Homoclinic Bifurcations to Equilibria

Homoclinic Bifurcations to
Equilibria
I. Theory and applications
Alan Champneys University of Bristol
Department of Engineering Mathematics
MRI Master Class Utrecht – p. 1

Outline: Lecture 1
1. Introduction: solitary waves & global bifurcations
2. Tame and chaotic homoclinic bifurcations to equilibria
Shil’nikov’s theorems
application: excitable systems
3. Reversible and Hamiltonian systems
hyperbolic cases ⇒ one codimension less
saddle-centre homoclinics
4. Simple strategies for continuation of homoclinic orbits in
AUTO-07P.
MRI Master Class Utrecht – p. 2

1. solitary waves.
1834 J Scott-Russell observed barge on aqueduct
. . . a boat drawn along a narrow channel . . . suddenly stopped . . . the mass
of water in the channel . . . accumulated around the prow [and] rolled forward
with great velocity, assuming the form of a large solitary elevation, a rounded,
smooth heap of water, which continued its course along the channel [at 8 or 9
miles an hour for 1.5 miles] preserving its original feature some thirty feet
long and a foot and a half high . . .
Explanation: 1870s Bousinesq & Lord Rayleigh, theory
of wall of water = ‘solitary waves’
MRI Master Class Utrecht – p. 3

The ‘soliton’
1895 Korteweg & de Vries derived KdV equation
∂u
∂t
= ∂3 u
∂x
+ 6u ∂u
3 ∂x
solution speed c
u(x,t) = (−c/2)sech 2 ( √ c/2)(x − ct)
sech 2 (x)
x
1960s Zabusky & Kruskal showed its ‘completely’
stable
⇒ New name for particle-like solitary waves; solitons
1970s because KdV equation is integrable (Lax,
Gardner, Zakarov ... & the ‘Clarkson mafia’)
MRI Master Class Utrecht – p. 4

The ‘soliton’
1895 Korteweg & de Vries derived KdV equation
∂u
∂t
= ∂3 u
∂x
+ 6u ∂u
3 ∂x
solution speed c
u(x,t) = (−c/2)sech 2 ( √ c/2)(x − ct)
sech 2 (x)
x
1960s Zabusky & Kruskal showed its ‘completely’
stable
⇒ New name for particle-like solitary waves; solitons
1970s because KdV equation is integrable (Lax,
Gardner, Zakarov ... & the ‘Clarkson mafia’)
Nb. ‘Solitary killer’ waves e.g. Tsunami
MRI Master Class Utrecht – p. 4

Optical solitons
Optical fibres; means of transatlantic communication.
Pulses travel at speed of light c
MRI Master Class Utrecht – p. 5

Optical solitons
Optical fibres; means of transatlantic communication.
Pulses travel at speed of light c
Problem: denigration due to dispersion
One idea (Hasegawa & Tappert 1973) use natural Kerr
nonlinearity to self-focus the light
MRI Master Class Utrecht – p. 5

⇒ Nonlinear Schrödinger (NLS) equation
i ∂v
∂t + ∂2 v
∂x
+ Q|v| 2 v = 0
2
also integrable & explicit ‘envelope of waves’ soliton
v(x,t) = V m e ikt √
sech(V m |Q|/2U(x − ct))
envelope of wave
oscillating wave
Use solitons as bits of information
1
0
1 1 1 0 1
1 1 0 1 1 0 1 1
MRI Master Class Utrecht – p. 6

... but
KdV and NLS are leading-order approximations: ‘Most’
(probability 1) nonlinear wave equations NOT integrable
can exist a ‘zoo of solitary waves’:
u(x,t) = U(x − ct) ⇒ ODE for U(ξ)
How big is the zoo? their dynamics? (not these lectures)
MRI Master Class Utrecht – p. 7

Another motivation: global bifurcation
Ed Lorenz 1963 discovered chaos in simple system
ẋ = σ(y − x)
ẏ = x(ρ − z) − y
ż = xy − βz
σ = 10, β = 8/3, ρ increasing
homoclinic bifurcation triggers chaos at ρ ≈ 24
( Sparrow 1982 )
MRI Master Class Utrecht – p. 8

