CA2132194A1 - Method for control of chaotic systems - Google Patents
Method for control of chaotic systemsInfo
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- CA2132194A1 CA2132194A1 CA002132194A CA2132194A CA2132194A1 CA 2132194 A1 CA2132194 A1 CA 2132194A1 CA 002132194 A CA002132194 A CA 002132194A CA 2132194 A CA2132194 A CA 2132194A CA 2132194 A1 CA2132194 A1 CA 2132194A1
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- chaotic systems
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
- G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
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Abstract
A method for determining the amount of coupling between interacting chaotic systems arranged in a distribution that will produce concordant aggregate action. The method uses a subgroup of chaotic systems in the distribution that are operated with a selected set of initial seed condition values. Non-linear differential equations describing actions by the chaotic systems are integrated to determine required values for the coupling constant.
Description
wr, 94/17478 PCT/US93100360 2~ 32:J '~1 i ~ETHOD FOR CONTROL
OF CHP~QTIC 5Y~EMS
Teçbnica.l Field The present i nvention relates to methods for effectirl~ c:or~cordant action of a c:ollection of interacting, i.e., coupledr chaoti~:: sy:t~ms organized in a distrlbution. More ~;pQcif~cally, the presen~ invention relates to methods for determining c~p~rating parameters in con~unc:tion wit~ sel~ck~d ini~i~l conditions for controlling int;eractin~ cha~tic sy~it~ms organized in a distribution so as to l3f ~ t c:onc:ordant action.
Back~rQun~
Chaotic systems a~e ubi~itous in the natural world.
They are found on all scales o~ time ~nd space, from-astrc:no~iaal, e .g., asteroid distri~ution and motion ~G~.J. Sussman and J~ Wi~dom, Scienc~ 257 (lg92) 56);
thrQugh the terrestrial, ~ . g . m~chanical p~ndulu~ns: to th~ molecular, e.g. turbulent flow in stirr d fluids H. L. Swinney, and J ~ P~ Gc~llub, i'The Transition To Turbulence, " Physics Today 31, No. 8 ~Au~ust 1~78) 41), including levels of~chemical c:onstituents seen in the Belouso~-Zhabotinski~ reaction ~ . Roux, "Exparimental Studies of Bifurcations Leading To Chaos In The Belousov-Zhabotinski Reactionn 1 Phy ica 7D ~lg83) 57~; to the atomic, e.g., pinn~ng site fre~uency and distribution in high-Tc ~YBCO~ superconducting films (M. Hawley, I.D.
:Raistrick, J.G. Berry, R.J. Houlton, "Growth Mechanism of Sputtered Films of YBa2 Cu3O7 Studied By Scan~;ing Tunneling Microscopy," Reports (29 March 19~1) 1587), and radiation emitters such as lasers and masers (R.A.
~: :
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~ :
WO94/l7478 PCT~S93/~0360 '~ ~ 3 ~
Elliott, R.K. DeFreez, T.L. Paoli, R.O. Burnham, and W.
Streifer, IEEE J. Quantum Electron. ~E-21, 598 (I985);
and even to the subatomic, e.g., chaotic quantum mechanical systems ~M.C. Gutzwiller, Chaos in Classical and ouant~m ~echaniç~, (l990) Springer-Verlag, New York).
For more than a century chaot~c behavior in physical systems has been a recognized phenomenon. Two characteristics of chaotic behavior are that it is deterministic, i.e.j for precisely selected initial conditions the resul~in~ chaotic behavior can be predicted, but this determini~tic behavior is non-periodic. Chaotic beh~vior was first re~Qgnized as such when simple ~echanical ~y t~ms were shown to have very complicated motiol~s. Not only is su~h behavior exceedingly sensitive to precise values for starting or initial conditions, but chaotic behavi~r never settles into~predictable final state~ via recognixable patterns~
: :Chaotic beha:vior: is p~rv~siYe, as outlined above, and~ can be foun~ even in the pr~sence of d~erministic periodic behavior. For example, laser li~ht output intensities are now r~cognized as having an intrinsically unstable component~ with chastic intensity fluctuations.
This chaotic~behavior exists in ~pite of th~ fact that laser~outputs are coherent with exceedingly narrow band widths. Specifically, individual }asers can exhibit undamped chaotic~intensity output spiking behavior on the order of 100 picoseconds ~ps} intervals. (R.A. Elliott, R.K. DeFreez, T~L. Paoli, R.D. Burn~am, and W. Streifer, IEEE J. Quantum Electron QE-21, 598 (1985); and S.SO Ward and H.G. Winful, Appl. Phys. Lett. 52, ~774 ~19~8~).
~ ~ , Such behavior is unavoidably detrimental in applications : where temporal stability is important.
: :
WO94/17478 PCT~S93/00360 ~J l.321~
Chaotic behavior can be spatiotemporally extended by arranging individual chaotic systems in one, two and three spatial dimensional distributions. Of particular interest here is the ~ituation where multiple chaotic sy5tems are ~ sp~tlally arranged in distribution~;
and, (ii) interact with ~ach oth~r by means such as evanescent or di~u~v~ c~upling. As used here the term distribution is int~nded to b~ synonymous with other :: :
terms Por groupings and arrBn~ments such as array and st:ructure. Thè ag~regate output acti~n from such distri~u~ions can exh~lbit so~e le~el of concordance or : can be chaotic a~ if individual systems were independently func~ioning wit~out interaction or coupling~. Most probabl~ aggregate actions are either spatiotemporal chaos, which o~ten results from ~xces~ive :interacti~n, or asynchronous chaos, which o~ten results from:insu~ficient interaction.
SummarY ~f the Invention ;Principal ~bjects~ of this invention are to provide a :method for: determining how to achieve concordant aggregate~ acti~n~:fro~ distributions of interacting chaotic ~systems~:ana to then use ~uch control Xor achieving :specific~ levels of ~oncordant action.
Distributions o~ interacting chaotic systems here include spatially~and temporally periodic substructures, that ~ay : ;exist in a homogeneous media, where normal aggregate : behavior is chaotic, irregular or turbulent. ~oncordant ac~ion here is~achieved through setting up a particular mode of initial operation for a select number of chaotic systems in a portion of the distribution, along with using predetermined coupling strengths between individual ~ ~:
/17478 PCT~S93/00360 ~ 4 -chaotic systems in the distribution, and by also using predetexmined dimensional arrangements for the chaotic systems in distributions. ~esulting distributions can be skinny bars, i.e., such skinny ~ars can be considered one-dimensio~al; or they can be flat plates, i.e., two-dimensional; or c ~ es, i.e~, three-dimensional. Achieved concordant ou~put actions for interacting cha~tic ~ystems ~ontrolled using the method o~ th~ invention are stable and resistant to per~ur~tlons ~nd noise.
