heteroclinic networks: stability, switching and...

HeteroclinicNetworks

ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Heteroclinic Networks: stability, switching and

memory

Claire Postlethwaite

with: Peter Ashwin, Jonathan Dawes, Vivien Kirk,Alastair Rucklidge, Mary Silber

Department of MathematicsUniversity of Auckland

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Outline

1 Introduction

2 Stability of Heteroclinic CyclesThe Guckenheimer–Holmes cycle

3 Heteroclinic NetworksThe Kirk–Silber networkDecision network

4 Summary

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Outline

1 Introduction



4 Summary

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Side-blotched lizard

The side-blotched lizard (Uta stansburiana) lives in thePacific Coast Range of California.

There are three types of male lizard, each with a differentcoloured throat indicating three types of geneticallydetermined mating strategy.

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Side-blotched lizard

Strategy 2: Be sneaky.

Yellow-throated males aresneaky and can mimic thebehaviour of females.

Strategy 3: Guard your mate.

Blue-throated males defend smallterritories holding just a few fe-males. Because the territories areso small they can guard their matescarefully.

Strategy 1: Have a lot of territory.

Orange-throated males establish large territories,with several females. The more females the moreoften they can mate.

Blue-throated males can take over apopulation of mainly yellow-

throated males.

Yellow-throated

males

cansneakinto

the

territoriesof

orange-throatedmales.

Orange-

throatedmales

cantake

territoryfrom

blue-throatedmales.

[Cartoon taken from http://www.sciencenewsforkids.org]

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Heteroclinic cycles

Consider the ODE:

x = f (x), x ∈ Rn. (1)

An equilibrium ξ of (1) satisfies f (ξ) = 0.

A solution φj of (1) is a heteroclinic connection from ξj toξj+1, if it is backward asymptotic to ξj and forwardasymptotic to ξj+1.

A heteroclinic cycle is a set ofequilibria {ξ1, ..., ξm} and orbits{φ1, ..., φm}, where φj is aheteroclinic connection between ξjand ξj+1, and ξ1 ≡ ξm+1.

ξ1 ξ2

ξ3

ξ4

ξ0

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Heteroclinic cycles in systems with symmetry

Field 1980, Krupa and Melbourne 1995, and many others

In systems with invariant subspaces, heteroclinicconnections can exist robustly.Invariant subspaces arise naturally in systems withsymmetry.

ξ1

ξ2

x1

x2

x3

Z2 symmetry: κ3 : (x1, x2, x3) → (x1, x2,−x3)Fix(κ3) = {(x1, x2, 0)}

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Examples

Population models

Strategy 2: Be sneaky.

Yellow-throated males aresneaky and can mimic thebehaviour of females.

Strategy 3: Guard your mate.

Blue-throated males defend smallterritories holding just a few fe-males. Because the territories areso small they can guard their matescarefully.

Strategy 1: Have a lot of territory.

Orange-throated males establish large territories,with several females. The more females the moreoften they can mate.

Blue-throated males can take over apopulation of mainly yellow-

throated males.

Yellow-throated

males

cansneakinto

the

territoriesof

orange-throatedmales.

Orange-

throatedmales

cantake

territoryfrom

blue-throatedmales.

Fluid dynamics

N. Becker and G. Ahlers

8 / 29


ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Classification of eigenvalues

Krupa and Melbourne, 1995

It can be useful to classify the eigenvalues near eachequilibrium.

Radial

Contracting

Expanding

Transverse

ξ1

ξ2

x1

x2

x3

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Analysis of flow near heteroclinic cycles

ξ1

ξ1

ξ2ξ2

ξ3H in

1 H in

2

Hout

1

Local flow: H in

1 → Hout

1

x2 = λux2,

x3 = −λsx3.

Global flow: Hout

1 → H in

2

Linearise flow aroundheteroclinic connections.

Combine local and global flow to get a Poincare map.

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles

GH cycle


Summary

Outline

1 Introduction



4 Summary

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles

GH cycle


Summary

The Guckenheimer–Holmes cycle

Guckenheimer and Holmes, 1988

��

��

��

ξ1

ξ2

ξ3

The cycle exists in a system with symmetry Z3 ⋉ (Z2)3.

Contracting eigenvalue −λs , expanding eigenvalue λu.

Local map gives x2 → xδ3 , δ = λs/λu.

Global map x3 → Ax3.

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles

GH cycle


Summary


��

��

��

ξ1

ξ2

ξ3

Poincare map: x → Axδ.

Fixed points exist at x = 0 and x = A1/(1−δ).

A resonance bifurcation at δ = 1 produces a long-periodperiodic orbit.

