Stability of synchronous chaos and on-off intermittency in coupled map lattices
Mingzhou Ding *and Weiming Yang ?
Program in Complex Systems and Brain Sciences,Center for Complex Systems and Department of Mathematical Sciences,
Florida Atlantic University,Boca Raton,Florida 33431
?Received 25April 1997?
In this paper we consider the stability of synchronous chaos in lattices of coupled N -dimensional maps.For global coupling,we derive explicit conditions for computing the parameter values at which the synchronous chaotic attractor becomes unstable and bifurcates into asynchronous chaos.In particular,we show that after the bifurcation one generally observes on-off intermittency,a process in which the entire system evolves nearly synchronously ?but chaotically ?for long periods of time,which are interrupted by brief bursts away from synchrony.For nearest-neighbor coupled systems,however,we show that the stability of the synchronous chaotic state is a function of the system size.In particular,for large systems,we will not be able to observe synchronous chaos.We derive a condition relating the local map’s largest Lyapunov exponent to the maximal system size under which one can still observe synchronous chaos and on-off intermittency.Other issues related to the characterization of on-off intermittent signals are also discussed.?S1063-651X ?97?07210-3?PACS number ?s ?:05.40.?j,05.45.?b
I.INTRODUCTION
Consider a coupled map lattice ?1?.Assume that there is a range of parameter values for which this system exhibits a unique attractor of synchronous chaos where every element evolves chaotically and is in synchrony with every other el-ement.Suppose that,as the parameter is varied past a certain bifurcation point,this synchronous chaos loses stability,and is replaced by an asynchronous chaotic state.It can be shown that,under rather general conditions,immediately after the bifurcation,the dynamics exhibits on-off intermittency ?3–7?in which long episodes of nearly synchronous evolution ?laminar phase ?are interrupted by a certain element or ele-ments in the system bursting away from the synchronous state.The bursts become more and more frequent as the pa-rameter is moved further and further away from the bifurca-tion point.Eventually,the system reaches fully developed heterogeneous chaos where no clearly identi?able episodes of synchronous chaos are seen.According to the terminology of a recent paper ?2?,this bifurcation is a spatiotemporal example of a nonhysteretic blowout bifurcation.Blowout bi-furcations occur in systems with symmetry,which in our case is the spatial translational invariance due to identical elements used at each space site.It is nonhysteretic because there are no other attractors in the phase space coexisting with the synchronous chaos attractor.Hysteretic blowout bi-furcations occur if there are simultaneously more than one attractor in the system.In this case one may observe riddled basins on the side of the parameter axis where the synchro-nous state is still stable ?8?.
Assume that the local map is N dimensional and is cha-otic in the absence of coupling.In Sec.II we carry out a stability analysis of synchronous chaos for globally coupled systems of such maps.We show that,when the synchronous chaos becomes unstable,the system exhibits on-off intermit-tency.Section III presents numerical results on the charac-terization of on-off intermittent signals using quantities like laminar phase distribution plots and power spectra.In Sec.VI we consider the stability of synchronous chaos in coupled map lattices where the coupling is nearest neighbor.We point out that,in such systems,we can only expect to see stable synchronous chaos and the accompanying on-off in-termittent behavior if the number of coupled maps is small.Section V concludes this paper.
II.STABILITY OF SYNCHRONOUS CHAOS
IN GLOBALLY COUPLED MAPS
AND ONSET OF ON-OFF INTERMITTENCY
Coupled map lattices,as discrete analogs to coupled os-cillators and partial differential equations,have in recent years become the model of choice for developing intuitions and concepts in the study of spatiotemporal dynamical sys-tems ?1,9?.Assuming global ?mean ?eld ?coupling we ex-press our model as
x n ?1?i ???1???f …x n ?i ?…?
?
L
?
j ?1
L
f …x n ?j ?…,?1?
where x is an N -dimensional column vector,n denotes the time step,i ,j are labels of lattice sites,f ?x ?is an N -dimensional nonlinear mapping function,?is the coupling strength satisfying 0р??1,and L is the total number of coupled elements.The local N -dimensional map
x n ?1?f ?x n ?
?2?
is assumed to be chaotic.From Eqs.?1?and ?2?one can see that x n (i )?x n (j )?x n ,i ,j ?1,2,...,L ,is a solution to Eq.?1?,indicating that fully synchronized chaotic states are pos-sible.For this synchronous chaotic state to be observable the corresponding N -dimensional manifold must be attracting or stable.Below we derive the criterion for the stability of this
*Electronic address:ding@https://www.doczj.com/doc/9b8686631.html,
?
Electronic address:yang@https://www.doczj.com/doc/9b8686631.html,
PHYSICAL REVIEW E OCTOBER 1997
VOLUME 56,NUMBER 456
1063-651X/97/56?4?/4009?8?/$10.00
4009
?1997The American Physical Society
synchronization manifold.Stability analysis for synchro-nized periodic orbits in coupled map lattices can be found in ?10?.
A.The case of N ?1
We begin by considering the simplest case where the lo-cal map is one dimensional.Rewrite Eq.?1?and Eq.?2?as
x n ?1?i ???1???f …x n ?i ?…?
?L
?
j ?1
L
f …x n ?j ?…,?3?
and
x n ?1?f ?x n ?.
?4?