To use the method of the inv¢n~ion) dyna~ics for chaotic systems in the dist~ibution ne~d to be described using non-linear diff:erential equations with boundary conditlons inserted prior to integr2~ion of the equations. These boundary c~nditlons are d~termined fro~
both physica~ character~stic~ for the dis~ribution, i.e., size, shape and num~er of c~aotic ~ystams, and for achieYable va}ues for chaotic system op rating parameters including coupling constants. Be~re integration i5 begun, a subgroup of interacting rhaotic ~ystems from within the dis~ribution i5 select~d. ~or those selected chaotic systems in the subgroup a set of operating parameter values deflning an initial seed condition is input to their: corresponding non-lin~ar differ~ntial equations. Initial seed condition~ defin~ a set of operating parameter values that initiate a concordant output from chaotic systems in ~he subgroup. A variety of initial seed ~onditions is appropriate for the present invention as dis~cussed below. Integration is now accomplished using a computer~ with the boundary conditions and initial seed condition operating parameter values set in the non-linear differential equations.
Stability measu~es are used to ~valuate aggregate output WO94/1747~ PCT~S93100360 1~ 32 l~?;~
as determined by integration of the non-lineax differential equations. These stability measures include viewing aggregate output performance on ~re appropriate device such as a cathode ray tube (CRT) or by use of mathematical tests as discussed b~ow~ When stability is achieved integration i~ ~erminated, and identified optimum ~perating ~lues for ac~ieving determined concQrdant action ar~ used to op~rate chaotic systems in selected distributions.
Brief DescriPtio~ .o~ winq~
The various o~jectives, ~dvant~ges and novel features of t~e pr~n~ invention will become more readily apprshended from the following detailed description when taken in conjunction with the appended drawings, in which:
Fig. 1 show~ in field ~pac~, using gray tones, an initial~se~d condition in the for~ of a minimum spiral :seed (MSS~ for a ~ x 2 array of interac~ing chaotic systems as a substructure in a larger planar distribution of chaotic systems;~:
Fig. 2 shows instantaneous phase se~d conditions in x, y phase space ~a~sociated with the initial seed conditi~ns depicted ~in ~igure l;
Fig. 3 shows~in field space, using gray tones, a concordant aggregate ~output in the ~orm of an asynchronous periodic spiral (ASPS) from a two dimensional distribution of interacting chaotic systems, that can resul~ from:initial seed conditions depicted in Figures 1 and 2 with low value coupling constants;
Fig. 4 shows in field space~ using gray tones, a concordant aggregate ou*put in the form of periodic W~94/17478 PCT~S93/00360 ~ 3 ~ 6 banding structure (PBS) for a two dimensional distribution of interacting chaotic systems resulting from initial seed conditions d~picted in Figures l and 2 with high value coupling constants.
Fig. ~ shows inst~ntaneous phase values in x, y phase space for the PBS aqgregate output depict~d in Fig.
4; and, ~: ~ Fig. 6 is a sche~atic for a two-dimensional array of : lasers.
~:
Det3iled ~escripti~n_o ~h~ Inv~ntion The present inv~ntion provides a method for determining how to control in~era~ting chaotic systems organized in distri.~utions 50 t~ey aqgregately act in concordant fashion. ~ll typ~ o~ di tributions, namely ; one, two and three spatial dimensions, involving action of~one, two, three and even more variables for individual chaotic systems in the distributionst are within the scope of this invention. Each chaotic syste~ ~rganized in a distribution for ths present invention, though, must be ~apable of interacting with at least one other chaotic ; sy tem: in the distribution. Using distribution parameters, e.g., spacing b~tween chaotic systems, ~or controlling coupling int~ractions between individual : chaotic systems is an available aspect of the present invention. So, not only are all types of distributions ~: within the scope of this invention, but also all chaotic systems capable of both being orqanized in distributions and interacting with each other are within the scope of this invention.
: In general the method of the invention begins with describing actions of individual chaotic systems with ~;:
WO 94/1747~ PCT/US93/00360 .i¢ ~. c~ ,~` L ~ i non-linear differential ~3quatic~ns that can be of the ~orm:
dVl~dtSRv~Vl~ V2,, . Vn) ~D", ~V
nYl to ~ nf in i ty ~: where Vi are loc:al state variables, I~y are cs:)upling : constaTIts, and E~ , V2,.~.Vn~ are functions describing kinet:ics for the i th ~:haotic system. ~t is ns~ted that it may not be necessary to clescrîbe the action of each chaotic system ~n the distri~ution with a corr~sponding : indi~idual non-linear di~erential e~tion if parameter :: :
Yalues fc:r initial s~ed ~antipodal phase apposition) collditions capable of leading to concordant aggr~gate action ~re ussd wi~:h the non-linear differential e~ations Por those chaot~c syste~s in ~ selected subgroup within~.the dis~ribution~ ~cc~urrence of this s~ituat:ion,~ howe~rer, will be dependent on characteristics of ~those chaoti~ systems in a selected distribution.
The next: step~ o~ this imrentic)n is to se}ect both a finite nu~i.ber of chaotic systems in a subgroup of the distribution and s~lect a cet of initial seed parameters for the chaotic systems in the su~group at the time of initiation. 5elected initial seed parameters are then input to their respective non-linear differential e~u~tions for ths chosen chaotic systems and the entire field of all non-linear differ~ntial equations describing the distribution are used for calculating operating values~ The calculations, involving integraeions, can be W094/17478 PCT~S93tO0360 ij ~ 3 ~ 8 -made on a computer using known programming techniques.
As the integrations p~oceed as an itera~ion process, the optimum coupling constants are determined~ Determined coupling constants may be a funct~on of distribution spacing.
An example of ~n appropr~ate initial seed an : include mini~um ~piral se~d (MSS~ con~tions a~ shown in Figures ~ and ~. H~r~ th~ selected array is a 30 x 30 :~: two-dimensional dist~ibution/ an~ th~ achieved concordant output is an asynchronou$ periodic spir~l ~A5PS) ~s shown in~Figure 3, which shows ASPS in f i81d space. The coupling oonstant ~or ~b~s example i5 O- 2 To achieve this ASPS result a c~ntral 2 x ~ su~group of chaotic ystems are set to hav~ initial ~eed conditiQns sh~wn in fi~eld ~space in Figure 1, and ~nstantan~ous phase seed ;conditions:shown in x, y phase spac~ ~s depicted in Pigur~ 2. Speci~ically the selected in~tantanQous values for phase for those chaotic systems in the ~ubgroup are : shown as dots in relation to steady st~te indicated by a ; "+~"~in~:phase~:space. Stated in words, a MSS can be a subgroup of four;chaotic systems forming a closed path, whose ~x and y ~alues~ are such that in clockwise or counter~clockwise~directions, they approximate values, .e. state space ~variables of the non-linear dynamic system, at 90~ degree phase differences. Now the :resulting output, given proper coupling ~atween chaotic systems, can be ASPS in character, which has the general property of having a constant aggregate QUtpUt and i5 a function of the number: of units in the distribution.
This constant output is additionally re~is~ant to perturbation from both external and intrinsic (or deterministic) noise which is damped. Such a result i5 ~:
WO94t17478 ~ PCT~S93/~0360 , ....
realized because sta~le phase relationships are fixed in the output and all phases are represented at any instant in time.
