Nonlinear dynamics and pattern formation w ith applications to ecology M eiji U niversity, N ov. 25-28, 2008

Ehud M eron Blaustein Institutes for D esert R esearch & Physics D epartm ent Ben G urion U niversity, Israel

T he w orld w e perceive w ith our senses is highly nonlinear: Clim ate phenom ena (tornados, hurricans), patterns in nature (clouds, vegetation), oscillations and synchronization (heart beats, hands clapping, stock-m arket dynam ics), are all nonlinear phenom ena. T o study phenom ena of this kind w e need the tools of dynam ical system theory

Focus on pattern formation phenomena, their nonlinear aspects, and their relevance to the ecology of w ater-limited systems G eneral outine B

Part I: Basic m echanism s of pattern form ation e n

G

Basic concepts: dynam ical system s, instabilities, etc. u r i o

Pattern form ing system s n U

1. Class I – instabilities of uniform states n i v 2. Class II – m ultistability of states, front dynam ics e r s i t y ,

Part II: Controling patterns by tem poral and spatial periodic forcing E h

Frequency locking, w avenum ber locking in spatially extended system s u d

Class I and class II pattern form ation m echanism s induced by the M e r

Forcing o n - w

Part III: M ultim ode localized structures w w . Localized structures, m ultiple instabilities, instabilities of single- b g u

m ode to dual-m ode localized structures . a c . i l /

Part IV : A pplications to ecology ~ e h

V egetation pattern form ation in w ater-lim ited system s, plant u d interactions and biodiversity. O utine of Part I: Basic mechanisms of pattern formation B e

Basic concepts: O pen non-equilibrium system s, dynam ical system s, n

G

nonlinear system s, m ultiplicity of states, instabilities, hysteresis u r i o n

Pattern form ation in nature: S ym m etry breaking, universality U n i v e r s

Canonical m odels of pattern form ing system s: S w ift-H ohenberg, i t y ,

FitzH ugh-N agum o

E h u d

Pattern form ing system s class I: Instabilities of uniform states M e

Exam ples, linear stability analysis, nonlinear analysis – am plitude r o equations, secondary instabilities n - w w w

Pattern form ing system s class II: M ultiplicity of stable states . b g u

Exam ples, fronts, front instabilities, labyrinthine patterns, spiral . a c .

w aves, spiral turbulence. i l / ~ e h u d Basic concepts: open non-equilibrium systems B e

T he natural system s w e m entioned represent open non-equilibrium n

G

system s, that continuously exchange m aterials and energy w ith u r i their environm ents. o n U n i O ne of the first tasks in exploring such system s is defining w hat v e r constitutes the “system ” and w hat constitutes the system ’s s i t y

“environm ent”. ,

E h u T he distinction betw een the system and its environm ent depends d

M

on the processes they involve, w hich can be divided into tw o e r o general groups: n - w

1. G roup I: Processes that m utually affect one another through w w . various feedbacks. b g u .

2. G roup II: Processes that affect G roup-I processes but are a c . i

hardly affected by them . l / ~ e h

U sing this process classification w e can introduce the concept of a u dynam ical system . d Basic concepts: defining a dynamical system

D ynamical system - a set of group-I processes, quantified by B e n dynam ical variables, U (t), that evolve in tim e according to som e law s i G u of m otion. T hese law s describe the m utual relationships am ong the r i o dynam ical variables and the m anners by w hich they are affected by n U param eters, P (t), representing group-II processes, w hich define the n

i i v e

system ’s environm ent. r s i t y ,

E

P1(t) h u d

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D ynamical U2(t) b g

S ystem u . a

G roup-I processes c . i l (dynam ical variables) Environment / ~ e h

G roup-II processes (param eters) u d Ben G urion U niversity, Ehud M eron - w w w .bgu.ac.il/~ehud x t

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m s n i e i c a a e o = d s i e a y s ) h r h e e h f n t h h a x o ( D p B o G w a T B p w E T v B Basic concepts: instabilities

N onlinearity can induce instabilities to new states B W > WC e W < W n C T he buckling of a beam : G u r i o

T here is a critical w eight, WC, n U

beyond w hich the beam buckles. n i v e r

Let, s i t y ,

u - m easure the degree of buckling

E h l = (W - W ) W - the deviation from the critical point u

C C d

M e

Close to the critical point the system can be described by the r o n

nonlinear dynam ical system -

3 w w

u# = lu - u = f (u) (u# ô du dt ) u w + . u b g u

(u = 0): .

