PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 5- S C Chapman
2nd order non-linear DE – Pendulum
Generalize what was done for ID, 1st order nonlinear DE – non- trivial differences.
Start by considering a familiar example: pendulum
L
d ! g
+ sin ! = 0
L
dt 2
2
!
g
with linear equation
g
d 2!
+ " 2! = 0
2
dt
- oscillates
for sin ! ! ! , ie: small perturbations about rest
position ! = 0
!2 =
! = Aei"t
g
.
L
How to deal with 2nd order DE? Write as two coupled 1st order DEs:
y=
d!
,
dt
dy
= "# 2 sin !
dt
Now we can find fixed points, linearize as before.
Fixed points
y = 0, sin ! = 0 or ! = n"
n = !3, ! 2, ! 1, 0, 1, 2....
Stability/Classification
y = y + ! y(t ) = ! y
Linearize, we write
! = ! + "! ( t ) = n# + "!
with ! y, !" small. Then linearized equations are:
d !"
=!y
dt
d! y
n
= $# 2 ( $1 ) !"
dt
since sin ( n! + "# ) = sin ( n! ) cos "# + cos n! sin "#
n
cos ( n! ) = ( !1)
sin !" ! !"
don't know behaviour of y ( ! ) yet.
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 5- S C Chapman
Now, can write:
d 2!"
n
= #$ 2 ( #1) !"
2
dt
- the linearized 2nd order ODE. There are 2 possibilities:
n even
d 2!"
= #$ 2!"
dt 2
!" = Aei$ t + Be# i$ t
oscillatory solution
d 2!"
= +# 2!"
dt 2
n odd
!" = Ae# t + Be$# t
- behaviour depends on the integration consts A, B
d ( !" )
dt
!y?
!y =
n even
dy = i! Aei! t " i! Be" i! t = ! Aei ( ! t + # / 2 ) + ! Be" i ( ! t + # / 2 )
i = ei! / 2 ,!i = e!i! / 2
ie: out of phase ! / 2 with !"
oscillatory – about centre fixed point.
" y = ! Ae!t # ! Be #!t
"# = Ae!t + Be!!t
n odd
this is either attracted or repulsed from fixed point.
!y
t!"
" A" term dominates
t ! #"
" B" term dominates
t!"
A, B > 0
y,!
!"
from t = !"
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 5- S C Chapman
Consider all possible combinations of signs of A, B – four possibilities:
ƒy
A > 0, B < 0
A, B > 0
!"
A < 0, B < 0
Saddle point:
Separatrix has lines given by
# y % Ae% t
t ! ",
=
=%
#$
Ae% t
$ y "& Be"& t
t ! "#,
=
= "&
$%
Be"& t
A < 0, B > 0
to make phase plane diagram (global solution) we need some global properties.
Will now do a sketch in y, ! phase plane.
Global properties for the pendulum:
Constant of the motion:
Equation of motion
d 2!
+ " 2 sin ! = 0
dt 2
can integrate once
d! d 2!
d!
+ " 2 sin !
=0
2
dt dt
dt
!
d!
dt
2
1 " d! %
y2
2
or
# ! 2 cos " = E .
(
)
cos
!
=
E
2
2 $# dt '&
Also symmetry property:
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 5- S C Chapman
! ! "!
y ! "y
E = #! 2 cos " 0
Different IC give different E. Say we start the pendulum at rest – at !0
- this is the maximum ! of the motion. ! is symmetric
+!0 = !!0
*NB: ! ! "! gives the same equation of motion, ! symmetry property.
Topology
We have
So
So, unless
of y, ! plane
d!
dy
=y
= #" 2 sin !
dt
dt
dy #! 2 sin "
=
d"
y
y = 0, # ! 2 sin " = 0 (the fixed points)
dy
is uniquely defined. This is tangent to y ( ! ) lines
d!
therefore, y ( ! ) lines cannot cross- General property.
Sketch of the phase plane
y
separatrix
n×= !2
×
n = !1
4
×
n=0
×
n =1
E = Ec
×
n=2
!
PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 5- S C Chapman
Properties of the phase plane:
- lines don't cross except at fixed points
y = 0, ! = n"
- n = 0 , n even are centres (oscillatory solutions)
- n odd are saddle points
y2
constant
- symmetric in y, !
E=
# ! 2 cos "
2
d!
- y is bounded for a given !0 at y =
=0
E = E0 = "# 2 cos! 0
dt
- now the separatrix corrects the saddle points, e.g. passes through !0 = ! = ", y = y = 0
This is the line where E = Ec = ! 2 , critical value of E
- critical value of E separates bonded from unbounded orbits.
E < Ec
oscillates about stable (circle) fixed point ! < "
! = " unstable (saddle) fixed point
!
E > Ec
unbounded
motion
Global behaviour obtained without integrating equation
2
1 # d! $
% " 2 cos ! = E
NB: integral of &
'
2 ( dt )
Can be done analytically- Jacobian Elliptic functions – look up if interested.
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 5- S C Chapman
Another way to think of E – in terms of potential function V
E=
y2
+ V (! )
2
V = #! 2 cos "
×
0
oscillatory
E > E int
×
!
×
separatrix
circle (centre) fixed point
If E > Ec - unbounded motion.
This is conservative system but now also know what happens for weak damping.
'weak' ≡ on timescale > 1 / !
particle slowly loses energy.
See handout for sketch
All of the above generalizes to any 2nd order nonlinear DE- for general properties see the handout
accompanying the lecture.
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