Application: Chua’s electronic circuit
smooth version (due to A. Khibnik 93)
˙U = α(V − (1/6)(U − U 3 ))
˙V = U − V + W
Ẇ = −βV
(α,β) = (0, 0): Z 2 -symmetric Takens Bogdanov point at
0: σ(0) = {0, 0, −λ} ⇒ tame homoclinic bifurcation.
Large enough α, β ‘double scroll’ chaotic attractor
3
2
1
z(t)
0
–1
–2
–3
–2
0
y(t)
0.2
0.4
2
1
0
x(t)
–1
MRI Master Class Utrecht – p. 9

tame and chaotic homoclinic orbits
1. at (α, β) = (1.13515, 1.07379), neutral saddle;
2. at (α, β) = (1.20245, 1.14678), double real leading eigenvalue
(with respect to stable eigenspace, which is non-determining);
4. at (α, β) = (1.74917, 1.76178), neutral saddle-focus;
δ = 1 ⇒ transition to chaotic bifurcation
5. at (α, β) = (6.00000, 7.191375), neutrally-divergent saddle-focus
δ = 1/2
⇒ divDf(0)x < 0 ⇒ attracting (strange attractor)
MRI Master Class Utrecht – p. 10

Later we will study a similar simple electronic circuit due to
Friere et al 1993
...
MRI Master Class Utrecht – p. 11

2. Homoclinic orbits to equilibria
Heteroclinic orbit Γ connecting equilibria x 1 ,x 2 ∈ R n
ẋ(t) = f(x(t),α)
x(t) → x 1 , x 2
as t → ±∞.
‘generic’ system: no symmetry or first integrals
Homoclinic orbit special case x 1 = x 2 = x 0
x 0 hyperbolic ⇒ codim 1, i.e. at isolated α = α 0
W
u
Γ
W
u
Γ
W
s
p
O
q
Z
W
s
p
O
q
Z
W
ss
W
ss
(a)
(b)
MRI Master Class Utrecht – p. 12

Homoclinic bifurcation as α varies
Suppose ∃ Hom orbit Γ at α = α 0 . Linearisation at x 0 :
σ(x 0 )
Im
determining
Re
leading
Theorem 1 (Shil’nikov’s tame homoclinic bifurcation)
Real determining eigenvalue ⇒ unique periodic orbit
destroyed at infinite period as α → α 0 .
Theorem 2 (Shil’nikov’s chaotic homoclinic bifurcation)
Complex determining eigenvalue ⇒ ∞-many high-period
periodic orbits in neighbourhood of Γ and α 0 .
MRI Master Class Utrecht – p. 13

Exercise
Proof of tame homoclinic bifurcation in 2D system using
Poincaré maps.
Consider
ẋ = λx + nonlinear
ẏ = −µy + nonlinear
Make assumption that at parameter value α = 0, there is a
homoclinic orbit that connects this equilibrium to itself.
MRI Master Class Utrecht – p. 14

In chaotic case ∃ ∞-many N-pulse homoclinic orbits at
nearby α-values. For each N
Γ
W u
e.g. 2-pulse homoclinic orbits Gaspard
more recently: Turaev , Sandstede 00 see (Shil’nikov et
al 1992, 1998)
Theorem 3 (Homoclinic ‘centre manifold’ theorem)
There exists a C 0 manifold of dimension of the leading
eigenspace in the neighbourhood of Γ that captures all
nearby recurrent dynamics. MRI Master Class Utrecht – p. 15
W
s
O

Sketch proof of Shil’nikov chaotic case
saddle focus case in R 3 (Glendinning & Sparrow 84)
assume ∃ α 0 = 0 at which hom orbit Γ to x 0 = 0 WLOG
σ(0) = {−µ ± iω,λ} (WLOG reverse time if nec.)
Construct Poincaré map close to Γ in α and x
Fixed points ⇒ periodic orbits
MRI Master Class Utrecht – p. 16

construct Poincaré map
Γ
W u
W s
Step 1: Set up Poincaré sections
Σ in = {θ = 0}, Σ out = {z = h}
MRI Master Class Utrecht – p. 17