S teady state solutions a # u0 c . i l / = - the unbuckled state l ~ u0 0 e h u

u d u ê = ê l - tw o sym m etric buckled states - Basic concepts: linear stability

T he steady state solutions tell us that beyond the critical w eight, B e n

or w hen l>0 , the buckled states appear, but they do not tell us G u r

that the system w ill necessarily evolve tow ards these states. T his i o type of inform ation is contained in the stability properties of the n U n

various states. i v e r s

Linear stability: A solution (state) u is stable if any sm all i

s t y , perturbation of decays in the course of tim e. us E h u

A graphical view : A solution us is stable (unstable) if the slope d

3 M f’(us) of u = f (u) = lu - u at us is negative (positive) e # r o n -

l < 0 l > 0 w u# u# w f(u) w . b g u

f(u) . a c . i l / u u ~ e h u d S table U nstable S table Basic concepts: bifurcation diagrams

W e describe such an instability by a l < 0 l > 0 B e n

diagram that show s the various G u steady-state solutions as functions r i o of the control param eter l, and n U n

their stability properties: i v

~ l e r

+ l s i

u t y ,

E h u d

D iagram s of this kind are M

l e called bifurcation diagram s r o n -

T his particular instability w

- l w

is called the pitchfork w . b

Stable Unstable bifurcation g u . a c .

In practice the tw o buckled states need not be perfectly sym m etric, i l / and the instability m ay favor one state over the other. ~ e h

M athem atically, such im perfections are described by additional u d term s that break the sym m etry u ç -u Basic concepts: the cusp singularity

3 2 B

u = lu - u + au + b = 0 e

# n

G

u Cusp singularity u r i o n

b U n i v e r s i t y ,

E h u d

M

l e r o n - w

Cuts at constant b values: w w . b g u .

u u u a c . i l / ~ e h u b > 0 b = 0 b < 0 d l l l Basic concepts: the cusp singularity

3 2 B

u = lu - u + au + b = 0 e

# n

G

u Cusp singularity u r i o n

b U n i v e r s i t y ,

E h u d

M

l e r o n - w

Cuts at constant l values: w w . b g u .

u u u a c . i l / ~ e h u l < 0 l = 0 l > 0 d b b b Basic concepts: hysteresis B e

u n

H ysteresis phenom ena: u+(b) G u r i

D ifferent transition points upon o n

increasing and decreasing b. U n i v

Introduced in the context of e r

u-(b) s m agnetism , exists in num erous i t y ,

other contexts

E h

b b u bmin max d

M e r

Coexistence of o n

S table states - w w

In ecology - catastrophic shifts (S cheffer et al. Nature 2001): w . b g

D esertification – sudden decrease in biological productivity as a u . a c

result of clim ate fluctuations (droughts) and hum an- related . i l / activities (over-grazing). ~ e h

S udden loss of transparency and vegetation in shallow lakes u d subjected to hum an-induced eutrophication. Basic concepts: hysteresis B

H ysteresis in Escher’s art e n

Birds G u r i o n U n i v e r s i t y ,

E

Fish to birds h u d

M

Coexistence e r o of birds & fish n - w

Birds to fish w w . b g u . a c . i l / ~ e h u Fish d Pattern formation in nature B e n

G u r i o n U n i v e r s i t y ,

E h u d

M e r o n - w w w . b g u . a c . i l / ~

o Symmetry breaking e h u o Universality d

o Emergence Salt formation in the Atacama desert (Marcus Hauser) Canonical models of pattern forming systems

N um erous experim ental and theoretical studies support the B e n interpretation of natural patterns as sym m etry breaking phenom ena. G u r i

