Master Stability Functions for Synchronized Coupled Systems
Louis M.Pecora and Thomas L.Carroll
Code6343,Naval Rearch Laboratory,Washington,D.C.20375
(Received7July1997)雅思口语救生圈
We show that many coupled oscillator array configurations considered in the literature can be put into a simple form so that determining the stability of the synchronous state can be done by a master stability function,which can be tailored to one’s choice of stability requirement.This solves,once and for all,the problem of synchronous stability for any linear coupling of that oscillator. [S0031-9007(98)05387-3]
PACS numbers:05.45.+b,84.30.Ng
A particularly interesting form of dynamical behavior occurs in networks of coupled systems or oscillators when all of the subsystems behave in the same fashion; that is,they all do the same thing at the same time. Such behavior of a network simulates a continuous system that has a uniform movement,models neurons that synchronize,and coupled synchronized lars and electronic circuit systems.A central dynamical question is:When is such synchronous behavior stable,especially in regar
d to coupling strengths in the network?Interest in this question has been high over the last veral years in both chaotic[1–11]as well as limit cycle systems[12–14].Such studies typically assumed a particular form of coupling in the network and then analyzed the features of, stability of,and bifurcations from the synchronized state. We have made progress toward developing a general approach to the synchronization of identical dynamical systems,building on the ideas of scaling in our previous work[15].The conquence of this is a master stability equation,which allows us to calculate the stability(as determined from a particular choice of stability measure, like Lyapunov or Floquet exponents)once and for all for a particular choice of ,Rössler,Lorenz,etc.) and a particular choice of component ,x,y, etc.).Then,we can generate the stability diagrams for any other linear coupling scheme involving that system and component.
Any one system can have a wide variety of desynchro-nizing bifurcations.Using the master stability diagram, we can predict a diversity of spatial-mode instabilities including bursting or bubbling patterns[8].The master stability diagram makes it obvious why particular cou-pling schemes may have an upper limit on the number of oscillators that can be coupled while still retaining a stable,synchronous state.
We assume the following:(1)The coupled oscillators (nodes)are all identical,(2)the same function of t
he components from each oscillator is ud to couple to other oscillators,(3)the synchronization manifold is an invariant manifold,and(4)the nodes are coupled in an arbitrary fashion which is well approximated near the synchronous state by a linear operator.Numbers(1) and(3)guarantee the existence of a synchronization hyperplane in the pha space and number(2)makes the stability diagram specific to our choice of oscillators and the components.Number(4)is the choice of many studies of coupled systems since it is often a good approximation and can be considered prototypical.
In determining the stability of the synchronous state, various criteria are possible.The weakest is that the maximum Lyapunov exponent or Floquet exponent be negative.This is a universal stability standard,but it does not guarantee that there are not unstable invariant ts in the synchronous state[8]or areas on the attractor that are locally unstable[1,16,17],both of which can cau attractor bubbling and bursting of the system away from synchronization when there is noi or parameter mismatch.The theory we develop below will apply to almost any criterion that depends on the variational equation of the system.Each stability criterion will lead to its own master stability function.For that reason,we develop the theory in the context of Lyapunov exponents as a stability criterion and show in the conclusions how the other criteria can be ud.
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Let there be N nodes(oscillators).Let x i be the m-dimensional vector of dynamical variables of the i th node.Let the isolated(uncoupled)dynamics beᠨx iF͑x i͒for each node.H:R m!R m is an arbitrary
function of each node’s variables that is ud in the coupling.Thus,the dynamics of the i th node areᠨx iF͑x i͒1s
P
j
G ij H͑x j͒,where s is a coupling strength. The sum
P
j
G ij0,so that assumption(3)above holds. The N21constraints x1x2···x N define the synchronization manifold.
Let x͑x1,x2,...,x N͒,F͑x͓͒F͑x1͒,F͑x2͒,..., F͑x N͔͒,H͑x͓͒H͑x1͒,H͑x2͒,...,H͑x N͔͒,and G be the matrix of coupling coefficients͕G ij͖,then
ᠨxF͑x͒1s G≠H͑x͒,(1) where≠is the direct product.Note,we could start with a more general,nonlinear form in the coupling term and then assume that evaluation of the Jacobian of(1)leads to a constant matrix on the synchronization manifold.Either way,the analysis from here on follows the same pattern and we prent(1)for its greater clarity.