Again selecting a MSS initial seed condition, but altering coupling constants, in particular using high value coupling constant~, can pr~duce a pQriodic banding struature (PBS) ~ggre:gata output. A property o~ PBS is a synchronous periodic amplitude output as shown in the field space depiction se~ out in ~`igur~ 4. Instantaneous phase values or this PBS output are ~;1QWn by the four separate values lOQ, 112, 114 and 116 ~orming an isochron as set out in the x, y ph~s~ spa~e depic~îon in Figure 5~
Other initia~ s~ed conditions that can be u~ed in~lude: (i) rando~ uniformly distributed state variables, whose local state variable, V;, values are oentered about a ~teady state; and, (li) Fandom uni~ormly distributed state variables in which ~ra~sients are destroyed ~y ~running the non-linear diferenti~1 equations through a l~rqe nu~ber of iterations, e.g., lOh iteration~ ~lOO iterations/unit time~, before initiating coupling. : :
As an exampl~ f~r~using t~e pre~ent invention, a two dimensional distribution of lasers is de~cribed below.
Positioning o~ lasers~in the distribution determines the extent of evanescent field overlap which is the inter-laser coupling mechanism used here in part to effect the invention. This two dimensional distribution of interacting lasers is used for description purposes only and is not intended to identîfy an exclusive application of the invention.: Other distri~utions of chaotic systems capable of interaction can also be used with the present invention. For example, ~ven arbitrarily discrete WO94l17478 PCT~S93100360 '~t~3~
portions of continuous chaotic systems can be used with the present invention. However, for such application~
where a continuous chaotic system is divided there must also be a means for controlling magnitudes of coupling between su~divided portions~ This i~ a necessary condition for using th~ pres~t invention.
Group~ of laser~, ~.g. t two or more ~emiconductor laæers, may be arran~ed spat~ally ~ ~heir evane~cent fields ovQrlap and effect coupling betw~en adjacent lasers. Such a spatial arrangem~nt is shown in Fig. 6, and this spatial arrang~m~nt or distribution is generally designated by :r~ference number lO. The iasexs 12 are positioned with respect to each other in the d~stri~ution lO so evanescent Pields from ea~h lase: 12 overlaps with that of at least one adjacent laser 12. ~ach laser 12 in the distributivn 10 is operated in the same longitudinal and transverse ~ode, e.g., TEM ~O,O), TEM (l,O) or TEM
In spite of coupling between lasers 1~ effected by overlapping evanescent fields, thQ a~grega~e radiation output from th~ spatial arrangement lO without additional control will, except:for a few uniqu2 ca~e~ be chaotic~
~i.e., irregular and unordered even as if, for large enough ~arrays, each of the lasers 1~ were emitting chaoti~ intensity radiation patterns. Such chaotic aggregate radiation output patterns will occur even when individual lasers 12 are operated in the fundamental TEM
(O,~) mode.
The distribution lO of lasers 12, with each operated in the same ~ongitudinal and transverse mode, provide~ an example of where the present invention can be applied for determining the magnitude of coupling between lasers 12 :
WOg4/17478 PCT~S93/0036 required for effecting a non-chaotic aggregate system output.
For each laser 12 in the spatial distribution 10 the time varyinq electric ~ield ~ar the guided mode i5 describable as E~(t~e~ , w~ere ~h~ complex a~plitude (E~(t); varies slowly compared to the optical frequency . This function i5 id~ntified as b~ing ~or the jth laser 12. Assuming ne~rest-neighbor coupling, there is an evolution ~f the ~ode ~mplitude (Ej~ and the population (Nj) in the jth laQer that is described by the following:equations~respec~iYe~y:
=iG~3~ ia)~j ~2) 'P-~-G~ , t2) whére G is~ gain,~tp ~(~lpsl is photon lifetime, rs (~
2nanosecond (ns:))::is:lifetime of the active population, P~ is:~pump~rate,~and ~K i5 coupling strength betwae~
ad~acent~lasers.~ The parameter ~ is~ known as the line width~enhance~ent~;~factor ~in~semiconductor lasers and is a~measure of carrier-density-dependent refractive index.
For operation not too far from laser threshold of uncoupled lasers,~ the: gain may be expressed ~s G(N~)=G(Nth)+g(Nj; -~N~h),~ where Nth is the aarrier density at threshold, G(Nth)~ p,~and g= ~G~N is differential gain. It shauld be~noted that if the population Nj is W094/t7478 PCT~S93/00360 ~ 3 ` ~-~ 12 -adiabatically eliminated, Eq. (1) represents a set of coupled van der Pol oscillators and is also identical to the discrete Ginzburg-Landau equation which has often been used as a model for spatiotemporal complexity.
:It is convenient t~ ~ran~form Eqs. (1) and ~2) into dimensionless form for the no~malized magn~tude (Xj) and phase (0J) o~ ~he electric ~ield, and normalized excess ~;carrier density Z~ in:th~ jth la~r ~. Th~se e~uations ~are:
~=Z~-JI f~ f-a~
-x~ ,sin~DJ I-~J) t, (3) ::
=aæ~-~[~ cos ~'0j-0J-l~
(xj lf~ cos ~ -0,~
: ~ ~
TZ~-p-Z~ Z2~) ~ ,j-1,2,..... ,N, (5~
with XO_X~1=O. ;Here the overdots signify derivatives with~:;respect to~a reduced ti~e ~/7p and we deine the ollowing varia~les and parameters:
2 s j' j 2~th~N~t~--l~, P= 2 gNth~p ~P/Pth--1~, ~ =K~p, T= c";/tp.
.
Equations ~3)-~5):represent an oscillator assembly, which : in the absence of coupling (~=O) would evol~e toward a :~ :
WO~4/17478 PCT~S93/0036 steady state with Zj=0, Xj = ~p, and arbitrary phases 0j.
: : For nonzero coupling, the output of spatial distribution lO of las~rs 12 can organize it~elf into a macroscopically coherent s~ruc~ur~ with well-defined phase relationships. Prior ~o the pr~en~ in~ention it was believed that th~ self-organizing principle ; underlying such collectlve ~avisr rQs~ d only from or~ed synchr~nization or mutual entrainm~nt. But with : : the present in~ention it is determinable that select~d ranges of values for coupling are required to achieve the ~: collective behavîor.
When working with small laser arrays, e.g. two lasers, there are two~l~v~ls of ~ynchronization involved~
The ~fiFst represents a~ guiescent state in which amplitudes X; and carrier Zj are constant in time while : phases 0j e~oIY~ linearly in time at the same xate possibly ~zero~ ~r all lasers. For weakly coupled lasers ~lOs), a stable ph~s:e-locked or quiescent state is~aGhievable in which amplitude distributions a~ross th~
distribution lO are nearly uniform. This uniform phase-locked: ætate, howeverl is not alway~ stable. Above a :critical coupling~ strength the quiescent state l~ses ta~ility th~ough;a~supercritical Hopf bifureation. It has~been thought delayed response of carriers leads to phase~ lags between~ oscillators and destroys phase locking. Dynamical variables--amplitudes, pha~es, and ~ ~ carrier d~nsities--all pulsate in time. Th~se pulsations :~ :occur in different elements of a distribution ~nd may or ~: may not be~ in time step with each other.
Typical behavior for small distributions of coupled : interacting lasers, e.g., l x 3, is chaotic for most pump :
WO94/17478 PCT~S93/0~360 ., ~ 3 ~ - 14 -currents and coupling co~stants above ~=10-3-5. This chaotic a~gregate output continues for larger distributions of Iasers in the a~sence of preferred initial seed conditions, coupling strengths and spatial distri~utions.