O f these, studies based on m athem atical m odels have been very o n

instrum ental in understanding the m echanism s underlying these U n i phenom ena. v e r s i

In som e contexts the m odel equations are based on firm theories and t y ,

reproduce observed behaviors w ith quantitative accuracy. Exam ples E h

are the N avier-S tokes equations of fluid dynam ics, and the M axw ell u d

equations in nonlinear optics. M e r o

In m ost other contexts, especially w hen the system ’s com plexity n increases, m athem atical m odels are less accurate. H ow ever, they can - w w

still be very helpful in capturing the qualitative behavior of a system , w . b

and in understanding the specific m echanism s responsible for it. g u . a c .

A long w ith m odels that are geared to reproduce and predict i l / behaviors of specific system s, sim plified m odels, focusing on general ~ e h aspects of pattern form ation rather than on the details of a specific u d system , have appeared. A few of them have becom e canonical m odels. Canonical models of pattern forming systems

T he S w ift-H ohenberg m odel B e n

2 G 3 2 2 u r

u = l u - u - (∞ + k ) u i t c o n U n i v T he FitzH ugh-N agum o m odel e r s i t y

3 2 , 3

u t = l u - u - v + ∞ u E

t h u

2 d

v t = e ( u - a 1 v - a 0 ) + d ∞ v M e r o n -

2 2 w ïu 2 ï ï - Ñ < x < Ñ w w

u = ∞ = + . t , 2 2 b -Ñ < y < Ñ g

ït ïx ïy u . a c . i l / ~ e

u(x,y,0) u(x,y,t) h u v(x,y,0) v(x,y,t) d Classes of pattern forming systems

M odels of this kind help us understand basic m echanism s of pattern B e n

form ation and classify pattern form ing system s accordingly. G u r i o W e w ill distinguish here am ong tw o classes w hich w e illustrate w ith n U

the follow ing m odel sim ulations: n i v e r

S andstede s i t y ,

E h u d

M e r o n - w w w . b g u . Class I: Instabilities Class II: M ultistable Class III: Excitable a c . i l of uniform states system s system s / ~ e h u d Class I: Instabilities of uniform states

T herm al convection (R ayleigh-Bénard): heating a fluid layer from B e n

below G u r i Below a critical tem perature o n

Ttop difference, U n i v

Tbot e d DT = T - T < DT r Flui bottom top C s i t y ,

the fluid is at rest – heat is E h

transferred by m olecular conduction u d

M e

A bove DTC convection sets in the form of ordered parallel roles r o n - w w w . b g u . a c . i l /

Ttop ~ e

Cold h bot > top V isualization by illum inating from Tbot T T u

H ot d fluid below (index of refraction changes fluid w ith tem perature periodically). Class I: Instabilities of uniform states B

W hy do w e have an instability w hen the tem perature difference is e n large enough? G u r

top plate i o n U n i T v e e m r D s p

d e T1+ T i ç e Colder fluid layer heavier t T2

t t

y u E r h e

T : A fluid layer w ith a hotter fluid particle u 1 T1+dT d

M e r o n

Bottom plate - w w

Im agine a fluctuation in w hich a fluid particle at som e height has a w . b

tem perature w hich is slightly higher than the surrounding fluid at g u . that height. Because of therm al expansion the fluid particle w ill a c . i have a density low er than that of the surrounding fluid (i.e. w ill be l / ~ lighter) and w ill tend to m ove upw ard. A s it m oves upw ard the e h u

surrounding fluid becom es yet colder and the buoyancy force d upw ards increases. Class I: Instabilities of uniform states

W hy does not the instability occur w hen the tem perature B e n difference is not large enough? G u r i

T here are processes that stabilize the rest state: o n U

Fluid viscosity: induces transfer of linear m om entum from the up- n i v m oving fluid particle to its neighborhood, thus reducing its e r s m om entum and speed. i t y ,

T herm al conduction: induces diffusion of heat from the fluid E h particle to its colder neighborhood, thus reducing the buoyancy u d force that drives the fluid particle upw ard. M e r o n