The synchronous chaotic attractor with x n (i )?x n (j )?x n lies along the one-dimensional diagonal in the L -dimen-sional phase space spanned by the vector z
??x (1),x (2),...,x (L )?T .In other words,the synchroniza-tion manifold is the diagonal which is invariant under the dynamics.The stability of this invariant manifold can be assessed by computing its Lyapunov exponent spectrum.Differentiating Eq.?3?and evaluating the derivatives along the synchronization trajectory leads to
?x n ?1?i ???1???f ??x n ??x n ?i ??
?
L
?
j ?1
L
f ??x n ??x n ?j ?.
?5?
This means that the tangent vector
?z n ???x n (1),?x n (2),...,?x n (L )?T
evolves,along the chaotic trajectory x n (1)?x n (2)?????x n (L )?x n with x n ?1?f (x n ),according to
?z n ?1?J n ?z n ,
?6?
where
J n ?f ??x n ?
?
1??L ?1??/L ?/L
?/L ????/L ?/L
1??L ?1??/L
?/L
????/L
?
?
?
???/L ?/L ?/L
???
1??L ?1??/L
?
L ?L
?f ??x n ?J .
?7?
Note that the constant matrix J is a cyclic matrix and it commutes with the following shift matrix S :
S ?
?0
1
???0
1
???
0????
?
???
1
1
???
?
L ?L
,?8?
namely,
JS ?SJ .
This implies that these two matrices share the same set of eigenvectors.The eigenvectors for S are known to be
E m ??exp ?
2?i m ?1L ?,exp ?4?i m ?1L ?
,...,exp ?
2L ?i
m ?1
L
??
T
,?9?
where m ?1,...,L and T denotes matrix transpose.From
these eigenvectors we ?nd that J has an eigenvalue of one and an (L ?1)-fold degenerate eigenvalue of (1??).
For m ?1we get E 1?(1,1,...,1)T ?v 1as a real eigenvec-tor of the constant matrix J in Eq.?7?pointing along the diagonal direction.The corresponding eigenvalue is https://www.doczj.com/doc/9b8686631.html,-ing v 1and the de?nition of Lyapunov exponents ?11?,we obtain
?1?lim n →?1
n ln
???m ?1
n J m ??v 1/?v 1?
??lim n →?1n
ln ??m ?1
n
f ??x m ??
.
It is not surprising that the value of ?1,describing the
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MINGZHOU DING AND WEIMING YANG
stretching dynamics within the synchronization manifold,is the same as that of the one-dimensional map.In particular, from the chaos assumption earlier,?1?0.
Note that the constant matrix J is a symmetric matrix. This means that the eigenvectors of the eigenvalue(1??) span the(L?1)-dimensional subspace orthogonal to the di-agonal?synchronization manifold??12?.Choosing an arbi-trary set of mutually orthonormal vectors from this subspace, v2,v3,...,v L,we obtain the remaining L?1Lyapunov exponents,
?2??3??????L??1?ln?1???.
We refer to the Lyapunov exponents?2??3??????L as transversal Lyapunov exponents since they characterize the behavior of in?nitesimal vectors transversal to the synchro-nization manifold.In other words they determine the linear stability of synchronous chaos.Note that the Lyapunov ex-ponents found here do not depend on the lattice size L.This is in contrast to the case where the coupling is nearest neigh-bor?see Sec.IV?.
When?is relatively large such that0??1??ln(1??), all the transversal Lyapunov exponents are negative,and the synchronous chaos state is stable and is the only observed system behavior.This result makes intuitive sense since strongly coupled systems tend to behave in unison.
As the coupling gets weaker,especially when?1??ln(1??),all the transversal Lyapunov exponents become positive and the system undergoes a blowout bifurcation, through which the asynchronous state is born.For a given local one-dimensional map,the critical value of coupling is
?c?1?e??1.?10?From this formula it is clear that,the more chaotic the local map,the larger the value of?c.This is again an intuitively reasonable result.
For?slightly less than?c the synchronous chaos is no longer stable.However,from the assumption that the syn-
chronization manifold is the unique attractor for???c,we know that points far away from the synchronization manifold are still attracted to it immediately after?becomes smaller than?c.This combination gives rise to the situation that, after the synchronous chaos becomes unstable,the variable that describes the distance between the system state and the synchronization manifold exhibits on-off intermittency.Be-low we illustrate this point with a numerical example.
Let f(x)?1?ax2be the logistic map.For a?1.9we have?1?0.5490.Consider a globally coupled system of L ?100such maps.From Eq.?10?we get?c?0.4225.Figure
1?a?shows the dynamics for??0.43??c.The variable plot-ted at the lattice site i is x n(i)?xˉn where the overline indi-cates the spatial average of the variable x.The absolute value of this quantity measures the distance between the sys-tem state and the synchronization manifold.As expected,for the40units displayed in the?gure,we observe synchronized chaos in which the plotted quantity is uniformly zero.For ??0.41,which is slightly below?c,clearly identi?ed epi-sodes of synchronized behavior are seen in Fig.1?b?which is interspersed with bursts away from the synchronization at-tractor,suggesting the occurrence of on-off intermittency. This intermittent dynamics is eventually replaced by fully developed asynchronous chaos as the parameter??0.385is far removed from the critical value,as shown in Fig.1?c?.