Now a speci~ic example ~or controlling and eliminating such chaotlc b~hav~or in small, e.g., 2 x 2, or lar~e, e.g., 160 x 160~ d~stributions is given. For semiconductor la ers spatially arrang~d in a distribution it is possible through sel~cti~n of initial conditions to obtaîn an ASPS output~across t~e entir~ di~tribution.
Other output patterns, to include PBS, are achievable through use of the invention, but hexe achieYement of ASPS will be considered.
A distribution lO of las~rs 12 that is larger than a;:2~X 2 distribution is ~onsidered. Within this larger distribution 10 a 2 X 2 subgroup oP ~our la~ers 1~ is arbitrarily selected. Lasers 12 in the diætri~ution lO, including the four:lasers 12~in the identi~ied subgroup, are: selected to have a 1 ns natural frequency. As an initial seed ~condi:tion, the four lasers 12 in the selected 2 X 2 subgroup are positioned and operated so :they are coupled by t~eir evanescent fields and form a closed path with~ maxi~um intensity occurring at approximately 250;~ps intervals around the orthogonal ne~ighborhood în a clockwise or counte~ clockwise fashion.
Adjusting positioning and operati~n parameters for the :four lasers 12 in the distribution 10 to function in such a~ manner provides MSS initial saed conditions to the distribut-ion 10. However, just selecting a 2 X 2 subgroup within the distribution lQ and operating the four lasers 12 in ~the subgroup under M5S initial seed :
W~94/17478 PCT~S93/00360 ., ~ , .
conditions doss not necessarily, even probably, result in concordant aggregate output from the distribution lO much less an ASPS aggregate output.
A set of non-lin~ar differential equations, as described above, for ea~h ~ the lasers ~2 in the distribution 10 i5 creat~d~ Be~ore calculations using these non-linear di~f~rential equation~ can be mad~, boundary conditions as dict~ted by physical paramet~rs : for lasers 12 and di~ri~ution lO must be put in the differential ~quations. Specifically th~se boundary conditions are~ the size, shape and number of chaotic systems, i.e., l~sers 12~ in distribution 10, ~ii) identification of which lasers 12 interact with each laser 12 in distribu~ion lO; ~iii3 value ranges for each physical parameter, including coupling Gonstants ~ that are achi~vable for Iasers 12 in distribution lO; and, :~ (iv) s~ecific values for each physical param~ter required to establish MSS~initlal seed conditions in the selected 2~ ~:2 subgroup.
With this description of la~ers 12 in the defined distribution lO, the non-linear differential e~uations are integrated at m~ltiple points in time and over a range of coupling constants to determine aggregate : :c~aracteristics for the distribution lQ. ~esulting aggregate characteristics can range fr~m unordered chaos to some form of concordant action or stability~ As part : of the method of this invention stability measures are ~i~ used to evaluate performance o~ initial seed conditions com~ination with laser 12 operating parameters : including coupling constant values. Such stability measures can include:
(i) Displaying results of integrating the non-, :
WOg4/17478 PCT~S93/00360 ~ - 16 -linear differential equations on a cathode ray tube (CRT) 50 aygregate output action ~an be directly viewed to determine if stability has been achi~ved, (ii~ Mapping f~ (Uj) V5 ~ (U~) where n is the nth crossinq of a Poincare s~ction, U is a sel~cted loc~l state var~able (Vi) : r~pr~sentativ~ of ag~regate output from distributi~n 10. Then dispe~sion is measured as standa~d ~eviation~ from th~ m~p ~n(Uj~ = f"
Uj~ 5tability is achieved at minimum disp~rsion.
: ~iii) Calculating the sum of absolute net diffusion or th~ ~ystem and monitorin~ ~h~se values until they rea~h a cons~ant v~lue : charact~ristic of the dynamic sy~t~m and its ~ :
param~ter~.
When stability i& achieved in~@gra~ion o~ the non-linear dif~ rential equ~tions is term~nated~ If ~tability is :: :
n~t~ achievedj the range of values ~or opera~ing parameters and couplin~ constants ~ust be changed or the initisl seed conditions must be ch~nged. Po~sibly changing both is r~guired.
A~ter it is~:determined that st~bility is achieYed, values ~or a distribution arrangement, including shape and size, are iden~ifiable from the integration r~sults along with op~rating paramekers and coupli~g constant values. Accordingly a distribution of chaotic sys~ems in Gonformity wi~h thes~ results can be built and operated, beginning with the identified initial seed conditions for those chaotic syste~s in an identified subgroup, to produce the selected corlcordant output.
'' W~94/17478 PCT~S93J00360 . ~ 3 ~
The above discussion and related illustrations of the present invention are directed primarily to preferred embodiments and practices. However, it is believed numerous chan~s and modifications in actual implementation o~ d~crihsd conc~pts will be apparent to those skill~d ln th~ art, ~nd it is cont~mplated that such chang~s and m~difications may be made without departing from the scope of the invention as defined by the fo`~lowing claims.
~:
:
::
~::
OF CHP~QTIC 5Y~EMS
Teçbnica.l Field The present i nvention relates to methods for effectirl~ c:or~cordant action of a c:ollection of interacting, i.e., coupledr chaoti~:: sy:t~ms organized in a distrlbution. More ~;pQcif~cally, the presen~ invention relates to methods for determining c~p~rating parameters in con~unc:tion wit~ sel~ck~d ini~i~l conditions for controlling int;eractin~ cha~tic sy~it~ms organized in a distribution so as to l3f ~ t c:onc:ordant action.
Back~rQun~
Chaotic systems a~e ubi~itous in the natural world.
They are found on all scales o~ time ~nd space, from-astrc:no~iaal, e .g., asteroid distri~ution and motion ~G~.J. Sussman and J~ Wi~dom, Scienc~ 257 (lg92) 56);
thrQugh the terrestrial, ~ . g . m~chanical p~ndulu~ns: to th~ molecular, e.g. turbulent flow in stirr d fluids H. L. Swinney, and J ~ P~ Gc~llub, i'The Transition To Turbulence, " Physics Today 31, No. 8 ~Au~ust 1~78) 41), including levels of~chemical c:onstituents seen in the Belouso~-Zhabotinski~ reaction ~ . Roux, "Exparimental Studies of Bifurcations Leading To Chaos In The Belousov-Zhabotinski Reactionn 1 Phy ica 7D ~lg83) 57~; to the atomic, e.g., pinn~ng site fre~uency and distribution in high-Tc ~YBCO~ superconducting films (M. Hawley, I.D.
:Raistrick, J.G. Berry, R.J. Houlton, "Growth Mechanism of Sputtered Films of YBa2 Cu3O7 Studied By Scan~;ing Tunneling Microscopy," Reports (29 March 19~1) 1587), and radiation emitters such as lasers and masers (R.A.
~: :
~::
~ :
WO94/l7478 PCT~S93/~0360 '~ ~ 3 ~
Elliott, R.K. DeFreez, T.L. Paoli, R.O. Burnham, and W.
