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Physics Letters A www.elsevier.com/locate/pla

Travelling waves associated with saddle-node bifurcation in weakly coupled CML Ma Dolores Sotelo Herrera a,∗ , Jesús San Martín a,b a b

Departamento de Matemática Aplicada, E.U.I.T.I., Universidad Politécnica de Madrid, Ronda de Valencia 3, 28012 Madrid, Spain Departamento de Física Matemática y de Fluidos, U.N.E.D., Senda del Rey 9, 28040 Madrid, Spain

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 24 November 2009 Received in revised form 25 May 2010 Accepted 2 June 2010 Available online 9 June 2010 Communicated by A.R. Bishop

Weakly coupled CML can be analytically solved by using perturbative methods. This technique has been recently used to deduce analytical expressions for travelling waves. Nonetheless, the results were limited for periodic solutions far away from saddle-node bifurcation. In this Letter, this problem is solved and periodic solutions, arising from the individual dynamics, are totally characterised. © 2010 Elsevier B.V. All rights reserved.

Keywords: Coupled map lattice Travelling waves Saddle-node bifurcations

1. Introduction Many systems in physics, chemistry, biology and engineering consist of a group of coupled elements, but at the same time every element has its own individual evolution. These systems are normally described by coupled ordinary differential equations or partial differential equations, but researchers do not know how to approach them systematically. A simple approach to these dynamical systems are Coupled Map Lattices (CML), whose evolution equations [1] are given by

X i (n + 1) = (1 − α ) f X i (n) +

m α

m

f X j (n)

(1)

j =1

where X i (n) represents the state of the so-called oscillator located at node “i” of a lattice, in the instant “n”. The function f rules the individual dynamics of every oscillator, and the parameter “α ” weights the coupling among oscillators, given by the coupling term. The α m f ( X (n)) and neighbour coupling α ( f ( X coupling terms can be classiﬁed into two types: global coupling m j i −1 (n)) + f ( X i +1 (n))). j =1 2 When α = 0, there is not coupling between the oscillators and they evolve individually according to

X i (n + 1) = f X i (n); r

(2)

where r is the control parameter ruling the individual dynamics. The system (2) can be interpreted as a discrete dynamical system obtained from a continuous one by using a Poincaré section; so f p is then the pth recurrence map. CML show a rich sort of behaviours, as a consequence of the interaction among their constitutive elements: synchronisation, travelling waves, period-doubling cascade, intermittency, patterns [2–7], that are usual in spatially extended systems. Facing and understanding these complex and important behaviours is easier in CML hence its importance. Furthermore, as CML are exceptional in modelling interaction among constitutive elements of a system, they have been extendend to an extraordinary variety of unrelated ﬁelds. They have been used in: evolution of genetic sequences [8], cell differentiation [9], biological networks [10], phase transition [11], turbulence [12], reaction– diffusion equation [13,14], habitat fragmentation [15], economic dynamics [16], fatigue cracks in beams [17], to mention only a few in biology, physics, chemistry, sociology and engineering (see [18,19] for more details). In spite of the many and interesting works in CML most of them only show numerical results. Just a few provide theoretical results [20– 24] and unfortunately some of the theoretical results are based on particular functions [21]. In former papers [25,26] the authors found

*

Corresponding author. E-mail addresses: [email protected] (M.D. Sotelo Herrera), [email protected] (J. San Martín).

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.06.002

M.D. Sotelo Herrera, J. San Martín / Physics Letters A 374 (2010) 3292–3296

3293

Fig. 1. 3-periodic S-N orbit of the system X (n + 1) = f ( X (n); r ), corresponding to the ﬁxed points of f 3 . The more points the S-N orbit has, the smaller the distance between C and x∗1 .

