Chaotic Coupled Map Lattices

Chaotic Coupled Map Lattices Author: Dustin Keys Advisors: Dr. Robert Indik, Dr. Kevin Lin 1 Introduction When a syste of chaotic aps is coupled in a way that allows the to share inforation about each other s state, there is a possibility that they ay synchronize. Each individual ap ay still be chaotic but has exactly the sae state as its synchronized partner. An entire network of aps ay be arranged in such a anner as to allow for not only the possible synchronization of all ebers, but also ore coplicated odes of quasi-synchronization, where each eber is synchronized with at least one partner but not necessarily all partners. Soe possible ethods of coupling include diffusive (or nearest neighbor) coupling, and one-way coupling, but it is easy to conceive of uch ore coplicated networks. One-way coupling involves choosing a direction along the lattice for each eber to counicate, an anology is the gae Telephone where a person whispers a essage to the person on his right and the essage is transitted down a line of people until the last person tells it to the person who created the essage, usually sounding little if anything like the original essage (in which case the network would be closed, but this need not be the case). Diffusive coupling is the sae concept except that instead of going one way, the essage is told to everyone in the iediate vicinity. Think of breaking news spreading in a tightly packed crowd. In the case of these networks we ay control the coupling strength, which is soething like how tightly packed the crowd is (without changing the nuber of people). It is interesting to look at the behavior of the syste as this coupling paraeter changes. Theory So far we have studied the circle lattice of logistic aps under diffusive coupling. The ap corresponding to this can be written as x (i) n+1 = f((1 )x(i) n + (x(i+1) n + x (i 1) n )) which says that the n + 1 state of the ith location on the lattice is a weighted ean of its own state and the i+1 and the i 1 locations, where is the coupling paraeter. In our case the function is the logistic ap x n+1 = ax n (1 x n ) We ay also represent the lattice as a vector where each diension is a lattice eber: x n+1 = F ((1 )I x n + (R x n + L x n )) where R and L are shift operators (corresponding respectively to Up and Down shifts on the vector, which are Right and Left shifts on the lattice), I the identity atrix, and it is understood that the function F acts on the vector eleentwise. 1

Then we ay represent the coupling as one atrix, C, which acts on the vector, x n. C = (1 )I + (R + L) So that in the case of a diffusively coupled circle lattice with 4 ebers (1 ) C = (1 ) (1 ) (1 ) This type of atrix is called a circulant atrix and it ay be diagonalized by perforing a discrete fourier transfor (DFT) on the atrix. This will be convenient in the derivation of the synchronization condition given below. Conditions for Stable Synchronization If we consider a phase space with each diension corresponding to one lattice eber s state, then there is a subspace which corresponds to synchronization: the diagonal x (1) = x () =... = x (). We wish to consider the behavior of the syste in the vicinity of the diagonal. Do trajectories on the diagonal stay on the diagonal? Of course, since we are dealing with copies of the sae deterinistic ap, a trajectory on the diagonal just corresponds to the siulatenous iteration of the sae ap with the exact sae values, and there is nothing to knock the syste out of synchronization. Is the diagonal linearly stable, i.e do trajectories close to the diagonal end up on the diagonal? That is a uch ore coplicated question. Rewording it slightly we can ask: for what values of the coupling paraeter, does the iteration of the lattice contract every direction in phase space except along the diagonal? Singular values give us a way to look at the directions which an operator stretches or contracts. For exaple if a atrix is applied to the unit circle transforing it into an arbitrary ellipse, the largest singular value is length of the sei-ajor axis, and the second singular value is the length of the sei-inor axis. So to see if the diagonal is linearly stable we can look at the singular values of the Jacobian, which are just the absolute value of the eigenvalues in decreasing order. In our case we can analytically approxiate the singular values in the vicinity of the diagonal. The jacobian for a circle lattice looks like J = f ( x)c 1, where C is the circulant atrix and is a f ( x) is the derivative of the ean of entries of the state vector x. The eigenvectors of C are the coluns of the DFT atrix which have the for: w k = 1 ζ k ζ ḳ. ζ 1 k where is the size of the lattice ζ k = e πik. Applying J to this vector J w k = f (y) (1 ) w k + ] (R w k + L w k ) 1 the reason we consider the ean is addressed in the notes