Streifer, IEEE J. Quantum Electron. ~E-21, 598 (I985);
and even to the subatomic, e.g., chaotic quantum mechanical systems ~M.C. Gutzwiller, Chaos in Classical and ouant~m ~echaniç~, (l990) Springer-Verlag, New York).
For more than a century chaot~c behavior in physical systems has been a recognized phenomenon. Two characteristics of chaotic behavior are that it is deterministic, i.e.j for precisely selected initial conditions the resul~in~ chaotic behavior can be predicted, but this determini~tic behavior is non-periodic. Chaotic beh~vior was first re~Qgnized as such when simple ~echanical ~y t~ms were shown to have very complicated motiol~s. Not only is su~h behavior exceedingly sensitive to precise values for starting or initial conditions, but chaotic behavi~r never settles into~predictable final state~ via recognixable patterns~
: :Chaotic beha:vior: is p~rv~siYe, as outlined above, and~ can be foun~ even in the pr~sence of d~erministic periodic behavior. For example, laser li~ht output intensities are now r~cognized as having an intrinsically unstable component~ with chastic intensity fluctuations.
This chaotic~behavior exists in ~pite of th~ fact that laser~outputs are coherent with exceedingly narrow band widths. Specifically, individual }asers can exhibit undamped chaotic~intensity output spiking behavior on the order of 100 picoseconds ~ps} intervals. (R.A. Elliott, R.K. DeFreez, T~L. Paoli, R.D. Burn~am, and W. Streifer, IEEE J. Quantum Electron QE-21, 598 (1985); and S.SO Ward and H.G. Winful, Appl. Phys. Lett. 52, ~774 ~19~8~).
~ ~ , Such behavior is unavoidably detrimental in applications : where temporal stability is important.
: :
WO94/17478 PCT~S93/00360 ~J l.321~
Chaotic behavior can be spatiotemporally extended by arranging individual chaotic systems in one, two and three spatial dimensional distributions. Of particular interest here is the ~ituation where multiple chaotic sy5tems are ~ sp~tlally arranged in distribution~;
and, (ii) interact with ~ach oth~r by means such as evanescent or di~u~v~ c~upling. As used here the term distribution is int~nded to b~ synonymous with other :: :
terms Por groupings and arrBn~ments such as array and st:ructure. Thè ag~regate output acti~n from such distri~u~ions can exh~lbit so~e le~el of concordance or : can be chaotic a~ if individual systems were independently func~ioning wit~out interaction or coupling~. Most probabl~ aggregate actions are either spatiotemporal chaos, which o~ten results from ~xces~ive :interacti~n, or asynchronous chaos, which o~ten results from:insu~ficient interaction.
SummarY ~f the Invention ;Principal ~bjects~ of this invention are to provide a :method for: determining how to achieve concordant aggregate~ acti~n~:fro~ distributions of interacting chaotic ~systems~:ana to then use ~uch control Xor achieving :specific~ levels of ~oncordant action.
Distributions o~ interacting chaotic systems here include spatially~and temporally periodic substructures, that ~ay : ;exist in a homogeneous media, where normal aggregate : behavior is chaotic, irregular or turbulent. ~oncordant ac~ion here is~achieved through setting up a particular mode of initial operation for a select number of chaotic systems in a portion of the distribution, along with using predetermined coupling strengths between individual ~ ~:
/17478 PCT~S93/00360 ~ 4 -chaotic systems in the distribution, and by also using predetexmined dimensional arrangements for the chaotic systems in distributions. ~esulting distributions can be skinny bars, i.e., such skinny ~ars can be considered one-dimensio~al; or they can be flat plates, i.e., two-dimensional; or c ~ es, i.e~, three-dimensional. Achieved concordant ou~put actions for interacting cha~tic ~ystems ~ontrolled using the method o~ th~ invention are stable and resistant to per~ur~tlons ~nd noise.
To use the method of the inv¢n~ion) dyna~ics for chaotic systems in the dist~ibution ne~d to be described using non-linear diff:erential equations with boundary conditlons inserted prior to integr2~ion of the equations. These boundary c~nditlons are d~termined fro~
both physica~ character~stic~ for the dis~ribution, i.e., size, shape and num~er of c~aotic ~ystams, and for achieYable va}ues for chaotic system op rating parameters including coupling constants. Be~re integration i5 begun, a subgroup of interacting rhaotic ~ystems from within the dis~ribution i5 select~d. ~or those selected chaotic systems in the subgroup a set of operating parameter values deflning an initial seed condition is input to their: corresponding non-lin~ar differ~ntial equations. Initial seed condition~ defin~ a set of operating parameter values that initiate a concordant output from chaotic systems in ~he subgroup. A variety of initial seed ~onditions is appropriate for the present invention as dis~cussed below. Integration is now accomplished using a computer~ with the boundary conditions and initial seed condition operating parameter values set in the non-linear differential equations.
Stability measu~es are used to ~valuate aggregate output WO94/1747~ PCT~S93100360 1~ 32 l~?;~
as determined by integration of the non-lineax differential equations. These stability measures include viewing aggregate output performance on ~re appropriate device such as a cathode ray tube (CRT) or by use of mathematical tests as discussed b~ow~ When stability is achieved integration i~ ~erminated, and identified optimum ~perating ~lues for ac~ieving determined concQrdant action ar~ used to op~rate chaotic systems in selected distributions.
Brief DescriPtio~ .o~ winq~
The various o~jectives, ~dvant~ges and novel features of t~e pr~n~ invention will become more readily apprshended from the following detailed description when taken in conjunction with the appended drawings, in which:
Fig. 1 show~ in field ~pac~, using gray tones, an initial~se~d condition in the for~ of a minimum spiral :seed (MSS~ for a ~ x 2 array of interac~ing chaotic systems as a substructure in a larger planar distribution of chaotic systems;~:
Fig. 2 shows instantaneous phase se~d conditions in x, y phase space ~a~sociated with the initial seed conditi~ns depicted ~in ~igure l;
Fig. 3 shows~in field space, using gray tones, a concordant aggregate ~output in the ~orm of an asynchronous periodic spiral (ASPS) from a two dimensional distribution of interacting chaotic systems, that can resul~ from:initial seed conditions depicted in Figures 1 and 2 with low value coupling constants;
Fig. 4 shows in field space~ using gray tones, a concordant aggregate ou*put in the form of periodic W~94/17478 PCT~S93/00360 ~ 3 ~ 6 banding structure (PBS) for a two dimensional distribution of interacting chaotic systems resulting from initial seed conditions d~picted in Figures l and 2 with high value coupling constants.
Fig. ~ shows inst~ntaneous phase values in x, y phase space for the PBS aqgregate output depict~d in Fig.
4; and, ~: ~ Fig. 6 is a sche~atic for a two-dimensional array of : lasers.
~:
Det3iled ~escripti~n_o ~h~ Inv~ntion The present inv~ntion provides a method for determining how to control in~era~ting chaotic systems organized in distri.~utions 50 t~ey aqgregately act in concordant fashion. ~ll typ~ o~ di tributions, namely ; one, two and three spatial dimensions, involving action of~one, two, three and even more variables for individual chaotic systems in the distributionst are within the scope of this invention. Each chaotic syste~ ~rganized in a distribution for ths present invention, though, must be ~apable of interacting with at least one other chaotic ; sy tem: in the distribution. Using distribution parameters, e.g., spacing b~tween chaotic systems, ~or controlling coupling int~ractions between individual : chaotic systems is an available aspect of the present invention. So, not only are all types of distributions ~: within the scope of this invention, but also all chaotic systems capable of both being orqanized in distributions and interacting with each other are within the scope of this invention.