Ω T here should be a critical - w

DT w tem perature difference, c, at w . w hich the stabilizing factors just b g u .

balance the destabilizing buoyancy a c . i

force. l / ~ e h

Bodenschatz lab. 1997: sm all Prandtl u num ber fluid experim ents d Class I: Instabilities of uniform states

Farady surface w aves B e n

(stripes and squares) G u r i o n U n i v e r s i t y ,

S tephen W . M orris

E h u d

Chem ical reactions M e r o n - w w

CIM A reaction: w . b

T uring patterns g u

V igil, O uyang and S w inney . a c . i l / ~ e h u d Class I: Linear stability analysis of the uniform state

A pow erful m ethod that provides inform ation about the instability B e n threshold of a uniform state and the nature of the m ode that grow s G u at the instability point is linear stability analysis. Let’s illustrate this r i o m ethod using the S w ift-H ohenberg (S H ) m odel: n U n i

2 v

2 e ï ≈ ï ’ r u 3 2 s i = l u - u - ∆ + k ÷ u - Ñ < x < Ñ t ∆ 2 0 ÷ y ,

ï t ï x « ◊ E h u d

Consider the stationary uniform solution us=0. T his solution is M e linearly stable if any infinitesim ally sm all perturbation decays in the r o n

course of tim e. It is linearly unstable if there exists a perturbation - that grow s in tim e. w w w . W riting u(x, t )= u s + du(x, t ), w here du(x, t ) is an infinitesim ally b g u . sm all perturbation, w e obtain after linearizing in du(x, t ): a c . i l

2 /

2 ~ ï d u ≈ ï ’ e

2 h

= ld u - ∆ + k ÷ d u u ∆ 2 0 ÷ d ï t « ï x ◊ Class I: Linear stability analysis of the uniform state

S ince any perturbation can be represented as a Fourier integral and B e n since the equation for du is linear, it is sufficient to study the G u stability of u =0 to the grow th of any Fourier m ode, r s i o n

ikx ikx U ≈ = ’ n du(x, t) = A(t)e + c.c. | A |ç 0 e i v

∆ ÷ e

∆ ÷ r cos kx + i sin kx s « ◊ i t y

In other w ords w e study the ,

2p E grow th or the decay of sinusoidal k = h u

l d

perturbations w ith w avenum ber k M or w avelength l=2p/ k: e l r o n T he am plitude of the perturbation satisfies the equation - w w

2 w 2 2 . b

A# = s (k ) A, s = l - (k 0 - k ) g u . a c . w here s is the grow th rate of the perturbation. T he solution to this i l / equation is ~ e

s (k )t h = u A(t) A(0)e d Class I: Linear stability analysis of the uniform state B

T he solution us=0 is stable if s(k)<0 for all w avenum bers k. Plotting e n the grow th rate s =s(k) for various l values w e find: G u r i o n

s U

l < 0 l = 0 l > 0 n i v e r s i t

0 0 0 y ,

E h u d

M e r k0 k0 k o n

s<0 for all k s(k0)=0 , us=0 is s(k)>0 for a band of - w

us=0 is stable m arginally stable m odes,us=0 is unstable w w . b

Ω T he solution u = loses stability at l = g s 0 0 u . a

to the grow th of a m ode w ith w avenum ber k w hich c

0 . i l /

breaks the translational sym m etry of the system . ~ e h u

T his is a m ajor m echanism of sym m etry breaking in d nonequilibrium system s Class I: Linear stability analysis of the uniform state

A nother w ay to describe the instability is by plotting a neutral B e

2 n

2 2 stability curve, by setting s = 0 l = (k - k ) G Ω 0 u r i o

l n U

A ll m odes w ithin the n i v

parabola grow e r s i t us=0 is unstable, a y ,

band of m odes E

Dk ~ l h u

around k0 grow d

M e

0 r

us=0 is stable, all o k0 k n

m odes decay - w w w

Linear stability analysis provides inform ation about instability . b g

thresholds, l = 0 in the case, and about the m odes that grow , a u . a c

finite-w avenum ber m ode k in this case. . 0 i l / ~

It does not provide inform ation about the new state that the e h

system evolves to because such inform ation can only be derived u d by considering large deviations from the original state us=0. Class I: Nonlinear analysis, amplitude equations

