Lecture 1

Transcription

Lecture 1

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.
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’
The ‘soliton’
1895 Korteweg & de Vries derived KdV equation
∂u
∂u
∂3u
+
6u
=
solution speed c
3
∂t
∂x
∂x
2 √
u(x, t) = (−c/2)sech ( 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’)
The ‘soliton’
1895 Korteweg & de Vries derived KdV equation
∂u
∂u
∂3u
+
6u
=
solution speed c
3
∂t
∂x
∂x
2 √
u(x, t) = (−c/2)sech ( 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
Optical solitons
Optical fibres; means of transatlantic communication.
Pulses travel at speed of light c
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
⇒ Nonlinear Schrödinger (NLS) equation
∂2v
∂v
i ∂t + ∂x2 + Q|v|2 v = 0
also integrable & explicit
‘envelope of waves’ soliton
p
v(x, t) = Vm eikt sech(Vm |Q|/2U (x − ct))
envelope of wave
oscillating wave
Use solitons as bits of information
1
0 1 1 1 0
1 1 0
1 1 0
1
1 1
. . . 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)
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 )
Application: Chua’s electronic circuit
smooth version (due to A. Khibnik 93)
U̇ = α(V − (1/6)(U − U 3 ))
V̇ = U − V + W
Ẇ = −βV
(α, β) = (0, 0): Z2 -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
–1
0
y(t)
0
1
0.2
0.4 2
x(t)
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)
Later we will study a similar simple electronic circuit due to
Friere et al 1993
...
2. Homoclinic orbits to equilibria
Heteroclinic orbit Γ connecting equilibria x1 , x2 ∈ Rn
ẋ(t) = f (x(t), α)
x(t) → x1 , x2
as t → ±∞.
‘generic’ system: no symmetry or first integrals
Homoclinic orbit special case x1 = x2 = x0
x0 hyperbolic ⇒ codim 1, i.e. at isolated α = α0
Γ
Wu
p
Ws
Γ
Wu
p
Ws
Z
Z
O
q
W ss
O
q
W ss
(a)
(b)
Homoclinic bifurcation as α varies
Suppose ∃ Hom orbit Γ at α = α0 . Linearisation at x0 :
σ(x0 )
Im
determining
Re
leading
Theorem 1 (Shil’nikov’s tame homoclinic bifurcation)
Real determining eigenvalue ⇒ unique periodic orbit
destroyed at infinite period as α → α0 .
Theorem 2 (Shil’nikov’s chaotic homoclinic bifurcation)
Complex determining eigenvalue ⇒ ∞-many high-period
periodic orbits in neighbourhood of Γ and α0 .
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.
In chaotic case ∃ ∞-many N -pulse homoclinic orbits at
nearby α-values. For each N
Γ
W
W
u
s
O
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.
Sketch proof of Shil’nikov chaotic case
saddle focus case in R3 (Glendinning & Sparrow 84)
assume ∃ α0 = 0 at which hom orbit Γ to x0 = 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
construct Poincaré map
Γ
Wu
Ws
Step 1: Set up Poincaré sections
Σin = {θ = 0},
Σout = {z = h}
Σout
z
θ
r
Σin
Step 2 Linearise flow near 0 to compute Πloc : Σin → Σout
ż = λz
θ̇ = ω
ṙ = −µr
+ h.o.t
Σout
Πloc
Σin
Step 3 z(T ) = z0 eλT , r(T ) = r0 e−µT , θ(T ) = θ0 + ωt
z0
1
‘time of flight’ T = λ ln h . δ = µ/λ < 1 for chaotic case
δ
h
ω
r
,h
, ln
⇒ Πloc : (r, θ, z) 7→ r
h
λ
z0
Σ
Πglob
out
Σin
Step 4:
Computation of Πglob : Σout → Σin
Assume diffeomorphism; expand as Taylor series



   
 
c d
r cos θ
a
r
r



   
 