B.The case of N>1
Now let us assume that the local map f in Eq.?1?is N?1dimensional.This local N-dimensional map
x n?1?f?x n??11?
admits N Lyapunov exponents denoted by h1уh2у???уh N.Here x is an N-dimensional column vector.Let
A n?D x f?x n??12?
be the Jacobian matrix of the local map.From the theory of Lyapunov exponents?11?,for a typical initial condition x1in the proper basin of attraction,we can?nd a set of N unit vectors e1,e2,...,e N such that
h i?lim
n→?
1
n
ln?
??
m?1
n
A m??e i?.?13
?FIG. 1.The time series,x(i)?xˉ,from40of100globally coupled logistic maps.The overbar indicates spatial average.?a???0.43??c?0.4225,?b???0.41,which is slightly less than?c, and?c???0.385,which is further away from the critical value?c.
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STABILITY OF SYNCHRONOUS CHAOS AND ON-OFF...
Note that,if we pick an N -dimensional unit vector e at ran-dom,then a calculation similar to that in the above formula will always yield the largest Lyapunov exponent h 1.
Consider the coupled system Eq.?1?.The phase space now is L ?N dimensional.Let us form the phase space vec-tor as
z ??x T ?1?,x T ?2?,...,x T ?L ??T .
In other words,z is an (L ?N )-dimensional column vector.
The synchronization manifold,de?ned by x n (i )?x n (j ),i ,j ?1,2,...,L ,is N dimensional and the dynamics in the mani-fold evolves according to Eq.?11?.
Consider an in?nitesimal deviation from this manifold ?z n .Then,from Eq.?1?and along a synchronization trajec-tory,we have
?z n ?1?J n ?z n ,
?14?
where
J n ?
?
?1??L ?1??/L ?A n
??/L ?A n
??/L ?A n ?????/L ?A n ??/L ?A n
?1??L ?1??/L ?A n
??/L ?A n
?????/L ?A n
?????
??/L ?A n
??/L ?A n
??/L ?A n
???
?1??L ?1??/L ?A n
?
?15?
is an (L ?N )?(L ?N )matrix.
The spectrum of all the Lyapunov exponents with respect to the synchronization solution can be evaluated in a fashion similar to that of one dimensional local maps.Considering vectors of the form
v i parallel ??e i T ,e i T ,...,e i T ?T
,where e i is the same vector as that used in Eq.?13?,we
obtain the Lyapunov exponents describing the dynamics within the synchronization manifold through
?i
parallel ?lim n →?1
n
ln ???m ?1n J m ?
?v i
parallel ?
?v i
parallel ??
?16??lim n →?1
n
ln ???m ?1
n A m ??e i ?
,
?17?
which,as expected,is h i .
To describe the dynamics transversal to the synchroniza-tion manifold let us form vectors as
v i vertical ??a 1e i T ,a 2e i T ,...,a L e i T ?T
,
where a i ’s are chosen such that ?a 1,a 2,...,a L ?is an
L -dimensional unit vector orthogonal to the vector ?1,1,...,1?.Note that such vectors are eigenvectors of the constant ma-trix J in Eq.?7?with eigenvalue (1??).A calculation simi-lar to that in Eq.?17?gives the distinct set of all transversal Lyapunov exponents,
?i
vertical ?h i ?ln ?1???.So the largest transversal Lyapunov exponent is
?1
vertical ?h 1?ln ?1???.From this we calculate the critical value of coupling ?c to be
?c ?1?e ?h 1.
?18?
Note the similarity between Eq.?10?and Eq.?18?.As a
result we conclude that,for ???c ,the synchronous chaotic state is no longer stable and we observe the onset of on-off intermittency.
We now illustrate the above theoretical criterion for the bifurcation of synchronous chaos with an example where the
local map is the (N ?2)-dimensional He
′non map,x n ?1?y n ?1?ax n 2,
?19?y n ?1?bx n .
?20?
For a ?1.4and b ?0.3the largest Lyapunov exponent is cal-culated to be h 1?0.4207.Consider a globally coupled sys-tem of L ?100such maps.From Eq.?18?we get ?c ?0.3434.The variable of interest here is the distance be-tween the system state and the synchronization manifold measured
by
FIG.2.The value of d ?(1/L )?i ?1L ?x (i )?x ˉ?as a function of
time n from L ?100globally coupled He
′non maps.Here ??0.34is slightly less than ?c ?0.3434.
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MINGZHOU DING AND WEIMING YANG
d n?1
L
?
i?1
L
?x n?i??xˉn?.
Here the overbar denotes the spatial average of the variable inside.For???c,the value of d n is uniformly zero in the long run for any initial condition,indicating the presence of
synchronized chaotic attractor.Figure2shows the on-off intermittent time series for??0.338which is slightly less than?c.
III.CHARACTERISTICS
OF ON-OFF INTERMITTENT TIME SERIES The characterization of on-off intermittent time series like
the one shown in Fig.2has been extensively studied in the
past?3–7?.We will not duplicate these analyses here.Instead
our goal is to point out the mathematical origin that enables
the past analyses and results to be applicable to the present
spatiotemporal on-off intermittency problem.