G oing back to the S H m odel (now in tw o space dim ensions) B e n

G u

3 2 2 2 r i = l - - ∞ + o u t u u ( k 0 ) u -Ñ

w e consider approxim ate solutions of the form r s i t y ,

u(x,t) @ A(x, y,t) exp(ik x) + c.c. 0 < A << 1 0 E h u d

w here the am plitude A is assum ed to be sm all but finite. W e M e further assum e that A varies w eakly in space and in tim e. S uch a r o n

solution describes w eak m odulations of a stripe pattern, either - w w w . longitudinal or transverse or both. b g u . a c . i l /

U sing various m athem atical m ethods, e.g. m ultiple tim e scales, a ~ e h

nonlinear equation for the am plitude A can be derived from the u original S H equation: d Class I: Nonlinear analysis, amplitude equations

T he equation, know n as the N ew ell-W hitehead-S egel (N W S ) B e n

equation, reads: G u r i

2 o ≈ 2 ’ n ïA ï ï 2 U

= lA + ∆ 2k0 - i ÷ A - 3 | A | A n ∆ 2 ÷ i v

ït « ïx ïy ◊ e r s i t y ,

Equations of this kind are called “am plitude

s E

l = 0 h

equations”. T hey have universal form s that are u d

0 determ ined by the types of instabilities the M e system s go through. A ny system that goes r o n

through a finite-k instability of the form - k0 k w w

w ill be described close to the instability point by a N W S equation. w . b g

M ore generally, different system s that experience the sam e type u . a c

of instability are described by the sam e am plitude equation and w ill . i l / behave sim ilarly near the instability point. ~ e h u

T his explains the universal nature of m any of the patterns observed d in nature. Class I: Nonlinear analysis, amplitude equations

T he am plitude equation has periodic solutions of the form : B e n

2 2 G u

A = A exp(iqx), A = (l - 4k q )/ 3 r

0 0 0 i o n U n

w hich exist w ithin the parabola i k = k ê q v

0 e

2 2 r l = s 4k 0 q i t y ,

l l E h u d

M

or e r o n -

0 q k k w

0 w w . b g

Close to the instability point (0

2 2 c

for the grow ing m odes obtained from the linear . l = (k - k ) i 0 l / stability analysis. T his does not im ply that all m odes that grow from ~ e h u

the zero state evolve to periodic solutions – som e of them m ay be d unstable. Class I: Nonlinear analysis, amplitude equations

T he am plitude equation can be used to study the linear stability of B e n periodic solutions w hich exist inside the blue parabola. S uch an G u analysis gives the follow ing boundaries of longitudinal and transverse r i o instabilities: n U n

Periodic solutions are stable: i v e r s i t y ,

l

Eckhaus instability: Periodic E h

solutions lose stability to u d

longitudinal m odulations M e r o n - w w w . 0 b g

k k u 0 . a c . i l /

Z igzag instability: Periodic solutions lose ~ e

stability to transverse m odulations h u d Read m ore in Cross & H ohenberg, Rev. M od. Phys. 1993. Class I: S ummary B e n

G

Instabilities m ay break translational sym m etry and give rise to u r i spatial patterns even w hen the system and the forces it is subjected o n to are uniform . U n i v e

T he spatial patterns induced by instabilities are universal; the sam e r s i t patterns appear in different system s that go through the sam e y ,

instability. E h u d

T he dynam ics close to instability points are described by universal M e equations – the so called am plitude equations , e.g. N W S equation for r o n

a system close to a stationary finite- w avenum ber instability. - w w

A m plitude equations are useful for studying secondary instabilities, w . b

such as Eckhaus and zigzag. g u . a c . i l / ~ e h u d Class II: M ulti-stable systems

