
.
.
.
x L  x L  x L  x x L  w x L  (x L  x R )(a  b(x L  x R ) 2 )
2
L
x R  x R  x R  x x R  w x R  (x R  x L )(a  b(x R  x L ) 2 )
2
R
Bimanual Oscillatory Solutions
Assume x k (t)  rk (t)coswt  f k (t)
and x k (t)  rk (t)w sin wt  f k (t)
.
.
.
Requires  rk (t)f k (t)sin wt  f k (t)  rk (t)coswt  f k (t)  0
(where k  L,R)
Solve two DEs plus two conditions for
Integrate each of these over a period
.
.
.
.
rL , rR , f L ,and f R
2
w.
Set equal to zero and solve for rL , rR , and f L - f R
Bimanual Oscillatory Solutions
In - phase oscillation ( f L  f R = 0)

x L (t)  x R (t)  2
cos(wt)
2
3w  
Out - of - phase oscillation ( f L  f R   )
-a
x L (t)  2
cos(wt)
2
3w   - 8b
-a
x R (t)   2
cos(wt)
2
3w   - 8b
Stability Analysis
To determine stability,
examine eigenvalues of the Jacobian
.
.
.
 (rL ,rR , f )
 (rL ,rR , f )
.
Case 1 (in - phase) :
a  2
a
0 
. . .


 (rL ,rR , f )  
rL  rR  2
, f 0:
  a
a  2
0 
2
3w  
 (rL ,rR , f ) w 


0
2a
 0

Case 2 (out - of - phase) :

#
.
. .
-a
 (rL ,rR , f ) 2 
rL  rR  2
, f  :

#
2
3w   - 8b
 (rL ,rR , f ) w 
0

#
#
0

0


0

4b  a(3w 2   ) 

3w 2    8b 
Loss of Stability Leads to Phase Transition
• Stability of the out-of-phase motion depends on the
sign of the eigenvalue
.
2
f 2 4b  a(3w   ) 



2
f w  3w    8b 
• Increasing frequency w beyond a critical value wcr
leads to change in stability of out-of-phase motion,
triggering switch to in-phase motion

w cr 
4b  a
3a
Energy Well Analogy
V
• Potential function V(f) defined via f  
f
.
• Minima of V correspond to stable phases
• Maxima of V correspond to unstable phases
.
2

V
f

2
2


(r

r

r
,
f
)

(2a

br
)cos
f

br
cos2f 

L
R
2
f
f
w
Integrate twice to obtain

 
1 2
2
V (f;r)   (2a  br )cos f  br cos2f 

w 
4
An Energy Well Model
B
 0.1
A
B
 2.0
A
Slow twiddling
frequency
B 1

A 4
Fast twiddling
frequency
B 1

A 4

 
1 2
2
V (f;r)   (2a  br )cos f  br cos2f  Acos f  Bcos2f

w 
4


Basic Twiddling Model
Potential function V
for phase difference f:
Stable states correspond
to

energy wells (minima of V):
dV 
f
df
V  Acos(f )  Bcos(2f )
dV
 Asin( f )  2Bsin( 2f )  0
df
d 2V
 Acos(f )  4Bcos(2f )  0
2
df
Conclusion: f=0 is always a stable state (minimum for any a and b),
while f= is a stable state only for parameter values B/A>1/4.
Sources
• H. Haken, J.A.S. Kelso, and H. Bunz. A theoretical
model of phase transitions in human hand
movements. Biol. Cybern., 51:347-356, 1985.
• A.S. Kelso, G. Schöner, J.P. Scholz, and H. Haken.
Phase-locked modes, phase transitions and
component oscillators in biological motion. Physica
Scripta, 35:79-87, 1987.
• B.A. Kay, J.A.S. Kelso, E.L. Saltzman, and G.
Schöner. Space-Time Behavior of Single and
Bimanual Rhythmical Movements: Data and Limit
Cycle Model. Journal of Experimental Psychology:
Human Perception and Performance, 13(2):178-192,
1987.