A.Theoretical considerations
The on-off intermittent time series,for parameter values
close to the critical point,spends most of the time in the off
state,suggesting that the system state is near the synchroni-
zation manifold.The time series in this case show universal
properties.
To see the reason,consider the case where the local map is one dimensional(L?1).Let d n denote the absolute value of the projection of?z n of Eq.?6?onto any unit vector v that is perpendicular to the diagonal,d n??v T z n?.The value of d n can be viewed as measuring the distance between the trajec-
tory and the synchronization manifold and evolves according
to
d n?1???1???f??x n??d n??n d n,?21?where?n??(1??)f?(x n)?.
Clearly,d?0is a?xed point for Eq.?21?,re?ecting the
fact that the synchronization manifold is invariant.Accord-ing to?6,7?this d?0?xed point is stable if?ln?z n??is nega-tive where??denotes temporal average.This is equivalent to saying that?1?ln(1??)?0,the same stability condition de-rived earlier using Lyapunov exponents.As past work has shown,it is the linear equation with parametric driving in Eq.?21?that underlies the observed universal characters of the on-off intermittent time series.This is why we should expect the same characteristics for on-off intermittent time series found in earlier works to be applicable to the present problem.
The case where the local map is N?1dimensional is not
as simple.To see what to expect here,we again examine the projection of?z n in Eq.?14?onto a unit vector of the form
v??a1e T,a2e T,...,a L e T?T,
where e is an N-dimensional unit vector picked at random and?a1,a2,...,a L?T is an L-dimensional unit vector or-thogonal to the diagonal.This vector v is orthogonal to the N-dimensional synchronization manifold.Letting d n denote the absolute value of the projection,from Eqs.?14?and?15?, we get
d n?1???1???y n?d?n??n d?n,?22?wher
e d?n?v?T?z n,v???a1e?T,a2e?T,...,a L e?T?T,and y n e?T ?e T A n.Note that e?is still a unit vector and y n is a multi-plicative scalar that describes the amount o
f stretchin
g due to the application of A n.Althoug
h d n?1and d?n represent pro-jections of the tangent vectors?z n?1and?z n onto two dif-ferent directions,both directions are orthogonal to the syn-chronization manifold.Thus we can still think of them as measures of the distance between the trajectory point and the synchronization manifold.In this sense Eq.?22?and Eq.?21?play a similar role and we may expect the on-off intermittent time series from coupled(N?1)-dimensional maps to share the same universal properties as that from coupled one-dimensional maps.
B.Numerical results
We illustrate the properties of on-off intermittent time se-ries with three quantities:laminar phase distributions,power spectra,and mean bursting amplitude as a function of the coupling strength?.
Suppose d is the variable plotted against time.Let?de-note the threshold value of d such that for d??the signal is considered on and for d??the signal is considered off.The length of the laminar phase,denoted by T,is de?ned as the length of the off state.In practice one should choose?in such a way that when d??the linear approximation in Eq.?21?or Eq.?22?is valid.
For a typical chaotic local map,analysis in?7,13??see also?6??,which is based on a model like Eq.?21?and a random walk analogy,shows that the distribution of the laminar phase T is in the form
P?T??T?3/2e?T/T s,?23?where
T s???c????2?24?gives the crossover point from a power law behavior with an exponent?3/2to an exponential behavior.
In Fig.3?a?we plot the numerically calculated histogram for the length of the laminar phase for100globally coupled logistic maps.The on-off intermittent time series used here is constructed in the same way as that shown in Fig.2.We choose??0.01and collect1000000distinct laminar phases for statistics.Two values of?are considered:?1?0.415?plus?and?2?0.4075?triangle?.Recall that?c?0.4225in this case.The straight line in the?gure has a slope of
?3/2and is plotted to guide the eye only.From the?gure we can see that the numerical results conform to the theoretical prediction in Eq.?23?.In particular,the crossover from power law to exponential occurs earlier as?is moved further away from?c.The quantitative prediction concerning this behavior is contained in Eq.?24?and is con?rmed for the coupled logistic maps in Fig.4where T s is obtained and plotted for a number of?values.
Figure3?b?shows the numerically calculated histogram for the on-off intermittent time series in Fig.2for100glo-bally coupled He′non maps.We use??0.01and the same set of parameters as that used for Fig.2.Again a straight line of
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STABILITY OF SYNCHRONOUS CHAOS AND ON-OFF...
slope ?3/2is drawn in the ?gure to guide the eye.Clearly,the same power law and exponential crossover behavior is observed here for coupled (N ?1)-dimensional maps.
Figure 5shows the power spectrum for the globally coupled logistic maps with a ?1.9and ??0.41.The two straight lines in the ?gure have slopes of ?1/2and ?2,respectively.It is predicted in ?4?that one should observe three distinct regions of scaling behavior in the power spec-trum of on-off intermittent time series.For small frequencies the spectral density should be a constant.For intermediate frequencies the spectral density S (f )scales with frequency f as
S ?f ???1/f ?1/2
.
For large frequencies S (f )should scale with f as
S ?f ???1/f ?2.
In Fig.5we see the predicted behavior for the intermediate and large frequency regions clearly.However,the predicted low frequency behavior is not observed.This could be due to the prohibitively long time series required for the appearance of such behavior.