M ultistability of uniform states can result in asym ptotic patterns B e n even if all the uniform states are linearly stable. G u r i

T he FIS (ferrocyanide-iodate-sulfite) reaction: A bistable system of o n

high (w hite) and low (black) pH . D evelops patterns by a transverse U n i front instability (class II). v e r s i t y ,

E h u d

M e r o n - w w w . Bistability and b g hysteresis in the u . a

FIS reaction c . i l / ~ e h u d Lee & S w inney, PRE (1995) Class II: Bistable systems

W e w ill study pattern form ation phenom ena in bistable system s using B e n

the FitzH ugh-N agum o (FH N ) m odel: G u r

3 2 i o

u t = u - u - v + ∞ u n U

2 n i v = e (u - a v - a ) + d ∞ v v

t 1 0 e r s i T his is an exam ple of an A ctivator-Inhibitor system : u activates the t y ,

grow th of itself and v, w hile v inhibits the grow th of itself and u. E h u d

T he FH N m odel captures all 3 classes of pattern form ing system s: M e r

v v v o

e << 1 n - (u+ ,v+ ) w w w . b

( ) g u0 ,v0 u (u ,v ) u u

0 0 u (u- ,v- ) . (u- ,v- ) a c . i l / ~ e

Class I Class II Class III h u d 3 u - a1v - a0 = 0 u - u - v = 0 (nullclines) Class II: Bistable systems - fronts

W e focus here on class II, corresponding to bistable system s that B e n consist of tw o stable uniform stationary states: u = u - , v = v- and G u u = u+ , v = v+ r i o

v n

S uch system s support in general U n i v

front solutions u(x,t) that are bi- e (u+ ,v+ ) r s

asym ptotic to the tw o stable states: i t y ,

u(x,t)çu+ v(x,t)çv+ as x ç-Ñ E ( ) h u0 ,v0 u u d u(x,t)çu- v(x,t)çv- as x ç+Ñ M (u- ,v- ) e r o n - y (u+ ,v+ ) + - (u- ,v- ) w w

U p state D ow n state w . b g u

x W e w ill see in the follow ing that . a c .

pattern form ation phenom ena in i l u+ / bistable system s strongly ~ v+ e h

depend on the properties of u v x d - these fronts solutions. u- Class II: Bistable systems - fronts

Let’s begin w ith a sim pler system , assum ing v is constant, B e n

3 G u

ut = u - u - v + u xx [Ω a gradient system ] r i o n

3 U

u -u - v = 0 n i

v v e

For a certain range of v there are tw o stable r s i t

stationary uniform solutions, u+(v) and u-(v), y ,

and an unstable solution u0(v) in betw een: E h

u u d

Front solutions that are biasym ptotic to the u- u0 u+ M tw o stable states propagate at constant e r o speeds, w hich depend on the value of v: n - w

v<0 v=0 v>0 w w

u u u+ . + + [ Fronts propagate so b g u

as to m inim ize a . a

u c u- u- - . Lyapunov functional ] i l /

(+) invades (–) (–) invades (+) ~ e h u N ote: front speed is uniquely determ ined by the value of v. d Class II: Bistable systems – front instabilities

T he situation becom es com pletely different w hen the equation for B e n

the inhibitor v is added: G u r i u o

+ n 3 v u - a1v - a0 = 0 u -u - v = 0 v+ U n i v

v e

- r

(u+,v+) s

u i

- t y

x ,

(u-,v-) u E h u d

M e

Front solutions connecting the up-state (+) (at xç -Ñ) to the dow n- r o xç -Ñ n state (-) (at ) can them selves undergo instabilities. - w w

T w o front instabilities: w . b g

A transverse instability A longitudinal instability u . a c .

u+ i l /

v+ ~

+ - e h u

v- d u- Class II: Bistable systems – front instabilities B

T o understand these instabilities and their effects on pattern e n form ation w e first need to understand how the curvature of a front G u r k , affects its norm al velocity, C (velocity in a direction norm al to the i o n

front line). U n i v

T he curvature affects the diffusion of the e r

- s i

activator and the inhibitor and thereby the t y ,

front velocity C: E

+ h

R u d

C = C(k ) k = 1 R M e

Radius of curvature r o n - w w

T he actual form of the velocity-curvature relation w ill be affected w . b by the param eters that control these diffusion processes: d and e. g u . a c . i