Σ out
z
r
θ
Σ in
Step 2 Linearise flow near 0 to compute Π loc : Σ in → Σ out
ż = λz
˙θ = ω
ṙ = −µr
+ h.o.t
MRI Master Class Utrecht – p. 18

Σ out
Π loc
Σ in
Step 3 z(T) = z 0 e λT , r(T) = r 0 e −µT , θ(T) = θ 0 + ωt
‘time of flight’ T =
λ 1 ln ( z 0
)
h
. δ = µ/λ < 1 for chaotic case
( ( r
) ( ) )
δ ω h
⇒ Π loc : (r,θ,z) ↦→ r ,
h λ ln ,h
z 0
MRI Master Class Utrecht – p. 19

Σ out
Π glob
Σ in
Step 4:
Computation of Π glob : Σ out → Σ in
Assume diffeomorphism; expand as Taylor series
⎛
⎜
⎝
r
θ
h
⎞
⎟
⎠ ↦→
⎛
⎜
⎝
r
0
0
⎞
⎟
⎠ +
⎛
⎜
⎝
a
0
b
⎞
⎟
⎠α +
⎛
⎜
⎝
c d
0 0
e f
⎞⎛
⎟
⎠
⎜
⎝
r cos θ
0
r sin θ
⎞
⎟
⎠ + h.o.t
MRI Master Class Utrecht – p. 20

Π glob
Π loc
Σ in
Step 5 Poincaré map Π : Σ in → Σ in = Π glob ◦ Π loc
( ) ( ) ( ) (
r r a c 1 rz δ cos(k 1 ln z + φ 1 )
↦→ + α +
z 0 b c 2 rz δ cos(k 2 ln z + φ 2 )
)
MRI Master Class Utrecht – p. 21

dynamics of the Poincaré map
search for fixed points: r-dynamics slaved to z
⇒ 1D map for z (nb. period ∼ − ln z)
(z − bα) = Φ(z) = Kz δ cos(k ln z + φ) + h.o.t
MRI Master Class Utrecht – p. 22

dynamics of the Poincaré map
search for fixed points: r-dynamics slaved to z
⇒ 1D map for z (nb. period ∼ − ln z)
(z − bα) = Φ(z) = Kz δ cos(k ln z + φ) + h.o.t
δ > 1 (tame case)
α < 0
α = 0
α > 0
⇒
period
Φ(z)
z
α
z − bα
unique periodic orbit bifurcates
MRI Master Class Utrecht – p. 22

dynamics of the Poincaré map
search for fixed points: r-dynamics slaved to z
⇒ 1D map for z (nb. period ∼ − ln z)
(z − bα) = Φ(z) = Kz δ cos(k ln z + φ) + h.o.t
δ < 1 (chaotic case)
z
Φ(z)
⇒
period
z
α
infinitely many periodic orbits for α = 0
MRI Master Class Utrecht – p. 22

dynamics of the Poincaré map
search for fixed points: r-dynamics slaved to z
⇒ 1D map for z (nb. period ∼ − ln z)
(z − bα) = Φ(z) = Kz δ cos(k ln z + φ) + h.o.t
δ < 1 (chaotic case)
S.N.
P.D.
⇒
period
z
α
∞-many saddle-node & period-doubling as α → α 0
MRI Master Class Utrecht – p. 22

dynamics of the Poincaré map
search for fixed points: r-dynamics slaved to z
⇒ 1D map for z (nb. period ∼ − ln z)
(z − bα) = Φ(z) = Kz δ cos(k ln z + φ) + h.o.t
δ < 1 (chaotic case)
Φ(z)
⇒
period
z − bα
z
α
single ‘wiggly curve’ of periodic orbits. Also symbolic
dynamics on ∞-many symbols
MRI Master Class Utrecht – p. 22