: In general the method of the invention begins with describing actions of individual chaotic systems with ~;:
WO 94/1747~ PCT/US93/00360 .i¢ ~. c~ ,~` L ~ i non-linear differential ~3quatic~ns that can be of the ~orm:
dVl~dtSRv~Vl~ V2,, . Vn) ~D", ~V
nYl to ~ nf in i ty ~: where Vi are loc:al state variables, I~y are cs:)upling : constaTIts, and E~ , V2,.~.Vn~ are functions describing kinet:ics for the i th ~:haotic system. ~t is ns~ted that it may not be necessary to clescrîbe the action of each chaotic system ~n the distri~ution with a corr~sponding : indi~idual non-linear di~erential e~tion if parameter :: :
Yalues fc:r initial s~ed ~antipodal phase apposition) collditions capable of leading to concordant aggr~gate action ~re ussd wi~:h the non-linear differential e~ations Por those chaot~c syste~s in ~ selected subgroup within~.the dis~ribution~ ~cc~urrence of this s~ituat:ion,~ howe~rer, will be dependent on characteristics of ~those chaoti~ systems in a selected distribution.
The next: step~ o~ this imrentic)n is to se}ect both a finite nu~i.ber of chaotic systems in a subgroup of the distribution and s~lect a cet of initial seed parameters for the chaotic systems in the su~group at the time of initiation. 5elected initial seed parameters are then input to their respective non-linear differential e~u~tions for ths chosen chaotic systems and the entire field of all non-linear differ~ntial equations describing the distribution are used for calculating operating values~ The calculations, involving integraeions, can be W094/17478 PCT~S93tO0360 ij ~ 3 ~ 8 -made on a computer using known programming techniques.
As the integrations p~oceed as an itera~ion process, the optimum coupling constants are determined~ Determined coupling constants may be a funct~on of distribution spacing.
An example of ~n appropr~ate initial seed an : include mini~um ~piral se~d (MSS~ con~tions a~ shown in Figures ~ and ~. H~r~ th~ selected array is a 30 x 30 :~: two-dimensional dist~ibution/ an~ th~ achieved concordant output is an asynchronou$ periodic spir~l ~A5PS) ~s shown in~Figure 3, which shows ASPS in f i81d space. The coupling oonstant ~or ~b~s example i5 O- 2 To achieve this ASPS result a c~ntral 2 x ~ su~group of chaotic ystems are set to hav~ initial ~eed conditiQns sh~wn in fi~eld ~space in Figure 1, and ~nstantan~ous phase seed ;conditions:shown in x, y phase spac~ ~s depicted in Pigur~ 2. Speci~ically the selected in~tantanQous values for phase for those chaotic systems in the ~ubgroup are : shown as dots in relation to steady st~te indicated by a ; "+~"~in~:phase~:space. Stated in words, a MSS can be a subgroup of four;chaotic systems forming a closed path, whose ~x and y ~alues~ are such that in clockwise or counter~clockwise~directions, they approximate values, .e. state space ~variables of the non-linear dynamic system, at 90~ degree phase differences. Now the :resulting output, given proper coupling ~atween chaotic systems, can be ASPS in character, which has the general property of having a constant aggregate QUtpUt and i5 a function of the number: of units in the distribution.
This constant output is additionally re~is~ant to perturbation from both external and intrinsic (or deterministic) noise which is damped. Such a result i5 ~:
WO94t17478 ~ PCT~S93/~0360 , ....
realized because sta~le phase relationships are fixed in the output and all phases are represented at any instant in time.
Again selecting a MSS initial seed condition, but altering coupling constants, in particular using high value coupling constant~, can pr~duce a pQriodic banding struature (PBS) ~ggre:gata output. A property o~ PBS is a synchronous periodic amplitude output as shown in the field space depiction se~ out in ~`igur~ 4. Instantaneous phase values or this PBS output are ~;1QWn by the four separate values lOQ, 112, 114 and 116 ~orming an isochron as set out in the x, y ph~s~ spa~e depic~îon in Figure 5~
Other initia~ s~ed conditions that can be u~ed in~lude: (i) rando~ uniformly distributed state variables, whose local state variable, V;, values are oentered about a ~teady state; and, (li) Fandom uni~ormly distributed state variables in which ~ra~sients are destroyed ~y ~running the non-linear diferenti~1 equations through a l~rqe nu~ber of iterations, e.g., lOh iteration~ ~lOO iterations/unit time~, before initiating coupling. : :
As an exampl~ f~r~using t~e pre~ent invention, a two dimensional distribution of lasers is de~cribed below.
Positioning o~ lasers~in the distribution determines the extent of evanescent field overlap which is the inter-laser coupling mechanism used here in part to effect the invention. This two dimensional distribution of interacting lasers is used for description purposes only and is not intended to identîfy an exclusive application of the invention.: Other distri~utions of chaotic systems capable of interaction can also be used with the present invention. For example, ~ven arbitrarily discrete WO94l17478 PCT~S93100360 '~t~3~
portions of continuous chaotic systems can be used with the present invention. However, for such application~
where a continuous chaotic system is divided there must also be a means for controlling magnitudes of coupling between su~divided portions~ This i~ a necessary condition for using th~ pres~t invention.
Group~ of laser~, ~.g. t two or more ~emiconductor laæers, may be arran~ed spat~ally ~ ~heir evane~cent fields ovQrlap and effect coupling betw~en adjacent lasers. Such a spatial arrangem~nt is shown in Fig. 6, and this spatial arrang~m~nt or distribution is generally designated by :r~ference number lO. The iasexs 12 are positioned with respect to each other in the d~stri~ution lO so evanescent Pields from ea~h lase: 12 overlaps with that of at least one adjacent laser 12. ~ach laser 12 in the distributivn 10 is operated in the same longitudinal and transverse ~ode, e.g., TEM ~O,O), TEM (l,O) or TEM
In spite of coupling between lasers 1~ effected by overlapping evanescent fields, thQ a~grega~e radiation output from th~ spatial arrangement lO without additional control will, except:for a few uniqu2 ca~e~ be chaotic~
~i.e., irregular and unordered even as if, for large enough ~arrays, each of the lasers 1~ were emitting chaoti~ intensity radiation patterns. Such chaotic aggregate radiation output patterns will occur even when individual lasers 12 are operated in the fundamental TEM
(O,~) mode.
The distribution lO of lasers 12, with each operated in the same ~ongitudinal and transverse mode, provide~ an example of where the present invention can be applied for determining the magnitude of coupling between lasers 12 :
WOg4/17478 PCT~S93/0036 required for effecting a non-chaotic aggregate system output.