Many coupling schemes are covered by Eq.(1).For example,if we u Lorenz systems for our nodes,m3.
0031-9007͞98͞80(10)͞2109(4)$15.00©1998The American Physical Society2109
If the coupling is through the Lorenz “x ”component,then the function H is just the matrix
E 0B @1000000001C A .
Our choice of G will provide the connectivity of nodes.Equation (2)shows G for nearest-neighbor diffusive cou-pling and star coupling [18].Similarly,all-to-all coupling has all 1’s for G ij ͑i fij ͒and 2N 11for G ii .The boundary conditions are all cyclic in Eqs.(2),but many others are possible.The majority of coupling schemes treated in the dynamics literature can be put into the form of Eq.(1)by choosing the right G matrix.
G 10B B B B B B @0...............10 (122)
1C C C C C C A
,G 20B B B B B B @0...............10 (0211)
C C C C C C A
.(2)We get the variational equation of Eq.(1)by letting
j i
be the variations on the i th node and the collection of variations is j ͑j 1,j 2,...,j N ͒.Then,
ᠨj
͓1N ≠D F 1s G ≠D H ͔j .(3)When H is just a matrix E ,D H E .Equation (3)is ud to calculate Floquet or Lyapunov exponents.We really want to consider only variations j which are transver to the synchronization manifold.We want tho variations to damp out.We next show how to parate out tho variations and simplify the problem.The first term in Eq.(3)is block diagonal with m 3m blocks.The cond term can be treated by diagonalizing G .The transformation which does this does not affect the first term since it acts only on the matrix 1N .This leaves us with a block diagonalized variational equation with each block having the form
ᠨj k ͓D F 1sg k D H ͔j k ,(4)where g k is an eigenvalue of G ,k 0,1,2,...,N 21.
For k 0,we have the variational equation for the syn-chronization manifold ͑g 00͒,so we have succeeded in parating that from the other,transver directions.All other k ’s correspond to transver eigenvectors.We can think of the as transver modes and we will refer to them as such.
The Jacobian functions D F and D H are the same for each block,since they are evaluated on the synchronized state.Thus,for each k ,the form of each block [Eq.(4)]is the same with only the scalar multiplier sg k differing for each.This leads us to the following formulation of the master stability equation and the associated master
stability function:We calculate the maximum Floquet or Lyapunov exponents l max for the generic variational equation
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ᠨz ͓D F 1͑a 1i b ͒D H ͔z (5)as a function of a and b .This yields the stabilitytaper
function l max as a surface over the complex plane [e Fig.1,int (a)].Complex numbers are ud since G may have complex eigenvalues.Then,given a coupling strength s ,we locate the point sg k in the complex plane.The sign of l max at that point will reveal the stability of that eigenmode—hence,we have a master stability function.If all of the eigenmodes are stable,then the synchronous state is stable at that coupling strength.
To illustrate,we cho chaotic Rössler systems [19]͑a b 0.2,c 7.0͒as the nodes and coupled them through the x component;thus,H E and E is as above.Figure 1shows a contour plot of the master stability function for this oscillator.We e that there is a region of stability defined by a roughly micircular shape.The plot is symmetric in the imaginary directions about the real axis.At a b 0,l max .0since this is just the ca of isolated,chaotic Rössler systems.As a increas (with b 0),l max cross a threshold and becomes negative.Further increa in a reveals another threshold as l
max
FIG.1.Master stability function for x -coupled Rössler oscil-lators.Lightly dashed lines show contours of negative ex-ponents and solid lines show contours of positive exponents.Circles show the eigenvalues for the diffusive coupling ex-ample.Stars show the eigenvalues for a star-coupled ex-ample.The bold,dotted miellip is the line of eigenvalues of an asymmetrically coupled Rössler system for particular cou-pling strengths.LWB,IWB,and SWB label long-wavelength,intermediate-wavelength,and short-wavelength bifurcations,re-spectively,that occur with diffusive-coupling schemes when eigenvalues cross the stability threshold.For the star configu-ration DHB labels a drum-head-mode bifurcation.Int (a)shows a typical surface for the master stability function.Int (b)shows the relation between the hub and spokes oscillators when a DHB takes place.