analytical solutions of weakly coupled CML, where the individual evolution of the oscillators was ruled by an arbitrary C 2 function, instead of particular functions. Nonetheless, the results were not valid close to saddle-bifurcations, what, obviously, constrained the generality of results. In this Letter we solve this problem, and show analytical solutions of waves in weakly coupling CML close to saddle-node bifurcation. 2. Saddle-node waves in global coupling CML Saddle-node (S-N) p-periodic orbits are the origin of p-periodic windows in a dynamical system X (n + 1) = f ( X (n); r ). These orbits have p S-N points x∗i , i = 1, . . . , p, and ( f p ) (x∗i ) = 1 at all of them (see Fig. 1). When the control parameter r is changed the S-N orbit splits into two new ones, one of node type and the other of saddle type. Both orbits move away as the control parameter varies. Right at the onset of the bifurcation ( f p ) (x∗i ) = 1. As the control parameter varies the derivative vanishes, then the p-periodic supercycle takes place within the p-periodic window. Further changes of the control parameter produce a decrease of ( f p ) (x∗i ) (x∗i a node point) until it reaches the value −1 and the period-doubling cascade starts. The outlined process from S-N bifurcation to period-doubling bifurcation is a continuous one, resulting from the continuous change of the control parameter. By continuity and using the chain rule, there is a point of the S-N orbit such that it will turn into a point of the supercycle. This point will coincide with the critical point C of f (see Fig. 1). We denote this point by x p ,C , which is the S-N point closest to C . For instance, if we take into account the logistic equation X (n + 1) = r X (n)(1 − X (n)), 0 r 4, it results that d(C , x3,C ) = 0.02 units (see Fig. 1), where the length of the interval [0, 1] is taken as the unit length. The bigger the period p of the S-N orbit the smaller the distance d(C , x p ,C ). In conclusion, for any ε there exists p ∈ N such that the p-periodic S-N orbit has a point x p ,C with d(x p ,C ; C ) < ε . In the make of it, when we expand f p (x p ,C ) around C , by using Taylor expansion, it will result |x p ,C − C | < ε 1 and these terms will be dismissed in our Taylor expansion when we study the S-N waves in CML in the following theorem. Theorem 1 (Saddle-node wave). Let f : I ⊂ R → I , f ∈ C 2 , be a unimodal map, depending on a parameter, with critical point at C , and be {x∗1 , x∗2 , . . . , x∗p } an S-N p-periodic orbit of f with 1 − f p (x∗i ) O (ε ). If |x∗1 − C | ε then the CML given by

X i (n + 1) = (1 − εα ) f X i (n) +

p αε

f X j (n) ,

p

i = 1, . . . , p ,

ε1

(3)

j =1

shows a p-periodic solution given by

X i (n + j ) = x∗i + j + ε A i + j + O where

A i +1 =

α [−x∗i+1 + α (−x∗i+1

+

1 p

p

2

ε , i = 1, . . . , p , j ∈ N

x∗ +

l =1 l p ∗ 1 x ), p l =1 l

i

∗

n=2 ((−xn

+

1 p

p

x∗ )

l =1 l

i

l=n

f (xl∗ ))],

i = 2, . . . , p i=1

with periodic conditions

x∗i + p = x∗i ,

A i+ p = A i ,

∀i

Remark 1. 1 − f p (x∗i ) O (ε ) mathematically speaking means that the orbit is an S-N one or is close to it. x∗1 plays the role of x p ,C

described above. When 1 − f p (x∗1 ) O (ε ) the analytic solutions given in [25,26] are not valid, because the term 1 − f p (x∗1 ) appears in 1

the denominator generating terms of O ( ε ) and the expansions in powers of

ε are not valid.

3294

M.D. Sotelo Herrera, J. San Martín / Physics Letters A 374 (2010) 3292–3296

Proof of Theorem 1. The periodic solution

X i (n + j ) = x∗i + j + ε A i + j + O

2 ε ,

i = 1, . . . , p

will exist when the system

⎧ X i (n) = x∗i + ε A i , ⎪ ⎪ ⎪ ⎪ ⎪ X i (n + 1) = x∗i +1 + ε A i +1 , ⎪ ⎪ ⎨ .. . ⎪ ⎪ ⎪ ⎪ X i (n + p − 1) = x∗i + p −1 + ε A i + p −1 = x∗i −1 + ε A i −1 , ⎪ ⎪ ⎪ ⎩ X i (n + p ) = x∗i + ε A i ,

i = 1, . . . , p

is compatible and determined. As the evolution of the CML is given by

X i (n + 1) = (1 − εα ) f X i (n) +

p εα

f X j (n)

p

(4)

j =1

after substituting X i (n) = x∗i + ε A i + O (ε 2 ) into (4) it yields

x∗i +1 + ε A i +1 + O

2

ε = (1 − εα ) f x∗i + ε A i + O ε2 +

p εα

f x∗j + ε A j + O

p

2

ε

(5)