For each laser 12 in the spatial distribution 10 the time varyinq electric ~ield ~ar the guided mode i5 describable as E~(t~e~ , w~ere ~h~ complex a~plitude (E~(t); varies slowly compared to the optical frequency . This function i5 id~ntified as b~ing ~or the jth laser 12. Assuming ne~rest-neighbor coupling, there is an evolution ~f the ~ode ~mplitude (Ej~ and the population (Nj) in the jth laQer that is described by the following:equations~respec~iYe~y:
=iG~3~ ia)~j ~2) 'P-~-G~ , t2) whére G is~ gain,~tp ~(~lpsl is photon lifetime, rs (~
2nanosecond (ns:))::is:lifetime of the active population, P~ is:~pump~rate,~and ~K i5 coupling strength betwae~
ad~acent~lasers.~ The parameter ~ is~ known as the line width~enhance~ent~;~factor ~in~semiconductor lasers and is a~measure of carrier-density-dependent refractive index.
For operation not too far from laser threshold of uncoupled lasers,~ the: gain may be expressed ~s G(N~)=G(Nth)+g(Nj; -~N~h),~ where Nth is the aarrier density at threshold, G(Nth)~ p,~and g= ~G~N is differential gain. It shauld be~noted that if the population Nj is W094/t7478 PCT~S93/00360 ~ 3 ` ~-~ 12 -adiabatically eliminated, Eq. (1) represents a set of coupled van der Pol oscillators and is also identical to the discrete Ginzburg-Landau equation which has often been used as a model for spatiotemporal complexity.
:It is convenient t~ ~ran~form Eqs. (1) and ~2) into dimensionless form for the no~malized magn~tude (Xj) and phase (0J) o~ ~he electric ~ield, and normalized excess ~;carrier density Z~ in:th~ jth la~r ~. Th~se e~uations ~are:
~=Z~-JI f~ f-a~
-x~ ,sin~DJ I-~J) t, (3) ::
=aæ~-~[~ cos ~'0j-0J-l~
(xj lf~ cos ~ -0,~
: ~ ~
TZ~-p-Z~ Z2~) ~ ,j-1,2,..... ,N, (5~
with XO_X~1=O. ;Here the overdots signify derivatives with~:;respect to~a reduced ti~e ~/7p and we deine the ollowing varia~les and parameters:
2 s j' j 2~th~N~t~--l~, P= 2 gNth~p ~P/Pth--1~, ~ =K~p, T= c";/tp.
.
Equations ~3)-~5):represent an oscillator assembly, which : in the absence of coupling (~=O) would evol~e toward a :~ :
WO~4/17478 PCT~S93/0036 steady state with Zj=0, Xj = ~p, and arbitrary phases 0j.
: : For nonzero coupling, the output of spatial distribution lO of las~rs 12 can organize it~elf into a macroscopically coherent s~ruc~ur~ with well-defined phase relationships. Prior ~o the pr~en~ in~ention it was believed that th~ self-organizing principle ; underlying such collectlve ~avisr rQs~ d only from or~ed synchr~nization or mutual entrainm~nt. But with : : the present in~ention it is determinable that select~d ranges of values for coupling are required to achieve the ~: collective behavîor.
When working with small laser arrays, e.g. two lasers, there are two~l~v~ls of ~ynchronization involved~
The ~fiFst represents a~ guiescent state in which amplitudes X; and carrier Zj are constant in time while : phases 0j e~oIY~ linearly in time at the same xate possibly ~zero~ ~r all lasers. For weakly coupled lasers ~lOs), a stable ph~s:e-locked or quiescent state is~aGhievable in which amplitude distributions a~ross th~
distribution lO are nearly uniform. This uniform phase-locked: ætate, howeverl is not alway~ stable. Above a :critical coupling~ strength the quiescent state l~ses ta~ility th~ough;a~supercritical Hopf bifureation. It has~been thought delayed response of carriers leads to phase~ lags between~ oscillators and destroys phase locking. Dynamical variables--amplitudes, pha~es, and ~ ~ carrier d~nsities--all pulsate in time. Th~se pulsations :~ :occur in different elements of a distribution ~nd may or ~: may not be~ in time step with each other.
Typical behavior for small distributions of coupled : interacting lasers, e.g., l x 3, is chaotic for most pump :
WO94/17478 PCT~S93/0~360 ., ~ 3 ~ - 14 -currents and coupling co~stants above ~=10-3-5. This chaotic a~gregate output continues for larger distributions of Iasers in the a~sence of preferred initial seed conditions, coupling strengths and spatial distri~utions.
Now a speci~ic example ~or controlling and eliminating such chaotlc b~hav~or in small, e.g., 2 x 2, or lar~e, e.g., 160 x 160~ d~stributions is given. For semiconductor la ers spatially arrang~d in a distribution it is possible through sel~cti~n of initial conditions to obtaîn an ASPS output~across t~e entir~ di~tribution.
Other output patterns, to include PBS, are achievable through use of the invention, but hexe achieYement of ASPS will be considered.
A distribution lO of las~rs 12 that is larger than a;:2~X 2 distribution is ~onsidered. Within this larger distribution 10 a 2 X 2 subgroup oP ~our la~ers 1~ is arbitrarily selected. Lasers 12 in the diætri~ution lO, including the four:lasers 12~in the identi~ied subgroup, are: selected to have a 1 ns natural frequency. As an initial seed ~condi:tion, the four lasers 12 in the selected 2 X 2 subgroup are positioned and operated so :they are coupled by t~eir evanescent fields and form a closed path with~ maxi~um intensity occurring at approximately 250;~ps intervals around the orthogonal ne~ighborhood în a clockwise or counte~ clockwise fashion.
Adjusting positioning and operati~n parameters for the :four lasers 12 in the distribution 10 to function in such a~ manner provides MSS initial saed conditions to the distribut-ion 10. However, just selecting a 2 X 2 subgroup within the distribution lQ and operating the four lasers 12 in ~the subgroup under M5S initial seed :
W~94/17478 PCT~S93/00360 ., ~ , .
conditions doss not necessarily, even probably, result in concordant aggregate output from the distribution lO much less an ASPS aggregate output.
A set of non-lin~ar differential equations, as described above, for ea~h ~ the lasers ~2 in the distribution 10 i5 creat~d~ Be~ore calculations using these non-linear di~f~rential equation~ can be mad~, boundary conditions as dict~ted by physical paramet~rs : for lasers 12 and di~ri~ution lO must be put in the differential ~quations. Specifically th~se boundary conditions are~ the size, shape and number of chaotic systems, i.e., l~sers 12~ in distribution 10, ~ii) identification of which lasers 12 interact with each laser 12 in distribu~ion lO; ~iii3 value ranges for each physical parameter, including coupling Gonstants ~ that are achi~vable for Iasers 12 in distribution lO; and, :~ (iv) s~ecific values for each physical param~ter required to establish MSS~initlal seed conditions in the selected 2~ ~:2 subgroup.
