PX391 Nonlinearity, Chaos, Complexity- General Phase Plane Analysis
2nd Order Nonlinear DE – Intro to General Phase Plane Analysis
We will generalize what was done in lectures for the single nonlinear pendulum (for a complete description
of fixed point topology see for example course textbooks Ott, Drazin)
We have 2 kinds of information: global and local.
Global information
1) Express 2nd order DE as two coupled 1st order DE (which we can’t in general integrate)
dx
= F ( x, y )
dt
dy
= G ( x, y )
dt
Some notation:
autonomous
nonautonomous
c.f. pendulum
x ≡θ
F, G not explicitly t dependent
F, G are explicitly t dependent
dθ
≡y
dt
F ≡ y,
G ≡ −ω 2 sin x
pendulum – dynamics given by trajectory in phase space i.e. y,θ space
2) Can then write
dy G
=
dx F
hence there is a uniquely defined tangent to the y ( x) “trajectory” everywhere except at F = 0 or G = 0 - the
fixed points x, y .
So generally – trajectories can’t cross except at the fixed points.
For the pendulum, other global information came from the symmetries of y (θ ) . So generally look at
symmetries in F , G to understand topology.
N.B.: This framework encompasses Hamiltonian systems where one has p(q) trajectories in p, q phase
space and for each component:
dpi
∂H
=−
dt
∂qi
p, q
dqi
∂H
=−
dt
∂pi
H - Hamiltonian here
are canonical coordinates and H = ( p, q ) = ∑
j
p j2
2m
(H constant)
- the pendulum is of this class.
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+ V (q j )
PX391 Nonlinearity, Chaos, Complexity- General Phase Plane Analysis
Local information:
We can –
i) find fixed points F = 0 , G = 0
: - x, y
ii) linearize about fixed points to classify them.
- To consider the general case – let’s consider that a fixed point x , y exists – classify all possible behaviour.
Classification of fixed point behaviour
Let F = 0 , G = 0 have solution x , y
consider vicinity of this fixed point
x = x +δ x
y = y +δ y
we can Taylor expand F, G
∂F ( x , y )
∂F ( x , y )
δx+
δ y + O(δ x 2 , δ y 2 , δ xδ y )
∂x
∂y
∂G ( x , y )
∂G ( x , y )
δx+
δ y + O(δ x 2 , δ y 2 , δ xδ y )
G ( x, y ) = G ( x , y ) +
∂x
∂y
F ( x, y ) = F ( x , y ) +
but F ( x , y ) , G ( x , y ) = 0
-
assumed that F , G are twice differentiable.
So now need to solve equations of the form:
F = aδ x + bδ y
G = cδ x + d δ y
- nonlinear problem now reduced to linear algebra.
a, b, c, d are the differentials from Taylor expansion a =
So in the vicinity of x , y
dδ x
= aδ x + bδ y
dt
dδ y
= cδ x + dδ y
dt
where
⎛δ x ⎞
⎟
⎝δ y ⎠
δx=⎜
or
dδ x
= J iδ x
dx
⎛a b ⎞
J =⎜
⎟
⎝c d ⎠
2
∂F ( x , y )
∂x
etc.
PX391 Nonlinearity, Chaos, Complexity- General Phase Plane Analysis
Use the method of normal modes – i.e. assume solutions of the form δ x = e st u , where
⎛u ⎞
is a constant vector (eigenvector) and s is eigenvalue (one eigenvalue per eigenvector).
u =⎜ ⎟
⎝v ⎠
then J ⋅ u = s u . To rearrange, write the RHS as a dot product, i.e.:
⎛ s 0⎞⎛u ⎞
su = ⎜
⎟ ⎜ ⎟ = s iu so that J iu = s iu
⎝0 s ⎠⎝ v ⎠
b ⎞
⎛a − s
or ( J − s ) . u = 0 so J − s = ⎜
⎟
d − s⎠
⎝ c
For nontrivial solutions ( u ≠ 0 ) we have J − s = 0
working out the determinant:
0 = ( a − s )( d − s ) − bc
= s 2 − s ( a − d ) + ad − bc
which we will write as s =
giving s =
1
2
{p ±
1
2
i.e. 0 =
{( a + d ) ±
a−s
b
c
d −s
(a + d )
2
− 4 ( ad − bc )
}
}
p 2 − 4q for short.