But shifting w right or left is the sae as ultiplying by ζ k and ζ 1 k respectively. J w k = f (y) (1 ) + ] (ζ k + ζ 1 k ) w k J w k = f (y) (1 ) + ] πki πki (e + e ) w k ( ( ))] πk J w k = f (y) (1 ) + cos w k Having found an expression for the eigenvalues, we see that the singular values, s k, have the for ( )] s k = πk f (y) (1 ) + (cos A sufficient condition for linear stability would be if for all iterations, all the singular values except to the one corresponding to stretching along the diagonal are less then one. In the case of the logistic ap we ay bound the derivative by the logistic paraeter f (y) < a. So the sufficient condition for synchronization becoes ( ) a πk (cos 1 + 1) < 1 ( ) πk (cos 1 + 1) < 1/a ( ( ) ) πk 1/a < 1 + cos 1 < 1/a (1 + 1/a) < ( cos ( πk ) 1 1 1/a 1 cos ( ) < < πk ) < 1/a 1 1 + 1/a 1 cos ( πk However this is a very weak condition in the sense that though it guaruntees synchronization, it only applies to a sall subset of what is observed to actually synchronize. The axiu value of a for which equation (1) ay be satisfied is a = 3 which in the case of the logistic ap is not even large enough to be chaotic, though the condition is irrespective of the exact ap. With that in ind the following analysis provides ore realistic bounds. Stronger Condition for Synchronization ) (1) We ust first describe a synchronized solution, y = f( x n ) 1, where 1 represents a vector of 1 s. The difference between the actual state of the lattice and the synchronized state is δ n = x n y n so that δ n+1 = F (C x n ) y n+1 It is convenient to rewrite this equation as see Notes y n+1 + δ n+1 = F ( y n + C δ n ) 3

By substituting y n + C δ n = x n so that the arguent of F becoes (1 )( y n + δ n ) + (1 )I + ] (R + L) ( y n + δ n = (1 + ) y n + (1 )I + ] (R + L) δn = y n + C δ n Recall that y n is a constant vector and therefore unaffected by the shift operators. We linearize around synchronization where δ n = y n+1 + δ n+1 F ( y n ) + F ( y n ) δ n +... where F ( y n ) is the Jacobian. Since y n+1 = F ( y n ), we ay cancel the two leaving δn+1 F ( y n )C δ n Now as the ap δ n is iterated N ties the factor F ( y n )C becoes the product N j=1 F (y j )C j, and we wish to see if on average δ n+1 is growing or shrinking. We know how to diagonalize C, call the diagonalized atrix D. iterations we take the Nth root, i.e. the geoetric ean N F (y j )D j j=1 N D F (y j ) j=1 1/N 1/N So for N In the liit as N, the geoetric ean of the product of the derivatives just becoes the lyapunov nuber for the logistic ap, e λ. This gives the equation δn+1 e λ D δ n We know the eigenvalues of D and this gives us an equation for the singular values ( ( )) πk s k = e λ 1 cos where k =, 1,..., 1. For synchronization we need all the singular values except s to be less than one. ( ) πk e λ (1 ) + cos < 1 Which gives the following condition for synchronization ( ) ) πk (cos 1 + 1 < e λ The different odes correspond to frequencies of quasi-synchronization along the lattice. It is an observation that the total synchronization ode has never been blocked out, but another ode ay becoe ore proinent. 4

For the 3-lattice we get the condition 3 (1 e λ ) < < 3 (1 + e λ ) For aps larger than = 3 the expression can be refined by first letting k = 1 and then k = / if the lattice-size is even or k = ( ± 1)/ if it is odd. These are key odes to supress. For a general even size lattice we find for which 1 e λ 1 cos( π ) < < 1 + e λ 3 cos ( ) π λ ax = ln 1 + cos ( ) π The condition for odd lattice size just substitutes ± 1 for. An illustrative exaple is the case of the 4-lattice, where we find that and respectively. Solving for we get 1 < e λ 1 < e λ 1 e λ < < 1 + e λ 1 e λ < < 1 + e λ Taking the strictest conditions fro both equations we have 1 e λ < < 1 + e λ Fro this expression we ay solve for the axiu value of λ for which a lattice, 4, ay reliably synchronize: λ ax = ln 3. This does not apply to the 3-lattice which ay synchronize for all λ, however the range of decays exponentially with respect to increasing λ. If r is the length of the interval of then r = 4 3 e λ. Intuitively, it akes sense that if a ap is less chaotic, having a saller λ, then it should be easier to synchronize. Indeed for larger aps, synchronization ay only be stable for very sall λ. It is iportant to ake explicit what this condition actually says. All the condition can say is that if synchronization happens and satisfies these conditions then synchronization will be stable. However the data sees to relate very well to these bounds with the noted exception of the circle ap, which eans that there is still ore to be understood. There could be a fault in the progra or an unforseen phenoenon. 3 Data Matlab was used to odel a few lattices for a wide range of paraeters. The logistic ap and a torus ap x n+1 = ax n od 1 5