U sing a singular perturbation analysis in the range e/d << 1 , w hich l / ~ is often m et in actual system s, velocity-curvature relationships can e h u

be derived. d Class II: Bistable systems – front instabilities B e n

G u r i o n U n i

C v e r s i t y ,

E h

k u d

M e r o n - 0 w w w

0 . b g u . a c . i l / ~ e h u d

H agberg & M eron, Chaos (1994), N onlinearity (1994) Class II: Bistable systems – stationary patterns B

Front stability to transverse perturbations e n

G u

C C = C0 - Dk r i + - + - o Front is stable n U

(d small - slow v n i

Ω v 0 k diffusion) e r s i t y ,

3

d large C C = C0 + D1k - D3k E + - + - h u

Front is unstable d

(d large - fast v Ω M e

k diffusion) r

0 o n - w w

U nder these conditions stationary labyrinthine patterns develop: w . b g u . N ote: dom ains do not m erge a c . i

due to fast v diffusion l / ~ e h u d vf<0 vf=0 M im ura 2000 Class II: Bistable systems – traveling w aves

Front stability to longitudinal perturbations B e n

G u r i o

S tationary labyrinthine n U

patterns are found here n i d v C w here is large (fast v e r

diffusion) and e not too s i t

sm all. y ,

k

E h

Rem iniscent of the behavior u d

near a cusp singularity or M

pitchfork bifurcation e r o n - w w w . b g u . a c . i l / T he solid line is the threshold of a longitudinal front instability of ~ e h the pitchfork type com m only referred to as the “N on-equilibrium u d Ising-Bloch” (N IB) bifurcation. Class II: Bistable systems – traveling w aves

T he N on-equilibrium Ising-Bloch (N IB) bifurcation: B e n

u+ Ising front (Coullet) G u

v r

+ i o v v n

- U

u- n i v e r s i t y

h = ed ,

u E h u Bloch fronts d

M e

Ikeda, M im ura, N ishiura, 1989 r o

H agberg and M eron 1994 n -

In Ising regim e (h>hc) fronts have In Bloch regim e (h

unique direction of propagation propagate in opposite directions w . b g

C C C u . a c . i l / ~

hc h hc h hc h e h u d (+) invades (–) or (–) invades (+) (+) invades (–) and (–) invades (+) Class II: Bistable systems – traveling w aves B

W hat are the im plications for pattern form ation? e n

G

1. O nset of traveling w aves u r i o n

T he N IB designates a transition from stationary or uniform U n i

patterns to traveling w aves v e r s i Ising regim e: dom ains can t y ,

expand, shrink or becom e E h

stationary u

(+) invades (–) (–) invades (+) d

M e r o Bloch regim e: dom ains can n -

travel w

(–) invades (+) (+) invades (–) w w . b g

u . a

2. S piral w aves: c . i l / ~ e h u Point e d Gilli and Frisch (1995) Class II: Bistable systems – traveling w aves B

Close to the N IB bifurcation – m ore com plex dynam ics e n

H agberg & M eron, Chaos (1994), PRL (1994); Elphick, H agberg & M eron PRE (1995). G u

r

k=0 i 3. Bloch-front turbulence: Positive slope at o n

im plies transverse U n

instability i C v e r

Point s

d i t y ,

E h

V ortex u d

pair k M e r o n - w w w .

G row ing transverse A fter spiral vortex b g

perturbations nucleation u . a c . i l / ~

S im ulations of the FH N e h

m odel close to the front u d bifurcation Class II: Bistable systems – traveling w aves

B

4. S pot splitting: e n

G u r i

C Initial spot o n U n i v e r s i t y ,

0 k E h u d

M e

Local front reversals induce r o Point b n

nucleation of spiral-vortex pairs - w w w .