Our last numerical result is shown in Fig.6.Here we consider the globally coupled logistic maps with a ?1.9and plot the mean bursting amplitude as a function of the param-eter ?c ??.Theory in ?5?predicts a linear relationship be-tween the two quantities.This is clearly the case from the
?gure.
FIG.3.?a ?Log-log plot of the histogram for the laminar phase interval distribution for the on-off intermittent time series from 100coupled logistic maps.The time series used here is constructed in the same way as that used in Fig.2.?b ?Log-log plot of the histo-gram for the laminar phase interval distribution for the on-off time series shown in Fig.
2.
FIG.4.Log-log plot of the crossover time T s ?see Eqs.?23?and ?24??as a function of the parameter ?c ??.
FIG.5.Log-log plot of the power spectrum for on-off intermit-tent time series from 100globally coupled logistic maps.
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MINGZHOU DING AND WEIMING YANG
IV.NEAREST-NEIGHBOR COUPLING
AND THE STABILITY OF SYNCHRONOUS CHAOS
Consider the following coupled map lattice model where the coupling is nearest neighbor:
x n ?1?i ???1???f …x n ?i ?…??
2
?f …x n ?i ?1?…?f …x n ?i ?1?…?.
?25?
Here we assume one-dimensional local maps and impose the periodic boundary condition x n (0)?x n (L ).Similar discus-sions can be carried out for the case of (N ?1)-dimensional local maps.
Like the case of global coupling,from Eq.?25?,we see that the synchronous chaotic state is invariant under the dy-namics.We proceed to compute the Lyapunov spectrum for the synchronization attractor.
The Jacobian matrix for Eq.?25?at time n calculated along the synchronous chaotic trajectory x n (i )?x n (j )?x n with x n ?1?f (x n )is the following L ?L matrix:
K n ?f ??x n ?
?
?
?1???
?/2
0???0?/2
?/2
?1???
?/2???
00??????
00????1???
?/2
?/20
???
?/2
?1???
?
?f ??x n ?K .
The matrix K is also a cyclic matrix and commutes with the matrix S in Eq.?8?,i.e.,KS ?SK .Hence the eigenvectors for the Jacobian matrix K are still in the form of Eq.?9?,
namely,
E m ??exp ?
2?i m ?1L ?,exp ?4?i m ?1L ?
,...,exp ?
2L ?i
m ?1
L
??
T
,?26?
where m ?1,...,L and T represents matrix transpose.
From these eigenvectors and the de?nition of Lyapunov exponents ?11?we obtain the Lyapunov exponent spectrum for Eq.?25?as
?1?lim n →?1
n
ln
??
m ?1
n
f ?…X m ?i ?…?
,
?2??1?ln ?1????cos ?2?/L ??,
?
?L ?
?
?1?ln ?1?2??
if L even ?1?ln ?1????cos ?2?/L ??
if L odd
ordered in a descending fashion.
Unlike the Lyapunov exponents in globally coupled sys-tems,the Lyapunov exponents in the present system are
functions of the system size L .In particular,for large L ,the largest transversal Lyapunov exponent ?2is nearly the same as ?1which by de?nition is positive.This,coupled with the natural limitation that ?р1,means that a large nearest-neighbor coupled system does not have a stable synchronous chaotic attractor.For a given individual map,i.e.,?1is ?xed,let us calculate the maximum lattice size L m under which we can still observe synchronous chaos.Clearly,the larger the coupling strength,the larger the size L m .By letting ??1and ?2?0we have,
L m ?int
?
2?
cos ?1exp ??1?
,
?27?
where the function int ??gives the largest integer that is equal to or less than the argument.
To get a concrete idea of the value of L m ,suppose that the local map is the surjective logistic map,f (x )?1?2x 2.We know that ?1?ln 2in this case.From Eq.?27?we
get
FIG.6.Average bursting amplitude for on-off intermittent time series from 100globally coupled logistic maps as a function of the parameter ?c ??.
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STABILITY OF SYNCHRONOUS CHAOS AND ON-OFF ...
L m?6.That is,for a system with more than six nearest-neighbor coupled surjective logistic maps,one can no longer observe stable synchronous chaos.
V.CONCLUSIONS
The main results of this paper are as follows.
?1?For globally coupled map lattices,we derive explicit conditions for calculating the parameter value at which the synchronous chaotic state becomes unstable and bifurcates into an asynchronous chaotic state.
?2?We show that the on-off intermittent time series,im-mediately after the synchronous chaotic state becomes un-stable,has universal characteristics.
?3?For nearest-neighbor coupled systems,we show that the stability of the synchronous chaotic attractor is a function of the system size.In particular,we can only expect to ob-serve synchronous chaos in such systems if the number of coupled maps is small.
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?12?For example,the imaginary parts of the vectors in Eq.?9?for m?2,3,...,L can be the vectors to span such a subspace.
?13?For typical chaotic maps,the autocorrelation function for the variable?n in Eq.?21?decays exponentially.This means that we should use the ordinary Brownian motion exponent H ?1/2when applying the theory in?7?.