was used to test the synchronization conditions on the 4-lattice. In order to test for synchronization the standard deviation, σ, was taken of all the lattice points and then averaged over all iterations. Clearly if this σ avg is zero than the syste is fully synchronized. If we were interested in observing other synchronization odes, we can take the discrete fourier transfor over all the lattice ebers. Figure (1) shows a typical situation of a synchronizing lattice, in this case the lattice is a 4-lattice of logistic aps with logistic paraeter a = 3.9, corresponding to a lyapunov exponent of λ.516. Notice that in figure (1) there is an.4 Average STD vs. ε, n=4, a=3.9.35.3.5 σ avg..15.1.5.1..3.4.5.6.7.8.9 1. λ Figure 1: Average Standard Deviation vs. Coupling Paraeter for the circle lattice of logistic aps, n=4 a=3.9 interval for which the standard deviation drops to zero. This is the range of the synchronization paraeter predicted by the theory to give stable synchronization. Figure () shows an exaple of a lattice which was outside the theoretical bounds for synchronization, the 4-lattice of logistic aps with a = 4. In both cases shown there are regions in which the standard deviation fors a sooth curve. If the lattice were copletely out of synchronization there would be no correlation at all in this region of the plot. Therefore these probably represent regions of quasi-synchronization and other ore coplicated behavior such as the eergence of fixed points. If we consider 4-lattices of varying chaoticness we can test the bounds of our theory. Figure (3) shows the synchronizing values of the coupling paraeter versus the lyapunov exponent for the logistic and torus aps. It wasn t until we considered a faily of circle aps x n+1 = x + a sin (πx) 6

.45 Average STD vs. ε, n=6, a=4.4.35.3 σ avg.5..15.1.5.1..3.4.5.6.7.8.9 1. λ Figure : Average Standard Deviation vs. Coupling Paraeter for the circle lattice of logistic aps, n=6 a=4 od 1 that we found a counterexaple to the theoretical bounds, shown in figure (4). An interesting observation is that the there is a definite range of synchronizing paraeters which fit the theory and then there are outliers which don t. One ight be led to conclude that these ay be have an average standard deviation of zero but not be linearly stable and therefore the theory still stands, but in fact these were generated with initial conditions slightly perturbed fro synchronization so they are linearly stable points. 4 Conclusion Three different types of aps were looked at, two of which perfectly fit the bounds of synchronization. The third ap we looked at had an interval which fit the theory and outliers which didn t. Clearly there is ore to be understood here. 5 Future Plans For the next stage of the project we hope to answer soe of the questions which were raised in this report. Furtherore, we will look at the role that syetry plays in the synchronization of lattices. We have only looked at 7

1..9.8 ε sync vs λ Logistic Theoretical Bounds Circle Map Perturbed Logistic.7.6 ε.5.4.3..1..4.6.8 1 λ 1. 1.4 1.6 1.8 ax λ Figure 3: Synchronizing Coupling Paraeter Values vs. Lyapunov Exponent circular lattices under diffusive which have rotational syetry and this allowed us to diagonalize the jacobian to find explicit equations for the singular values. So it is natural to consider other cases where we can diagonalize the jacobian, and aybe be able to ake a general stateent about synchronization in such lattices. However, the ain focus will be to study the flow of inforation in general coupled ap lattices. 6 Notes Consider soe arbitrary state vector x, and decopose it into a vector on the diagonal, x x y =. x and one perpendicular to the diagonal, δ, where x is the ean of the entries of the state vector. If we consider our lattice ap, F, which acts on the state vector x giving another vector F ( x) we ust know if F ( x) F ( y) is also perpendicular to the diagonal. This is because the following analysis depends on the existence of a synchronized solution which is the orbit of y and in order for the state 8

1..9 ε sync vs λ Theoretical Bounds Circle Map.8.7.6 ε.5.4.3..1..4.6.8 1 1. 1.4 1.6 1.8 λ Figure 4: Synchronizing Coupling Paraeter Values vs. Lyapunov Exponent for the ap x n+1 = x + a sin (πx) od 1 with a =.16π solution to converge to the synchronized solution, δ n+1 = F ( x n ) F ( y n ) ust go to zero. If the singular values in every direction perpendicular to the diagonal are less than 1, and δ is perpendicular to the diagonal then it will go zero. To see that it is, consider F ( x) = F ( y + δ). Taylor expanding we find: which eans that F ( y + δ) = F ( y) + F ( y) δ +... F ( y + δ) F ( y) F ( y) δ And if this is perpendicular to the diagonal then the su of its eleents is zero, given that δ is already perpendicular. Observe F ( y) δ = f ( x)c δ (1 ) = f ( x) (1 ) (1 ) (1 ) δ δ 1 δ δ 3 = (1 )δ + δ 1 + δ 3 (1 )δ 1 + δ + δ (1 )δ + δ 1 + δ 3 (1 )δ 3 + δ + δ All factors cancel out and we are left with 3 k= δ k =, which eans that the new vector is perpendicular to the diagonal as required. With that in ind the preceding analysis applies. 9