There are 2 eigenvalues s+ , s− so there are also 2 eigenvectors u + , u −
⎛δ x ⎞
then the general solution for δ x = ⎜ ⎟ is δ x = C1e s+t u + + C2 e s−t u − and C1 , C2 are arbitrary constants.
⎝δ y ⎠
In components this is:
δ x = C1e s+t u+ + C2e s−t u− = Ae s+t + Be s−t
δ y = C1e s+t v+ + C2 e s−t v− = Ce s+t + De s−t
⎛u ⎞
and u ± both satisfy J ⋅ u = su so since u + = ⎜ + ⎟ etc, we have that
⎝ v+ ⎠
au± + bv± = s± u±
if you need to find the u + , u −
cu ± + dv± = s± v±
Now use this solution to classify all fixed points.
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PX391 Nonlinearity, Chaos, Complexity- General Phase Plane Analysis
Classification
All depends on nature of the roots:
s± =
1.
1
2
{p ±
p 2 − 4q
unstable
δy
}
p 2 > 4q, q > 0
δx
s± both real, distinct, same sign
distinct so take s+ > s− (say)
then for t → ∞, δ x
C1e s+t u + dominates, implying
s± > 0 -explosive growth, δ x → ∞
s± < 0 - decay δ x → 0 stable
for p > 0
for p < 0
unstable,
δy
[special case p 2 = 4q] → star point
δx
2.
p 2 > 4q q < 0
s± real, distinct –of opposite sign
consider δ x = C1e s+t u + + C2 e s−t u − and take s+ > 0 > s−
then as t → ∞
so
t → ∞,
δ x ≈ C1e s+t u + , as
δ x = c1e s+t u+
δ y = c1e s+t v+
δy=
t → −∞
v+
δx
u+
δ x ≈ C2 e s−t u −
ditto t → −∞, δ y =
v−
δx
u−
δy
moves from one characteristic to the
other as t → −∞ to t → +∞
δ y = v− μ δ x
−
Saddle point
δx
(only time you need to evaluate u,v)
δ y = v+ μ δ x
+
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PX391 Nonlinearity, Chaos, Complexity- General Phase Plane Analysis
3.
p 2 < 4q
p≠0
complex conjugate pair s± = α ± i β
p 2 − 4q
α = p2 β =
2
so that δ x = C1eα t e + iβ t u + + C2 eα t e − iβ t u − = eα t [C1e + iβ t u + + C2 e − iβ t u − ]
if p, α + ve then we have explosive growth
if
- ve -then decay
in addition - there is an oscillating part with ‘frequency’ β
1
2
spiral fixed point
δy
δx
4.
p 2 < 4q p = 0
s± = ± i q
now
(special case of (3)) so q > 0
Substitute into case 3 above α = 0 β = q then δ x = C1e + iqt u + + C2 e −iqt u − oscillates with ‘frequency’ q.
δy
δx
No growth or decay – centre fixed point
c.f. simple pendulum
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PX391 Nonlinearity, Chaos, Complexity- General Phase Plane Analysis
Summary
p=a+d
∂F ∂G
p=
+
∂x ∂y
Recall
i.e.
q = ad − bc
∂F ∂G ∂F ∂G
q=
⋅
−
⋅
∂x ∂y ∂y ∂x
all evaluated at the fixed point x , y
Nomenclature
Attractors, Attractive fixed points:
stable nodes
Repellors, Repulsive fixed points:
unstable nodes
centre, saddle– not an attractor or repellor
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