Period of orbit T ∼1

1− δ.

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles

GH cycle


Summary


ξ1

ξ2

ξ3

Poincare map: x → Axδ.

Fixed points exist at x = 0 and x = A1/(1−δ).

A resonance bifurcation at δ = 1 produces a long-periodperiodic orbit.

Period of orbit T ∼1

1− δ.

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles

GH cycle


Summary

Noisy heteroclinic cycles

Stone and Holmes, 1990

Consider additive noise to a heteroclinic cycle.

Mean passage time past an equilibrium

T ∼log η

λu

where η is the noise amplitude

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary

Outline

1 Introduction



4 Summary

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary

Heteroclinic networks

A heteroclinic network is a connected union of heterocliniccycles.

ξ1 ξ2

ξ3

ξ4

ξ0

Stability conditions of the network as a whole may bequite complicated.

Both resonance and noise in networks can givecomplicated behaviour.

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary

Kirk and Silber network

Kirk and Silber, 1994

A

Y

X

PB

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary


Subspace dynamics

A

Y

X

PB

A

B

x1

x2B

X

Y

P

x2

x3

y3

A

X

Y

P

x1

x3

y3

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary


Construction of maps

A

B

Y

X

Hin,AB

Hout,YB

Hout,XB

The position the trajectory hits H in,AB determines which

equilibrium is visited next.

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary


Construction of maps

Plot the Poincare section with polar coordinates.

0

log x2

2ππ2

πθ3

If the network is attracting trajectories can switch one waybut not the other.

If network is not attracting, there can be periodic orbitslying close to either or both of the sub cycles.

Kirk, Postlethwaite and Rucklidge, SIADS 201221 / 29


ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary

Noisy Kirk and Silber network

Armbruster, Stone and Kirk, 2003

Consider additive noise.

Depending on parameters, the ‘noise ellipse’ can becentered at the origin in H

in,AB .

Proportion of times each cycle visited proportional toshaded area.

−11 −10 −9 −8 −7 −6 −5−1.05

−1

−0.95

−0.9

−0.85

−0.8

−0.75

−0.7

−0.65

−0.6

−0.55

log (noise)

log(proportion)

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary

Noisy Kirk and Silber network

Armbruster, Stone and Kirk, 2003

For certain parameter sets, noise ellipse can move intobasin of attraction of one cycle or the other.

This is termed lift-off.

Lift-off can reduce switching.

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary

Decision network

Ashwin and Postlethwaite, 2013

replacements

v1

v2

v3 v4

v5 v6 v7 v8

Four sub-cycles, each with four equilibria.

Deterministic case is very complicated: numerics showswitching between sub cycles.

Addition of noise can allow memory.

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary

Decision network with memory

v1

v2

v3 v4

v5 v6 v7 v8

We can choose parameters to have lift-off occur in the x3direction as the trajectory passes ξ5.

−0.01 −0.005 0 0.005 0.010

200

400

600

800

1000

1200

1400

1600

x1(a) on H

in

5

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

50

100

150

200

250

300

x3(b) on H

out

5

−4 −3 −2 −1 0 1 2 3 4

x 10−4

0

50

100

150

200

250

300

x3(c) on H

in

2

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary

Decision network with memory

Without memory: With memory:

−3 −2 −1 0 1 2 3

x 10−5

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−5

x4

x3−2 −1 0 1 2 3

x 10−4

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−5

x4

x3

Red: trajectories which visited ξ5 on the previous loop.

Black, blue, green: trajectories which visited ξ6, ξ7 and ξ8.

Can see lift-off in the x3 direction for red points.

26 / 29


ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


KS network

Decision network

Summary

Transition matrices

Without memory: With memory:

0.46 0.26 0.18 0.100.47 0.23 0.20 0.100.49 0.23 0.18 0.100.50 0.22 0.18 0.10

0.63 0.33 0.03 0.020.49 0.27 0.16 0.080.54 0.24 0.13 0.090.52 0.27 0.14 0.07

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ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Outline

1 Introduction



4 Summary

28 / 29


ClairePostlethwaite

Outline

Introduction

HeteroclinicCycles


Summary

Summary

Heteroclinic cycles and networks can lose stability andproduce nearby long-period periodic orbits in resonancebifurcations.

Noisy heteroclinic cycles look like periodic orbits.

Noisy heteroclinic networks can have much morecomplicated behaviour, including switching and memory.

Acknowledgements

Jonathan Dawes, Alastair Rucklidge, Vivien Kirk, Mary Silber,Peter Ashwin.

London Mathematical Society.

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heteroclinic networks: stability, switching and...

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