 θ  7→  0  +  0  α +  0 0   0  + h.o.t
e f
r sin θ
b
0
h
Πglob
Πloc
Σin
Step 5 Poincaré map Π : Σin → Σin = Πglob ◦ Πloc
!
!
!
!
r
a
c1 rz δ cos(k1 ln z + φ1 )
r
+
α+
7→
0
b
c2 rz δ cos(k2 ln z + φ2 )
z
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)
period
α=0
α<0
α>0
⇒
Φ(z)
z
α
z − bα
unique periodic orbit bifurcates
δ < 1 (chaotic case)
z
period
⇒
Φ(z)
z
α
infinitely many periodic orbits for α = 0
period
S.N.
P.D.
⇒
z
α
∞-many saddle-node & period-doubling as α → α0
Φ(z)
period
⇒
z
z − bα
α
single ‘wiggly curve’ of periodic orbits. Also symbolic
dynamics on ∞-many symbols
Multi-pulses
(2)
αi ,
Infinitely many parameter values
i = 1, . . . ∞,
converging as α → α0 from both sides at which there exist
2-pulse homoclinic orbits.
. . . and N -pulses for all N . . .
A word about rigour
Linearisation to compute Πloc 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)
A word about rigour
in general.
equivalence
Set up Πloc as BVP. Use Imp. Fun. Theorem.
Shil’nikov co-ordinates ( see Shil’nikov et al 92, 98)
A word about rigour
in general.
equivalence
Set up Πloc 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)
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)
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)
vt = Dvxx + fα (v) − w + c
wt = ε(v − γw)
fα (v) = v(v − 1)(α − v), e.g. α = 0.1, γ = 1.0, ε = 0.001.
example FitzHugh Nagumo equations
A. spatially homogeneous (D = 0) dynamics
vt = fα (v) − w + p
wt = ε(v − γw)
isoclines: v = γw, fα (v) + p = w. ⇒ excitable
0.15
w
0.1
0.05
–0.2
0
0.2
0.4
0.6
0.8
v
B. travelling structures (for D > 0) z = x + st (· = d/dz )
v̈ − sv̇ = −fα (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
s
v1
1.4
0.8
0.6
0.4
1.2
0.2
0
1
-0.2
0.8
(b)
-0.4
0
Hom
200
400
600
800
T 1000
200
400
600
800
T 1000
0.07
v
0.6
Hopf
0.4
0.068
()
(a)
0
0.1
0.2
0.3
0.4
0.5
p
0.066
0.6
0
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)
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
0.11
0.03
0.04
0.05
0.06
0.07
0.08
0.11
0.1
0.1
0.09
0.09
0.08
0.08
0.07
0.07
0.06
0.4
0.06
0.4
0.3
0.05
-0.06
0.2
-0.04
0.1
-0.02
0
0
0.02
-0.1
0.3
0.05
-0.06
0.2
-0.04
0.1
-0.02
0
0
0.04 -0.2
two homoclinics for p ≈ 0.06, s ≈ 0.894386.
0.02
-0.1
0.04 -0.2
example 2: 8-variable Ca
2+
model
Sneyd, Yule et al Calcium waves in pancreatic acinar
cell
∂c
∂t
dce
dt
dG
dt
∂2c
D ∂x2
=
+ k1 (G)(c − ce ) + J1 (c, G)
= −k2 (G)(ce − c) + J2 (c, G)
= k3 (p, c)G
c(x, t) concentration of Ca2+ . ce (t) concentration in
boundary. G(t) ∈ R6 receptor variables.
Travelling waves ξ = x − st ⇒ 9D ODE system
Bif. pars: wavespeed s, IP3 concentration p
Similar C -U bifn diagram.
SL
s 14
Hom
12
Hopf
-
10
8
6
4
0
1
2
3
4
p
5
at upper end: Hom curve passes straight through Hopf!?
And at lower end, the homoclinic curve doubles back on
itself ∞ many times.
76.5
(a)
L2 -norm
76
75.5
75
74.5
74
1.4
(b)
e
1.6
1.8
2
2.2
()
e
74.8
74.7
2.4
2.6
p
2.8
75
74
74.6
74.5
73
74.4
72
74.3
74.2
71
74.1
70
74
73.9
0.03
0.04
0.05
0.06
0.07
0.08
0.09
69
0.02 0.04 0.06 0.08
0.1
0.12 0.14 0.16
4. Reversible and Hamiltonian systems
Reversible systems
ẋ = f (x), x ∈ R2n , Rf (x) = −f (Rx), R2 = Id, S = fix(R) ∼
= Rn .
⇒ symmetric homoclinic orbits are codim 0 (Devaney)
γ(t) → x0 as t → ±∞, γ(0) ∈ S,
where f (x0 ) = 0, x0 ∈ S.
Hamiltonian systems
ẋ = f (x) = J∇H(x),
x ∈ R2n ,
⇒ H(x(t)) = const.
⇒ W u & W s live in H −1 (x0 ) ⇒ codim 0
In either case, σ(x0 ) symmetric w.r.t. Im-axis
Everything happens with one codimension less
a fourth-order example
Buffoni, C. & Toland 96 uiv + P u′′ + u − u2 = 0