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cross over to become positive again.This implies that if the coupling is too strong the synchronous state will not be stable.If a is t to be in the stable range and b is incread,then l max can also cross a threshold and become positive,implying that a large imaginary coupling can destabilize the system.Imaginary eigenvalues ari from antisymmetric couplings(e below).
Diffusive coupling in a circular array[using the first G matrix in Eq.(2)]gives eigenvalues of g k4sin2͑
p k͞N͒,each twice degenerate and the eigenmodes are discrete sine and cosine functions of the node indices i[6,20].For a particular coupling strength s,we show the points s g k in Fig.1for an array of10Rösslers.The array has a stable synchronous state.As the coupling s increas from0,thefirst mode to become stable is the shortest spatial-frequency mode;the last mode to become stable is the longest spatial-frequency mode.Thus,in a stable,synchronous state,decreasing s will cau a desynchronization with the long-wavelength mode going unstablefirst,a long-wavelength bifurcation(LWB). Increasing s caus the shortest wavelength to become unstable,a short-wavelength bifurcation(SWB)[9,15]. Note,as more oscillators are added to the array,more transver modes are created and the distance(along the real axis a)between the longest and shortest wavelength modes increas.Eventually,the system will reach a point at which we will increa s to stabilize the long-wavelength mode only to have the short-wavelength mode become unstable at the same time.There will be an upper limit on the size of a stable,synchronous array of chaotic Rössler oscillators[9,15].Such a size limit will always exist in arrays of chaotic oscillators with such limited stable regimes.Such a size limit will not exist if the oscillators are limit cycles,but the stable range of s will be compresd down toward the origin as more oscillators are added to the array.
In all-to-all coupling schemes the transver eigenval-ues are all the same,g k2s N.The all-to-all sc
heme can support synchronous chaos for the Rössler oscillator example for the right s.Unlike diffusive coupling,all modes become unstable when the threshold is crosd. Star coupling[the cond matrix in Eq.(2)—e int (b)of Fig.1]results in two eigenvalues,g k2s and g k2s N.This yields two points on the master stability surface(e Fig.1for ven oscillators).If we decrea s,we get a desynchronizing bifurcation in which sinusoidal modes that are on the spokes of the star become unstable and grow.If we increa s,we get an interesting desynchronization bifurcation where the nodes on the spokes remain synchronous,but the hub node begins to develop motions of opposite sign to the former.We call this a drum-head bifurcation(e the int in Fig.1). There is also a size limit for the star configuration.For the x-coupled Rössler example,the maximum number of synchronized oscillators is45.
We now consider a more complex coupling scheme with asymmetric nearest-neighbor coupling.We also add all-to-all coupling.The x coupling term in the Rössler example becomes͑c s2c u͒x i111͑c s1c u͒x i212 2c s x i1c a
P
j
͑x j2x i͒.This is the sum of G1[in Eq.(2)],G2[Eq.(2)],and G3,an antisymmetric matrix with21on the row above the diagonal,11on the row be-low the diagonal,and zeros elwhere.With each matrix is associated a coupling strength c s,c a,and c u,respectively. The matrices are simultaneously diagonalizable using sinusoidal modes.The eigenvalues are complex(due to the antisymmetric part),g k22c s͓12cos͑2p k͞N͔͒1 2c u i sin͑2p k͞N͒2c a N,and they must lie on an ellip centered at22c s2c a N(e Fig.1).We can always adjust the coupling strengths so all transver eigenvalues lie in the stable region.Increasing c s will elongate the ellip along the real axis.Depending on where the ellip is centered,this can cau either a LWB or a SWB. Increasing c u can cau an intermediate wavelength bifurcation(IWB)for the Rössler situation,since the ellip can elongate in the imaginary direction causing the intermediate wavelengths to become unstable(IWB).