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MINGZHOU DING AND WEIMING YANG
Current-Mode Vs. Voltage-Mode Control In Synchronous Buck Converters by Brian Lynch, Texas Instruments Incorporated For the past twenty years, feedback current-mode control (CMC) has been the method of choice for many high-performance power supply applications. But is it the optimum choice for all power supplies? Here we discuss the pros and cons of CMC Vs. voltage-mode control (VMC) as applied to general purpose dc-dc synchronous buck converters. The choice of whether to implement CMC or VMC as the feedback control method in a synchronous buck dc-dc converter is based on a number of considerations. While the perceived advantage of CMC is better feedback loop response, today’s high-frequency VMC controlled converters closely rival their CMC counterparts. Fig. 1 illustrates the difference in feedback loop performance for a 3.3-V to 1.8-V dc-dc converter with a 10-A output using CMC and VMC and operating at 600 kHz. Since the conversion ratio is above 50% duty cycle, a small amount of slope compensation is added to the CMC loop. In textbook fashion the CMC control-to-output gain is higher at low frequencies and has a zero gain crossover frequency much higher than the VMC counterpart has. Fig. 1: CMC And VMC Control-To-Output Loop Gain/Phase The difference can be narrowed (see Fig. 2) by using Type 3 compensation around the VMC error amplifier when the overall loop performance matches that of a CMC system, albeit at the expense of requiring a higher-bandwidth amplifier.
EMC Guideline for Synchronous Buck Converter Design Keong Kam#1, David Pommerenke#2, Federico Centola*3, Cheung-wei Lam*4, Robert Steinfeld*5 #EMC Laboratory, Missouri University of Science and Technology 4000 Enterprise Dr. Rolla, MO, USA #1keong.kam, #2davidjp@https://www.doczj.com/doc/9b8686631.html, *Apple Inc. 1 Infinite Loop, Cupertino, CA, USA #3centola, #4lam, #5steinfeld@https://www.doczj.com/doc/9b8686631.html, Abstract— Synchronous buck converters generate broadband noise typically in 50 – 300 MHz range. In this paper, the root cause of this broadband noise and possible coupling mechanisms are analysed. Then, a list of EMC design guideline for minimizing the broadband noise is presented. The guideline contains circuit level guideline which involves input filtering, component selection, and an effective snubber strategy using a parallel resistor and inductor. The guideline also contains layout level guideline, which involves decoupling capacitor placement, layer stackup, and minimizing exposed phase voltage area. For each suggested guideline item, an example illustrates the impact of the particular strategy through experimental results or SPICE simulations, leading to a strong reduction of broadband noise from the synchronous buck converter. Key words: Broadband EMI, Synchronous Buck Converter, SMPS, Snubber I.I NTRODUCTION Switched-Mode Power Supply (SMPS), in spite of their superior efficiency, generates electrical noise due to its high dv/dt and di/dt from switching. One of the popular SMPS DC/DC converter topology is synchronous buck converter [1] [2]. The synchronous buck converter typically generates three types of EMI: low frequency conducted noise (<30 MHz) as harmonics of the switching frequency, broadband noise (50 – 300 MHz) from the phase voltage ringing, and high frequency noise (>300 MHz) as a result of reverse recovery [3]. The main focus of this EMC design guideline is to minimize the 50 – 300 MHz broadband EMI. Based on the root causes, a list of EMC design guideline for minimizing the broadband noise is presented. The guideline contains circuit level advice which involves input filtering, component selection, and patent-pending effective snubber strategy using a parallel resistor and inductor. The guideline also contains layout level advice which involves decoupling capacitor placement, layer stackup, and minimizing exposed phase voltage area. II.B RIEF DESCRIPTION OF S YNCHRONOUS B UCK C ONVERTER O PERATION Figure 1 shows the schematic of a typical synchronous buck converter [1] [2] 12 Figure 1. Typical Synchronous Buck Converter Schematic In an ideal operation, the two MOSFETs (high-side and low-side) switch alternatively. The switching of the MOSFETs is controlled by a controller IC which generates a PWM signal and drives the gates. Usually, two types of capacitors are placed at the input side: Electrolytic Bulk Capacitor and Ceramic SMT Capacitor. The bulk capacitors are generally very high-valued (>100 uF) and are used to decouple low-frequency switching noise (typically 100 – 300 KHz switching rate and its harmonics) and to provide charge. The ceramic SMT capacitors are placed closer to the switches (MOSFETs) to provide high-frequency current corresponding to the rise/fall time of the switching waveform (>30 MHz). III.B ROADBAND EMI FROM S YNCHRONOUS B UCK C ONVERTER Figure 2 shows the measured far-field EMI from a synchronous buck converter test board. Frequency [MHz] d B P V / m Buck Converter Testboard Baseline Figure 2.Far-field EMI from Synchronous Buck Converter Test Board Figure 2 shows the radiated broadband EMI centered at 120 MHz. Depending on the design of the converter, this center frequency can typically range from 50 to 300 MHz.