Multi-pulses
Infinitely many parameter values α (2)
i
, i = 1,...∞,
converging as α → α 0 from both sides at which there exist
2-pulse homoclinic orbits.
... and N-pulses for all N ...
MRI Master Class Utrecht – p. 23

A word about rigour
Linearisation to compute Π loc cannot be rigorously justified
in general.
Hartman-Grobman Theorem gives only C 0 topological
equivalence
Three rigorous approaches:
Sometimes (non-resonance) can justify linearisation
C 1 linearisation theorems (e.g. Belitskii)
MRI Master Class Utrecht – p. 24

A word about rigour
Linearisation to compute Π loc cannot be rigorously justified
in general.
Hartman-Grobman Theorem gives only C 0 topological
equivalence
Three rigorous approaches:
Sometimes (non-resonance) can justify linearisation
C 1 linearisation theorems (e.g. Belitskii)
Set up Π loc as BVP. Use Imp. Fun. Theorem.
Shil’nikov co-ordinates ( see Shil’nikov et al 92, 98)
MRI Master Class Utrecht – p. 24

A word about rigour
Linearisation to compute Π loc cannot be rigorously justified
in general.
Hartman-Grobman Theorem gives only C 0 topological
equivalence
Three rigorous approaches:
Sometimes (non-resonance) can justify linearisation
C 1 linearisation theorems (e.g. Belitskii)
Set up Π loc as BVP. Use Imp. Fun. Theorem.
Shil’nikov co-ordinates ( see Shil’nikov et al 92, 98)
Use normal vector to homoclinic centre manifold
(adjoint) to project
Lin’s method or HLS: ‘Homoclinic Lyanpunov-Schmidt’
‘Hale, Lin, Sandstede’ (Lin 2008)
MRI Master Class Utrecht – p. 24

Application: excitable systems
Small input ⇒ gradual relaxation
Large enough ⇒ burst + gradual relaxation
MRI Master Class Utrecht – p. 25

Application: excitable systems
Small input ⇒ gradual relaxation
Large enough ⇒ burst + gradual relaxation
A key feature of many electrophysical systems
heart tissue
neurons
cell signalling (calcium dynamics)
MRI Master Class Utrecht – p. 25

Application: excitable systems
Small input ⇒ gradual relaxation
Large enough ⇒ burst + gradual relaxation
A key feature of many electrophysical systems
heart tissue
neurons
cell signalling (calcium dynamics)
the ‘pendulum equation’ of excitable systems:
Fitz-Hugh Nagumo (FHN) system (1961-2)
v t
w t
= Dv xx + f α (v) − w + c
= ε(v − γw)
f α (v) = v(v − 1)(α − v), e.g. α = 0.1, γ = 1.0, ε = 0.001.
MRI Master Class Utrecht – p. 25

example FitzHugh Nagumo equations
A. spatially homogeneous (D = 0) dynamics
v t
w t
= f α (v) − w + p
= ε(v − γw)
isoclines: v = γw, f α (v) + p = w. ⇒ excitable
0.15
w
0.1
0.05
–0.2 0 0.2 0.4 0.6 0.8
v
MRI Master Class Utrecht – p. 26

B. travelling structures (for D > 0) z = x + st (· = d/dz)
¨v − s˙v = −f α (v) + w − p
ẇ = (ε/s)(v − γw)
node → saddle: 2 kinds of travelling wave:
periodic wave trains ⇐ Hopf bifurcation
pulse solution ⇐ homoclinic orbit to saddle
×
1.4
1.2
1
0.8
0.6
0.4
Ú1
0.8
0.6
ÀÓÑ
0.4
ÀÓÔ Ú´µ
Ì
0.2
0
-0.2
´µ
-0.4
Ô ´µ
0 200 400 600 800 1000
0.07
Ì
0.068
0.066
0 0.1 0.2 0.3 0.4 0.5 0.6 0 200 400 600 800 1000
MRI Master Class Utrecht – p. 27