S im ulations of the FH N m odel in the Ising b g

regim e close to the front bifurcation u . a c . i

U p state (+) D ow n state (-) l / ~ e h u

H agberg and M eron, PRL (1994), N onlinearity (1994), d PRL (1997), Physica D (1998) Class II: S ummary B e

Bistability of states allow s for dom ain patterns, stationary or n

G

traveling patterns, consisting of alternate dom ains of the different u r i stable states. o n U n i

D om ain patterns are strongly affected by front instabilities: v e r s i t

A transverse front instability that leads to stationary y ,

labyrinthine patterns. E h u d

A longitudinal instability – the N IB bifurcation – that designates M e

the onset of traveling w aves. r o n -

Coupling of the tw o instabilities leads to spiral turbulence, spot w w

splitting, etc. w . b g u . a

T he m ultistable states need not be stationary and uniform - c . i l periodic states in space or tim e, can be reduced to stationary / ~ e uniform states using am plitude equations. h u

A d A cknow ledgement

A ric H agberg, Christian Elphick (theory) B e n

H arry S w inney, Q i O uyang, Bill M cCorm ick (experim ents) G u r i o n

Funds U n i v e

BS F – U S -Israel Binational S cience Foundation r s i t y ,

E

R elevant references h u d

1. Meron E., “Pattern Formation in Excitable Media”, Physics Reports 218, 1-66 M e r

(1992). o n

2. M.C. Cross and P.C. Hohenberg, “Pattern Formation Outside Equilibrium”, Rev. -

Mod. Phys. 65, 851 (1993). w w

3. Hagberg A. and Meron E., “Domain Walls in Nonequilibrium Systems and the w . b

Emergence of Persistent Patterns”, Phys. Rev. E 48, 705-708 (1993). g u . 4. Hagberg A. and Meron E., “Pattern Formation in Non-Gradient Reaction Diffusion a c . i

Systems: The Effects of Front Bifurcations”, Nonlinearity 7, 805-835 (1994). l / ~

5. Hagberg A. and Meron E., “From Labyrinthine Patterns to Spiral Turbulence”, Phys. e h

Rev. Lett. 72, 2494-2497 (1994). u d R elevant references B e

6. Hagberg A. and Meron E., “Complex Patterns in Reaction Diffusion Systems: n

A Tale of Two Front Instabilities”, Chaos 4, 477-484 (1994). G u r

7. Elphick C., Hagberg A. and Meron E., “Dynamic Front Transitions and i o Vortex Nucleation”, Phys. Rev. E 51, 1899-1915 (1995). n U n

8. Lee K.J. and Swinney H.L., “Lamellar structures and self replicating spots in a i v e

reaction-diffusion systems”, Phys. Rev. E 51, 3052-3058 (1995). r s i

9. Hagberg A., Meron E., Rubinstein I, and Zaltzman B., “Controlling Domain t y , Patterns Far From Equilibrium”, Phys. Rev. Lett. 76, 427-430 (1996).

E

10. Haim D., Li G., Ouyang Q., McCormick W.D., Swinney H.L., Hagberg A., h u d

and Meron E., “Breathing Spots in A Reaction Diffusion System”, Phys. Rev. M

Lett. 77, 190-193 (1996). e r o

11. Hagberg A. and Meron E., “The Dynamics of Curved Fronts: Beyond n

Geometry”, Phys. Rev. Lett. 78, 1166-1169 (1997). - w

12. Hagberg A., Meron E., Rubinstein I, and Zaltzman B., “Order Parameter w w

Equations for Front Transitions: Planar and Circular Fronts”, Phys. Rev. E, 55, . b g

4450-4457 (1997). u . a

13. Hagberg A. and Meron E., “Order Parameter Equations for Front Transitions: c . i l Nonuniformly Curved Fronts”, Physica D 123, 460-473 (1998). / ~

14. Hagberg A. and Meron E., “Kinematic Equations for Front Motion and Spiral- e h u

Wave Nucleation”, Physica A 249, 118-124 (1998). d