j =1

The next stage in our calculations is to expand f (x∗i + ε A i + O (ε 2 )) in powers of ε and substitute the result into (5). To do so we must consider the relative positions of the points {x∗1 , x∗2 , . . . , x∗p } with respect to the critical point C , because the expansion in powers of ε will depend on such position. Two cases must be considered: i) |x∗1 − C | ε . The point x∗1 of the S-N orbit is very close to the critical point C (see Fig. 1). (We can suppose, without loss of generality, that x∗1 is the nearest point orbit to C that is, x∗1 = x p ,C .) Expanding in powers of ε we obtain

f x∗1 + ε A 1 + O

2

= x∗2 + O ε 2

ε

(6)

where | f (x∗1 )| ε | f (C )| has been used. Substituting (6) into (5) it yields ∗

x2 + ε A 2 + O

εα p

2

∗

∗

ε = x2 − εα x2 +

p

j =1

x∗j +1 + O

2

ε

Remark 2. Let us notice that f (x∗1 ) does not appear in the expansion above and eventually the problematic term 1 − f p (x∗1 ) will not appear in the denominator of ﬁnal result (see Remark 1). ii) |x∗i − C | > ε , i = 2, . . . , n. The points x∗i of the S-N orbit are far away from the critical point C . In this case, the expansion in powers of ε yields

f x∗i + ε A i + O

2

ε

= f x∗i + ε A i f x∗i + O ε 2

and substituting into (5) it produces

x∗i +1 + ε A i +1 + O

2

εα p

ε = (1 − εα ) x∗i+1 + ε A i f x∗i + O ε2 +

p

j =1

that is,

x∗i +1 + ε A i +1 + O

2

ε = x∗i+1 + ε A i f x∗i − εα x∗i+1 +

εα p

p

Therefore, to order ε , the system we must solve is

p − A i f x∗i + A i +1 = −α x∗i +1 + αp j =1 x∗j +1 , i = 2, . . . , p p x∗ , i=1 A 2 = −α x∗2 + αp j =1 j +1

whose matrix expression is:

j =1

x∗j +1 + O

2

ε

x∗j +1 + ε A j f x∗j

+ O ε2

M.D. Sotelo Herrera, J. San Martín / Physics Letters A 374 (2010) 3292–3296

⎛

⎞⎛

0 1 0 0 ··· 0 ⎜ 0 − f (x∗2 ) ⎟⎜ 1 0 ··· 0 ⎜ ⎟⎜ ⎜0 ⎟⎜ 0 − f (x∗3 ) 1 · · · 0 ⎜ ⎟⎜ ⎜ .. ⎟⎜ .. .. .. . . .. ⎝. ⎠⎝ . . . . . 1 0 0 0 · · · − f (x∗p )

⎛

⎞

−x∗2 +

1 p 1 p 1 p

p

x∗

j =1 j p x∗ j =1 j p x∗ j =1 j

3295

⎞

⎜ ∗ ⎟ ⎜ −x3 + ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ ∗ ⎟ ⎟ − x + ⎜ ⎟ ⎟=α⎜ 4 ⎟ .. ⎟ ⎜ ⎟ .. . ⎠ ⎜ ⎟ . ⎝ ⎠ Ap p −x∗1 + 1p j =1 x∗j A1 A2 A3

(7)

This system is compatible and determined because the determinant of the matrix is (−1) p +1 = 0. Furthermore, the solution of the system is different from the trivial one, since the independent term column is not null because x∗1 = x∗2 = · · · = x∗p . The solution of the system (7) can be obtained by inversion (see details in [25]) and reordering the rows, the following is obtained:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

A2 A3 ⎟ ⎟

⎛

0 1

0 0

.. .

.. .