This C U structure common for excitable models
But how does the Hom curve end as it approaches Hopf?
More details: (p vs. wavespeed s)
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0 0.05 0.1 0.15 0.2
(see Champneys, Kirk et al 07)
MRI Master Class Utrecht – p. 28

The hom curve doubles back on itself(!) and gains an extra
large loop in so doing (p vs. ‘norm’)
0.11
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.11
0.11
0.1
0.1
0.09
0.09
0.08
0.08
0.07
0.07
0.06
0.05
-0.06
-0.04
-0.02
0
0.02
0
-0.1
0.04 -0.2
0.1
0.2
0.3
0.4
0.06
0.05
-0.06
-0.04
-0.02
0
0.02
0
-0.1
0.04 -0.2
0.1
0.2
0.3
0.4
two homoclinics for p ≈ 0.06, s ≈ 0.894386.
MRI Master Class Utrecht – p. 29

example 2: 8-variable Ca 2+ model
Sneyd, Yule et al Calcium waves in pancreatic acinar
cell
∂c
∂t
dc e
dt
dG
dt
= D ∂2 c
∂x
+ k 2 1 (G)(c − c e ) + J 1 (c,G)
= −k 2 (G)(c e − c) + J 2 (c,G)
= k 3 (p,c)G
c(x,t) concentration of Ca 2+ . c e (t) concentration in
boundary. G(t) ∈ R 6 receptor variables.
Travelling waves ξ = x − st ⇒ 9D ODE system
Bif. pars: wavespeed s, IP 3 concentration p
MRI Master Class Utrecht – p. 30

× ËÄ ÀÓÑ
Similar C-U bifn diagram.
¹ ÀÓÔ
14
12
10
8
6
4
Ô
0 1 2 3 4 5
at upper end: Hom curve passes straight through Hopf!?
MRI Master Class Utrecht – p. 31

´µ
And at lower
ľ¹ÒÓÖÑ
end, the homoclinic curve doubles back on
itself ∞ many times.
76.5
76
75.5
75
´µ ´µ Ô
74.5
74
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
74.8
75
74.7
74.6
74.5
74.4
74.3
74.2
74.1
74
73.9
0.03 0.04 0.05 0.06 0.07 0.08 0.09
MRI
74
73
72
71
70
69
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Master Class Utrecht – p. 32

4. Reversible and Hamiltonian systems
Reversible systems
ẋ = f(x),x ∈ R 2n ,Rf(x) = −f(Rx),R 2 = Id, S = fix(R) ∼ = R n .
⇒ symmetric homoclinic orbits are codim 0 (Devaney)
γ(t) → x 0 ast → ±∞, γ(0) ∈ S, wheref(x 0 ) = 0, x 0 ∈ S.
Hamiltonian systems
ẋ = f(x) = J∇H(x), x ∈ R 2n , ⇒ H(x(t)) = const.
⇒ W u & W s live in H −1 (x 0 ) ⇒ codim 0
In either case, σ(x 0 ) symmetric w.r.t. Im-axis
Everything happens with one codimension less
MRI Master Class Utrecht – p. 33

a fourth-order example
Buffoni, C. & Toland 96 u iv + Pu ′′ + u − u 2 = 0
norm
Belyakov−
Devaney
P
−2
Ham.
Hopf
+2
saddle bi−focus centre
MRI Master Class Utrecht – p. 34

Application 1: wobbly bridges
Goldern gate bridge 1938, Chief Engineer R.G. Cone
a wind of unusual high velocity was blowing . . . normal to the axis of the
bridge . . . The suspended structure of the bridge was undulating vertically in
a wave-like motion of considerable amplitude, . . . a running wave similar to
that made by cracking a whip. The truss would be quiet for a second, and
then in the distance one could see a running wave of several nodes
approaching . . .
C. & McKenna 96 asymmetric beam model
Stable multi-humped solitary waves (Buryak & C 96)
U ′ (s)
U (s)
0 . 0 0 0 0 3 0
0 . 0 0 0 0 2 0
0 . 0 0 0 0 1 0
0 . 0 0 0 0 0 0
2(1)
2(2)
2(3)
2(4)
2(5)
2(7)
2(8) 2(9)
2(6)
- 0 . 0 0 0 0 1 0
- 0 . 0 0 0 0 2 0
0 . 5 . 1 0 . 1 5 . 2 0 . 2 5 . 3 0 . 3 5 . 4 0 .
s
seperation s
MRI Master Class Utrecht – p. 35