Digital Signal Processing Solution for Permanent Magnet Synchronous Motor Application Note Literature Number: BPRA044
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Experiment No. DATE: Rev. No. 1.00 AIM: - To study Serial Communication and Programmable Communication Interface (8251) in detail Difference between Synchronous and Asynchronous Transmission Serial communication occurs in either synchronous or asynchronous format. In synchronous transmission a receiver and a transmitter are synchronized and a block of character is transmitted along with the synchronization transmission while asynchronous transmission is character oriented means each character carries start and stop bits . Asynchronous transmission is used for high speed transmission (more than 20 kbps) while asynchronous transmission is generally used in low speed transmission. (less than 20 kbps). In asynchronous transmission when no data transmitted a receiver stays high at logic 1 called mark. Logic 0 is called space. Transmission begins with one start bit followed by a character and one or two stop bits. Difference between Simplex and Duplex transmission In simplex transmission data flow in only one directions while in case of duplex transmission data flow in both the directions. In duplex transmission if the transmission is only in one direction at a time then it is called half duplex if it transmission is both direction at a time then it is called full duplex transmission. RS-232C RS-232 (Recommended standard-232) is a standard interface approved by the Electronic Industries Association (EIA) for connecting serial devices. In other words, RS-232 is a long established standard that describes the physical interface and protocol for relatively low-speed serial data communication between computers and related devices. RS-232 is the interface that your computer uses to talk to and exchange data with your modem and other serial devices. The serial ports on most computers use a subset of the RS-232C standard.
Virtual Synchronous Machine Prof. Dr.-Ing. Hans-Peter Beck Clausthal University of Technology Institute of Electric Power Technology (IEE) Clausthal-Zellerfeld, Germany info@iee.tu-clausthal.de Dipl.-Ing. Ralf Hesse Clausthal University of Technology Institute of Electric Power Technology (IEE Clausthal-Zellerfeld, Germany ralf.hesse@tu-clausthal.de Demands in the area of electrical energy generation and distribution, as a result of energy policies, are leading to far reaching changes in the structure of the energy supply, which is characterised, on the one hand, by the substitution of conventional power stations by renewable energy generation, a decision which has already been made, and, on the other hand, by the changeover from centralised to decentralised energy generation. From an electrical engineering point of view, a new situation will arise for consumers concerning security of supply and power quality, which calls for further technical measures by the grid operators to ensure that the increasingly stringent supply criteria can be met. This article describes a new power electronics based approach which allows a grid compatible integration of predominantly renewable electricity generators even in weak grids making them appear to be electromechanical synchronous machines. As a consequence, all the proven properties of this type of machine which have so far defined the grid continue to do so, even when integrating photovoltaic or wind energy. These properties include, for instance, interaction between grid and generator as in a remote power dispatch, reaction to transients as well as the full electrical effects of a rotating mass. In addition, this new development can be operated in such a way that it provides primary reserve allowing, from a grid point of view, electricity generators such as wind and PV to be regarded as conventional power stations. I. I NTRODUCTION The changes in the grid structure can be described as a transition from the proven vertical structure of generation to consumption with a few, but powerful generators (figure 1, left), to a distributed, increasingly amorphous and strongly meshed structure with numerous renewable energy generators which have very different feed-in characteristics (fig. 1 right). Consequently, depending on the local distribution of generators and local grid capacity, there can be an increased line impedance and reduction in short circuit power, which may cause or promote the appearance and mitigation of various disturbances. The integration of electricity generators today is carried out using conventional converters and inverters, whose properties are the result of the implemented control principles. Frequently output voltage controlled inverters with subharmonic [1] or space-vector modulators [2, 3, 4] are used to locally stabilise the grid and supply the active and reactive power required by the grid depending on the local system management. The disadvantages of this operating principle lie in the large differences in grid behaviour between the grid coupled inverters and converters and conventional synchronous machines used for coupling traditional prime movers to the grid. Up to now the layout, properties and operating strategy of electrical grids, the potential for automatically balancing a power deficit between grid areas as well as the dynamic behaviour of the grid have been closely connected to the static and dynamic properties of the electromechanical synchronous machines. Figure 1. Change in structure of generation and distribution of electrical energy It therefore seems necessary to develop a process by which inverters are able to connect any type of electrical generator to the grid in such a manner that the combination of inverter and
Efficiency of synchronous versus nonsynchronous buck converters 同步于非同步BUCK转换器的效率 Choosing the right DC/DC converter for an application can be a daunting challenge. Not only are 对于一个应用选择正确的DC/DC转换器可能是一个使人气馁的挑战。 there many available on the market, the designer has a myriad of trade-offs to consider. Typical 设计者不仅要考虑市场上的模型，而且还要大量的权衡要去考虑。经典的power-supply issues are size, efficiency,cost, temperature, accuracy, and transient response. The 电源问题包括体积，效率，费用，温度，精度，和瞬态效应。为了满足能源之星规格need to meet ENERGY STAR ? specifications or green- mode criteria has made energy efficiency a 或者省电模式标准的要求，已经将效率变成了一个焦点。 growing concern. Designers want to improve efficiency without increasing cost, especially in a 设计者想要在不增加费用的情况下提高效率，特别是对大量的消费类电子high-volume consumer electronics application where reducing power consumption by one watt 产品，只要能够减小1瓦特，那么电网就可以减小上百万瓦特瓦特。半导体工业最近生产了can save megawatts from the grid. The semiconductor industry has recently developed low-cost 一种采用同步整流的低功耗DC/DC转换器，它被认为比非同步DC/DC转换器更有效率。DC/DC converters that employ synchronous rectification and that are thought to be more 这篇文章将在各种应用条件上面比较同步于非同步转换器的效率，体积，费用，以及efficient than nonsynchronous DC/DC converters. This article will compare the efficiency, size, 得失。它将展示同步buck转换器不是在任何时候都是最有效率的。 and cost trade-offs of synchronous and nonsynchronous converters used in consumer electronics under various operating conditions. It will be shown that synchronous buck converters are not always more efficient. To demonstrate the subtle differences between the two converter 为了展示两种拓扑微妙的差别，选择了一种经典的点负载电源。topologies, a typical point-of-load application was chosen. Many low-cost consumer applications 许多廉价的消费类电子产品使用从墙式适配器或者离线式电源上获得的12V轨电源。 use a 12-V rail that accepts power from an unregulated wall adapter or an off-line power supply. 输出电压通常在1到3 V，输出电流小于3A. Output voltages usually range from 1 to 3.3 V, with output currents under 3 A. The Texas Instruments devices in Table 1 were chosen to compare actual efficiency measurements under various output- current and output-voltage conditions. The rated output current, which is the level of output current each device is marketed to deliver, was taken directly from the data sheets (see References 1 and 2).