We experimentally tested the dependence of bifurcation type(LWB,IWB,or SWB)as a function of couplings c s and c u using a t of eight coupled Rössler-like circuits[6]which have individual attractors with the same topology as the Rössler system in the chaotic regime. We initially t c s0.2,c u0,and c a0.1so that the Rössler circuits were in the synchronous state.We controlled the coupling constants c s and c u using a digital-to-analog convertor in a computer.The circuits were started in the synchronous state and then the coupling was instantaneously ret to new values of c
s and c u.At the same time,we recorded the x signals from all eight oscillators simultaneously with a12-bit eight-channel digitizer card.We arbitrarily cho the threshold of the sum of modes1–4exceeding5%of the synchronous mode to determine when the oscillators were not in sync. More experimental information will be given elwhere. After we switched the coupling constants c s and c u from the synchronous state to a nonsynchronous state, wefit the transient portion of each mode-amplitude time ries to an exponential function tofind a growth rate l for each mode.We recorded the mode with the largest l as being the most unstable mode.Figure2(a)shows the experimental results.In Fig.2(b),we plot the least stable eigenmode found from the master stability function. Theory and experiment compare well.The synchronous region has a similar shape,including the sharp peak just before the SWB region.Other bifurcation regions agree reasonably well,including the small mode3region near the peak of the sync region.
We noted that other stability criteria are possible.Each will produce its own master function over the complex coupling plane.Among them are the following three: (1)Calculate the maximum Lyapunov exponent or Floquet multiplier for the least stable invariant t[8,17],e.g., an unstable periodic orbit in a chaotic attractor,(2)cal-culate the maximum(supremum)of the real part of the
2111
FIG.2.(a)Plot of experimental results for asymmetri-cally coupled Rössler-like circuits showing the class of desynchronizing bifurcations that occur when the sym-metric͑c s͒or antisymmetric͑c u͒part of the coupling is changed from a synchronous state to a state in which the theory predicts that one of the eigenmodes should be un-stable.The labeling scheme issynchronous mode,long wavelength(mode1)n nintermediate wavelength (mode2),white spaceintermediate wavelength(mode3), and3short wavelength(mode4).(b)Similar plot of theoretical prediction of which modes are least stable.
eigenvalues of the(instantaneous)Jacobian(including the coupling terms)at all points or some reprentative t of points on the attractor[16];e.g.,when negative,this function guarantees ultimate transver-direction contrac-tion everywhere on the attractor,and(3)calculate the maximum eigenvalue of the(instantaneous)symmetrized Jacobian(including the coupling terms)at all points or some reprentative t of points on the attractor[1];e.g., this guarantees monotone damping of transver pertur-bations[21].Using the same analysis as above,criteria (1)and(2)come down to Eq.(5),although the evalu-ation of the stability function will be on the special, unstable invariant t or of the real part of the eigen-value of the right-hand-side linear operator.Criterion(3) can also be analyzed in the same way provided there are some common restrictions.The,again,lead to a block
diagonalization of the variational equation in the same way as before with thefinal stability function being the maximum eigenvalue on the attractor of the linear op-eratorᠨz͓D F1D F T1͑a1i b͒D H͔z[22].Many other stability criteria,such as the recently introduced Brown-Rulkov criterion[23,24]will also produce a mas-ter stability function.Which one to u depends on one’s requirements.
The master stability function allows one to quickly establish whether any linear coupling arrangement will produce stable synchronous dynamics.In addition,it reveals which desynchronization bifurcation mode will occur when the coupling scheme or strength changes. Attractor bubbling or bursting behavior[8]shows up mainly as bursts of the particular mode or modes that are clost to instability.Using Eq.(5)for large a or b,we can explain why the synchronous state is unstable for certain systems in the asymptotic limit of large real or imaginary coupling.Finally,the coupling need only be locally linear for there to be a master stability function;
<,the form of the variational equation is similar to Eq.(3)near the synchronization manifold.The latter is a more common scenario.The issues in this last paragraph will be covered in more detail elwhere.
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