Nb. R.G. Cone was sacked for ‘disloyalty to the bridge’
Deck Amplitude
500
5
150
u
-8
-7
-6
-5
-4
-3
-2
-1 0123
400
450
t
u
0
-5
-10
-15
100
t
-50
0
Span
50
100
x
150
350
time
-50
0
50
100
x
150
0
50
5
0
-5
-10
-50 -15
0
50
100
150
0
50
100
150
2.5
1.5 2
0.5 1
-0.5 0
-1
-1.5
-2
-100
-50
0
50
100
150
200
250
0
50
100
MRI Master Class Utrecht – p. 36

... add competing nonlinearity
(Woods & C. 99)
u xxxx + Pu xx + u − u 2 + bu 3 = 0
2
9
< b < 38
27 . b = 0.29:
u
2.5
2
1.5
1
0.5
0
-0.5
x
L 2
1
0.8
primary
2(2)
u
3
2.5
2
1.5
1
0.5
0
-0.5
x
-1
-60 -40 -20 0 20 40 60
-1
-60 -40 -20 0 20 40 60
u
2.5
2
0.4
u
3
2.5
1.5
1
0.5
0
-0.5
x
2
1.5
1
0.5
0
-0.5
x
-1
-60 -40 -20 0 20 40 60
-1
-60 -40 -20 0 20 40 60
u
2.5
2
1.5
1
0.5
0
-0.5
x
2.5
u
2
1.5
1
0.5
0
-0.5
0
1.3 1.4 1.5 1.6 1.8 1.9 2
x
2
u
1.5
1
0.5
0
-0.5
P
x
u
2
1.5
1
0.5
0
x
u
3
2.5
2
1.5
1
0.5
0
-0.5
x
-1
-60 -40 -20 0 20 40 60
-1
-60 -40 -20 0 20 40 60
-1
-60 -40 -20 0 20 40 60
-0.5
-60 -40 -20 0 20 40 60
-1
-60 -40 -20 0 20 40 60
MRI Master Class Utrecht – p. 37

a small amplitude limit
b → (38/27) ⇒ narrow snake
1.2
L 2 2-pulse
L 2
primary
1
(a)
0.7
0.6
(b)
primary
2-pulse
0.8
0.5
0.4
0.3
0.4
0.2
0.2
0.1
0
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
p
0
1.9 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2
p
For P ≈ 2, b ≈ (38/27) normal form theory
envelope "spatial dyanamics"
decrease P
FRONT
...AND BACK
HUMP
But misses beyond-all-orders effects.
MRI Master Class Utrecht – p. 38

a related problem
Similar results for
α = 3/10 (Hunt et al 00)
u xxxx + Pu xx + u − αu 3 + u 5 = 0
See also Burke & Knobloch 07
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
MRI Master Class Utrecht – p. 39

spatial dynamics explanation of snake
Woods & C. 99 (cf. Coullet et al 2000)
Hamiltonian case. Take 2D Poincaré section within {H = 0}:
(a)
u
W (O)
O
s
W (L)
L
S
(b)
(c)
(d)
(e)
(f)
O
u
W (O)
s
W (L)
L
S
cf. Poincaré “heteroclinic tangle”
MRI Master Class Utrecht – p. 40

norm
(a) (b) (c) (d) (e) (f)
See Burke & Knobloch 06, Beck et al 09 for application to
localised patterns in Swift-Hohenberg eqns. (snakes and
ladders!)
λ
MRI Master Class Utrecht – p. 41

4. Computing homoclinic/heteroclinics
- 3 simple special cases approaches in AUTO
compute a periodic orbit to large period
case of 1D unstable manifold
reversible case
Next lecture: HomCont — general AUTO-07P method for
computing homoclinic orbits & detecting codimension-two
points
MRI Master Class Utrecht – p. 42