··· ··· .. .

f (x∗3 ) · · · f (x∗p −1 )

f (x∗4 ) · · · f (x∗p −1 )

···

1 f (x2 )

⎜ ⎜ .. .. ⎟ = α ⎜ ⎜ . ⎜ . ⎟ ⎟ ⎜ ∗ ∗ Ap ⎠ ⎝ f (x2 ) · · · f (x p −1 ) A1 f (x∗2 ) · · · f (x∗p )

f (x∗3 ) · · · f (x∗p )

f (x∗4 ) · · · f (x∗p )

⎛

⎞ −x∗2 + 0 ⎜ 0 ⎟ ⎜ −x∗3 +

0 0

1 p 1 p 1 p

p

∗

j =1 x j p x∗ j =1 j p x∗ j =1 j

⎞

⎟ ⎟ ⎟⎜ ⎟ .. ⎟⎜ ∗ ⎟ ⎟ ⎜ −x4 + . 0⎟ ⎟⎜ ⎟ ⎟⎜ ⎟ . 1 0 ⎠⎜ ⎟ .. ⎠ ⎝ ∗ f (x p ) 1 p −x∗1 + 1p j =1 x∗j

···

(8)

This is a lower triangular matrix, from which the looked for solution can be deduced as:

A i +1 =

α [−x∗i+1 + α (−x∗i+1

+

1 p

p

x∗ +

l =1 l p ∗ 1 x ), p l =1 l

i

∗

n=2 ((−xn

+

1 p

p

x∗ )

l =1 l

i

l=n

f (xl∗ ))],

i = 1 i=1

2

Remark 3. If there were more points of the S-N orbit close to C , apart from x∗1 , let us say x∗j , then |x∗j − C | ε , and the process to get the solution would be formally the same, the only thing we would have to do is to replace f (x∗j ) by 0 in the system (8) matrix.

Remark 4. According to Remark 2, if we pay attention to the matrix of system (7) we can observe that f (x∗1 ) has been replaced by zero, such a thing does not happen when the point x∗1 does not belong an S-N orbit (see (7) in [25]). In fact, the point i) of Theorem 1 proof is the key to avoid terms O ( ε1 ).

Remark 5. Theorem 1 means that CML given by (3) has travelling waves if oscillators of the CML are initially located at the points of an S-N orbit. The theorem rules when ε 0.1 and their solutions make an error O (ε 2 ). 3. Saddle-node waves in neighbour coupling CML Theorem proved for global coupling can immediately be reformulated for neighbour coupling. To turn a global coupling CML into a neighbour one we change the coupling term N ε

N

f X j (n)

(9)

j =1

by

ε 2

f X i −1 (n) + f X i +1 (n)

(10)

If we pay attention to the proof of Theorem 1, we can tell the coupling terms appear in the independent term columns, therefore the algebra of the proof does not change by substituting a coupling term by another, so it is enough to substitute (9) by (10) in the independent term columns of the proof of Theorem 1 to get the ﬁnal solution. In this way, proof is straightforward and the reformulated Theorem 1 goes as follows. Theorem 2 (Saddle-node wave). Let f : I ⊂ R → I , f ∈ C 2 , be a unimodal map, depending on a parameter, with critical point at C , and be {x∗1 , x∗2 , . . . , x∗p } a p-periodic orbit of f with 1 − f p (x∗i ) O (ε ). If |x∗1 − C | ε then the CML given by

X i (n + 1) = (1 − εα ) f X i (n) +

αε 2

f X i −1 (n) + f X i +1 (n) ,

i = 1, . . . , p ,

ε1

shows a p-periodic solution given by

X i (n + j ) = x∗i + j + ε A i + j + O where

A i +1 =

2

ε , i = 1, . . . , p , j ∈ N

α [−x∗i+1 + 12 (x∗i + x∗i+2 ) + α (−x∗i+1

+

1 ∗ (x 2 i

+ x∗i +2 )),

i

∗

n=2 ((−xn

+ 12 (xn∗−1 + xn∗+1 ))

i

l=n

f (xl∗ ))],

i = 2, . . . , p i=1

3296

M.D. Sotelo Herrera, J. San Martín / Physics Letters A 374 (2010) 3292–3296

with periodic conditions

A i+ p = A i ,

x∗i + p = x∗i ,

∀i

4. Discussion and conclusions We have shown how expansions in terms of the coupling weight have to be so that perturbative techniques are useful in weakly CML close to the S-N bifurcations. As a result, we have completed the problem partially solved in [25,26]. As in those works the solutions just presented are valid for an arbitrary number of oscillators in the CML and for an arbitrary function C 2 to rule the individual dynamics of the oscillators. In fact, the proofs of theorems hold if we assume that f is C 0 (instead of C 2 in the interval) and only C 2 around the points of S-N orbits. This remark can be very useful for researchers working on discrete networks with generalised functions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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