norm
Belyakov−
Devaney
P
Ham.
Hopf
+2
−2
saddle
bi−focus
centre
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. 000030
0. 000020
0. 000010
2(1)
2(2)
2(3)
2(4)
2(5)
2(7)
2(8)
2(9)
0. 000000
2(6)
- 0. 000010
- 0. 000020
0.
5.
10.
15.
20.
25.
30.
35.
s
40.
seperation s
De
ck
Am
pli
tud
e
Nb. R.G. Cone was sacked for ‘disloyalty to the bridge’
150
5
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
0
450
t
400
-50
350
0
50
Span
time
u
500
u
-5
100
-10
50
-50
0
50
100
100
x
x
150
150
5
0
100
-5
-10
-15
-50
50
0
50
100
150
0
t
-15
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-100
150
0
100
50
-50
0
50
100
150
200
250
0
. . . add competing nonlinearity
(Woods & C. 99)
uxxxx + P uxx + u − u2 + bu3 = 0
2
9
<b<
38
27 .
b = 0.29:
2.5
u
3
1
2
2
1.5
u2.5
primary
2(2)
2
L
1
1.5
1
0.8
0.5
0.5
x
0
-1
-0.5
-60
-40
-20
0
20
40
-1
60
2.5
u
x
0
-0.5
-60
-40
-20
0
20
40
60
3
u2.5
0.4
2
2
1.5
1.5
1
1
0.5
0.5
x
0
-1
-0.5
-60
-40
-20
0
20
40
-1
60
0
1.3
2.5
1.4
1.5
1.6
2.5
2
2
u2
1.5
1.5
1.5
1
1
u
1.8
P
u
1.9
2
3
u1.5
u2.5
1
1.5
-0.5
x
0
-60
-40
-20
0
20
40
60
-1
-60
-40
-20
0
20
40
60
-1
-20
0
20
40
60
0.5
0.5
1
x
0
x
0
-0.5
-0.5
-40
2
1
0.5
x
0
-60
2
0.5
0.5
-1
x
0
-0.5
x
0
-0.5
-60
-40
-20
0
20
40
60
-0.5
-60
-40
-20
0
20
40
60
-1
-60
-40
-20
0
20
40
60
a small amplitude limit
b → (38/27) ⇒ narrow snake
(a)
(b)
1.2
0.7
2
L
2
2-pulse
primary
L
1
primary
2-pulse
0.6
0.5
0.8
0.4
0.3
0.4
0.2
0.2
0
1.3
0.1
1.4
1.5
1.6
1.7
1.8
1.9
0
1.9
2
1.91
1.92
1.93
1.94
1.95
1.96
p
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.
a related problem
Similar results for
uxxxx + P uxx + u − αu3 + u5 = 0
α = 3/10 (Hunt et al 00)
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
See also Burke & Knobloch 07
spatial dynamics explanation of snake
Woods & C. 99 (cf. Coullet et al 2000)
Hamiltonian case. Take 2D Poincaré section within {H = 0}:
(a)
Wu(O)
O
(b)
Ws(L)
S
L
(c)
(d)
(e)
(f)
Ws(L)
u
W (O)
O
S
L
cf. Poincaré “heteroclinic tangle”
norm
(a)
(b) (c) (d) (e) (f)
λ
See Burke & Knobloch 06, Beck et al 09 for application to
localised patterns in Swift-Hohenberg eqns. (snakes and
ladders!)
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
Computing large-period periodic orbits
AUTO solves for periodic orbits via boundary-value
problem
Z 1
˙
ẋ = T f (x, α), x(0) = x(1), x(0)
(u(t)T ũ)dt
0
for x(t) in Rn , parameter α ∈ R, T ∈ R.
Homoclinic bifurcation ⇒ T → ∞
To compute homoclinic; fix T (large), solve for
α1 , α2 ∈ R2 .
Can show not the optimal choice (see next lecture)
this afternoon: AUTO demo pp2.
1D unstable manifold homoclinics case
Suppose A = Df (x0 , 0) has ns = 1 unstable eigenvalue
λ (and nu = n − 1 stable eigs)
Let Av1 = λv1 , AT w1 = λw1 .
Compute boundary value problem
ẋ = 2T f (x, α)
x(0) = x0 + εv1
0 = w1T (x(1) − x0 )
can show convergence as 2T → ∞ (see next lecture)
⇒ continuation problem with n + 1 boundary conditions
for n + 2 unknowns x(t) α1 , α2 .
Reversible case
Consider reversible homoclinic x(t)
ẋ = f (x, α),
x ∈ R2n ,
x(0) ∈ fix(R),
x(±∞) → x0
Truncate to [−T, 0] and solve the two-point BVP with n
B.C.s:
ẋ = f (x, α)
Lu x(−T ) − x0 = 0
Dfix(R)⊥ x(0) = 0
where Lu is projection onto unstable eigenspace of A (using
stable eigenspace of AT
e.g. for 4th -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 α1
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
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.
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Lecture 1

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