DS3476 Ver 1.5 Jul. 2010 1 DESCRIPTION The EUP3476 is a 500KHz fixed frequency synchronous current mode buck regulator. The device integrates both 135m ? high-side switch and 90m ? low-side switch that provide 3A of continuous load current over a wide operating input voltage of 4.5V to 28V.The internal synchronous power switch increases efficiency and eliminates the need for an external Schottky diode. Current mode control provides fast transient response and cycle-by-cycle current limit. The EUP3476 features short circuit and thermal protection circuits to increase system reliability. Externally programmable soft-start allows for proper power on sequencing with respect to other power supllies and avoids input inrush current during startup. In shutdown mode, the supply current drops below 1μA. The EUP3476 is available in SOP-8 package with the exposed pad. FEATURES z 3A Continuous Output Current z 110ns Minimum On Time z Integrated 135m ? High Side Switch z Integrated 90m ? Low Side Switch z Wide 4.5V to 28V Operating Input Range z Output Adjustable from 0.8V to 24V z Up to 95% Efficiency z Programmable Soft-Start z <1μA Shutdown Current z Available in 500KHz Fixed Switching Frequency z Thermal Shutdown and Over Current Protection z Input Under V oltage Lockout z Available in SOP-8 (EP) Package z RoHS Compliant and 100% Lead (Pb)-Free Halogen-Free APPLICATIONS z Distributed Power Systems z Networking Systems z FPGA, DSP, ASIC Power Supplies Typical Application Circuit Figure 1.
同步建模技术 2008年4月 协同产品开发合伙有限责任公司 Siemens PLM Software 制作的白皮书
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Synchronous Technology April200目录 技术突破 (1) 建模技术发展的巨大突破 (2) 业务影响 (3) 技术证明 (4) 特征树型结构变为特征集 (4) 在无约束模型上进行受控编辑 (7) 在参数约束模型上进行编辑 (8) 父/子结构 (10) 尺寸方向控制 (11) 程序特征 (13) 模型创建 (14) 快速进行“假设”变更 (15) 技术推广 (17) 总结和评价 (18)
Synchronous Technology April2008 同步建模技术 技术突破 图1：同步建模技术 在运行时间把当前 的几何模型状况与 永久约束合并在一 起。 2008年将见证三维CAD设计历史中的一个里程碑。 Siemens PLM Software推出了同步建模技术－交互式三维实体建模中一个成 熟的、突破性的飞跃。新技术在参数化、基于历史记录建模的基础上前进了一大 步，同时与先前技术共存。同步建模技术实时检查产品模型当前的几何条件，并 且将它们与设计人员添加的参数和几何约束合并在一起，以便评估、构建新的几 何模型并且编辑模型，无需重复全部历史记录。 在模型中找到的当前几何状况由用户有选择地添加约束和参数 同步建模技术 图2：一个普遍模型 编辑及其在基于历 史记录系统里面的 应用。 您可以设想这样带来的性能影响和设计灵活性－进行编辑而无需重新生成整 个模型，因为同步建模技术实时发现、定位和解析依赖关系。当设计人员不必再 研究和揭示复杂的约束关系以便了解如何进行模型编辑时，当他们也不用担心编 辑的下游牵连时，您可以想象对产品开发复杂性带来的正面利益。设计人员可能 要问，“当建模应用程序能够立即识别那些几何相互关系并且保持的时候，我们 为什么还要多余地再强制加上诸如两个模型面是共平面，或者是相切等约束条 件？” 同步建模技术突破了基于历史记录的设计系统固有的架构障碍。基于历史记录的 设计系统不能完全确定依赖相互的关系，从而必须依赖于全面重新执行顺序建模 历史记录。以上图2提出了相关问题。在目前基于有序历史记录的系统中，在需 要对历史记录清单中的特征进行 变更的任何时候，系统都需要删除 所有后续几何模型，回复模型到某 个特征再进行变更，然后重新执行 后续特征命令来重新建立模型。在 大型、复杂的模型中，特征损失可 能非常巨大，这取决于目标特征在 历史记录里面靠后有多远。同步建 模技术没有这个问题－系统实 时识别这些条件在哪里，并且使模 型重建仅仅局限于使模型的几何 条件保持正确所必要的那部分。 在目标编辑区域之后添加的模型几何模型 的这部分真正需要重建的可能性有多大？ 编辑直径 图片由Siemens PLM Software提供