Computing large-period periodic orbits
AUTO solves for periodic orbits via boundary-value
problem
ẋ = Tf(x,α), x(0) = x(1), x(0)
∫ 1
0
(u(t) T ˙ũ)dt
for x(t) in R n , parameter α ∈ R, T ∈ R.
Homoclinic bifurcation ⇒ T → ∞
To compute homoclinic; fix T (large), solve for
α 1 ,α 2 ∈ R 2 .
Can show not the optimal choice (see next lecture)
this afternoon: AUTO demo pp2.
MRI Master Class Utrecht – p. 43

1D unstable manifold homoclinics case
Suppose A = Df(x 0 , 0) has n s = 1 unstable eigenvalue
λ (and n u = n − 1 stable eigs)
Let Av 1 = λv 1 , A T w 1 = λw 1 .
Compute boundary value problem
ẋ = 2Tf(x,α)
x(0) = x 0 + εv 1
0 = w T 1 (x(1) − x 0 )
can show convergence as 2T → ∞ (see next lecture)
⇒ continuation problem with n + 1 boundary conditions
for n + 2 unknowns x(t) α 1 , α 2 .
MRI Master Class Utrecht – p. 44

Reversible case
Consider reversible homoclinic x(t)
ẋ = f(x,α), x ∈ R 2n , x(0) ∈ fix(R), x(±∞) → x 0
Truncate to [−T, 0] and solve the two-point BVP with n
B.C.s:
ẋ = f(x,α)
L u x(−T) − x 0 = 0
Dfix(R) ⊥ x(0) = 0
where L u is projection onto unstable eigenspace of A (using
stable eigenspace of A T
e.g. for 4 th -order example x = (u,u ′ ,u ′′ ,u ′′′ ),
fix R = (u, 0,u ′′ , 0). D fix (R) ⊥ = (0, 1, 0, 1).
⇒ computes a curve of solutions in one parameter α 1
MRI Master Class Utrecht – p. 45

What we have learnt so far:
Homoclinic orbits to equilibria can be tame or chaotic
Chaotic case leads to birth of multi-pulses
Everything depends on linearisation (+ twistedness -
see next lecture)
Topological ideas can be posed rigorously analytically
Hamiltonian (and reversible) case drops a co-dimension
Many applications
MRI Master Class Utrecht – p. 46

References
Beck, M., Knobloch, J., Lloyd, D., Sandstede B. and Wagenknecht, T. Snakes, ladders
and isolas of localized patterns, SIAM J. Math. Anal. (2009) appeared online
Burke, J. and Knobloch, E. Localized states in the generalized Swift-Hohenberg
equation, Phys. Rev. E, 73, (2006) 056211
Burke, J. and Knobloch, E. Snakes and ladders: Localized states in the
Swift-Hohenberg equation Phys. Lett. A, 360 (2007) 681-688.
Freire, E. Rodriguez-Luis, A.J., Gamero E. and Ponce E. A case study for homoclinic
chaos in an autonomous electronic circuit Physica D 62 (1993) 230-253.
Glendinning, P. and Sparrow, C. Local and Global Behaviour Near Homoclinic Orbits
J. Stat. Phys. 35 (1984) 645-696.
Hunt, GW; Peletier, MA; Champneys, AR, et al. Cellular buckling in long structures,
Nonlinear Dynamics 21 (2000) 3-29.
Lin, X.B. Lin’s method Scholarpedia, 3 (2008) 6972.
www.scholarpedia.org/article/Lin’s method
Sparrow, C. The Lorenz Equations: Bifurcations, Chaos and Strange Attractors.
Springer-Verlag, (1982)
Shilnikov, L.P., Shilnikov A.L., Turaev, D.V. and Chua, L.O. Methods of qualitative
theory in nonlinear dynamics. World Scientific. Part I (1992), Part II (1998). MRI Master Class Utrecht – p. 47