With this description of la~ers 12 in the defined distribution lO, the non-linear differential e~uations are integrated at m~ltiple points in time and over a range of coupling constants to determine aggregate : :c~aracteristics for the distribution lQ. ~esulting aggregate characteristics can range fr~m unordered chaos to some form of concordant action or stability~ As part : of the method of this invention stability measures are ~i~ used to evaluate performance o~ initial seed conditions com~ination with laser 12 operating parameters : including coupling constant values. Such stability measures can include:
(i) Displaying results of integrating the non-, :
WOg4/17478 PCT~S93/00360 ~ - 16 -linear differential equations on a cathode ray tube (CRT) 50 aygregate output action ~an be directly viewed to determine if stability has been achi~ved, (ii~ Mapping f~ (Uj) V5 ~ (U~) where n is the nth crossinq of a Poincare s~ction, U is a sel~cted loc~l state var~able (Vi) : r~pr~sentativ~ of ag~regate output from distributi~n 10. Then dispe~sion is measured as standa~d ~eviation~ from th~ m~p ~n(Uj~ = f"
Uj~ 5tability is achieved at minimum disp~rsion.
: ~iii) Calculating the sum of absolute net diffusion or th~ ~ystem and monitorin~ ~h~se values until they rea~h a cons~ant v~lue : charact~ristic of the dynamic sy~t~m and its ~ :
param~ter~.
When stability i& achieved in~@gra~ion o~ the non-linear dif~ rential equ~tions is term~nated~ If ~tability is :: :
n~t~ achievedj the range of values ~or opera~ing parameters and couplin~ constants ~ust be changed or the initisl seed conditions must be ch~nged. Po~sibly changing both is r~guired.
A~ter it is~:determined that st~bility is achieYed, values ~or a distribution arrangement, including shape and size, are iden~ifiable from the integration r~sults along with op~rating paramekers and coupli~g constant values. Accordingly a distribution of chaotic sys~ems in Gonformity wi~h thes~ results can be built and operated, beginning with the identified initial seed conditions for those chaotic syste~s in an identified subgroup, to produce the selected corlcordant output.
'' W~94/17478 PCT~S93J00360 . ~ 3 ~
The above discussion and related illustrations of the present invention are directed primarily to preferred embodiments and practices. However, it is believed numerous chan~s and modifications in actual implementation o~ d~crihsd conc~pts will be apparent to those skill~d ln th~ art, ~nd it is cont~mplated that such chang~s and m~difications may be made without departing from the scope of the invention as defined by the fo`~lowing claims.
~:
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Claims (10)
1. A method for controlling actions of multiple interacting chaotic systems positioned in a distribution where each chaotic system in said distribution interacts with and thereby affects action from at least one other chaotic system in said distribution so that a predetermined aggregate action is achieved, comprising the steps of:
using a non-linear differential equation for each chaotic system in said distribution to describe action by said chaotic system;
determining achievable boundary condition values for operating parameters of said chaotic systems in said distribution including coupling constants;
selecting a subgroup of chaotic systems in said distribution, said subgroup including at least two chaotic systems;
inputting initial seed conditions to each non-linear differential equation describing action from said chaotic systems in said subgroup, where said initial seed conditions are a set of operating parameter values that initiate a selected aggregate output by said chaotic systems in said subgroup;
programming a computer to integrate said non-linear differential equations describing action by all chaotic systems in said distribution and using said computer to integrate said non-linear differential equations for a given value of said coupling constant;
using said computer to determine if integration of said non-linear differential equations has resulted in a stable aggregate action from said chaotic systems in said distribution;
continuing integration of said non-linear differential equations at different values for said coupling constant until stable aggregate action from said chaotic systems is achieved; and using operating parameter values that produced integration results yielding stable aggregate action for operating said chaotic systems in said distribution to produce said stable aggregate action.
using a non-linear differential equation for each chaotic system in said distribution to describe action by said chaotic system;
determining achievable boundary condition values for operating parameters of said chaotic systems in said distribution including coupling constants;
selecting a subgroup of chaotic systems in said distribution, said subgroup including at least two chaotic systems;
inputting initial seed conditions to each non-linear differential equation describing action from said chaotic systems in said subgroup, where said initial seed conditions are a set of operating parameter values that initiate a selected aggregate output by said chaotic systems in said subgroup;
programming a computer to integrate said non-linear differential equations describing action by all chaotic systems in said distribution and using said computer to integrate said non-linear differential equations for a given value of said coupling constant;
using said computer to determine if integration of said non-linear differential equations has resulted in a stable aggregate action from said chaotic systems in said distribution;
continuing integration of said non-linear differential equations at different values for said coupling constant until stable aggregate action from said chaotic systems is achieved; and using operating parameter values that produced integration results yielding stable aggregate action for operating said chaotic systems in said distribution to produce said stable aggregate action.
2. The method of claim 1 further including the step of using said boundary conditions for describing positioning of said chaotic systems along a one-dimensional linear distribution.
3. The method of claim 1 further including the step of using said boundary conditions for describing positioning of said chaotic systems about a two-dimensional area distribution.
4. The method of claim 1 further including the step of using said boundary conditions for describing positioning of said chaotic systems within a three-dimensional volume distribution.
5. The method of claim 1 further including the steps of:
selecting four chaotic systems for said subgroup with each of said selected chaotic systems positioned for interacting with and thereby affecting action from at least one other of said selected chaotic systems, and said selected chaotic systems also positioned in said subgroup to permit a circular path for interaction between said selected chaotic systems; and, using specific operating parameter values for each of said selected chaotic systems in said subgroup so said initial seed conditions with said specific operating parameter values cause action from each of said selected chaotic systems that has an approximate 90 degree greater phase difference from action from the adjacent said selected chaotic system in a clockwise path.
selecting four chaotic systems for said subgroup with each of said selected chaotic systems positioned for interacting with and thereby affecting action from at least one other of said selected chaotic systems, and said selected chaotic systems also positioned in said subgroup to permit a circular path for interaction between said selected chaotic systems; and, using specific operating parameter values for each of said selected chaotic systems in said subgroup so said initial seed conditions with said specific operating parameter values cause action from each of said selected chaotic systems that has an approximate 90 degree greater phase difference from action from the adjacent said selected chaotic system in a clockwise path.
6. The method of claim 1 further including the steps of:
selecting four chaotic systems for said subgroup with each of said selected chaotic systems positioned for interacting with and thereby affecting action from at least one other of said selected chaotic systems, and said selected chaotic systems also positioned in said subgroup to permit a circular path for interaction between said selected chaotic systems; and, using specific operating parameter values for each of said selected chaotic systems in said subgroup so said initial seed conditions with said specific operating parameter values cause action from each of said selected chaotic systems that has an approximate 90 degree smaller phase difference from action from the adjacent said selected chaotic system in a clockwise path.
selecting four chaotic systems for said subgroup with each of said selected chaotic systems positioned for interacting with and thereby affecting action from at least one other of said selected chaotic systems, and said selected chaotic systems also positioned in said subgroup to permit a circular path for interaction between said selected chaotic systems; and, using specific operating parameter values for each of said selected chaotic systems in said subgroup so said initial seed conditions with said specific operating parameter values cause action from each of said selected chaotic systems that has an approximate 90 degree smaller phase difference from action from the adjacent said selected chaotic system in a clockwise path.
7. The method of claim 1 further including the step of selecting as said initial seed conditions random uniformly distributed operating parameter values having local sate variable values centered about a steady state.
8. The method of claim 1 further including the steps of: selecting as said initial seed conditions random uniformly distributed operating parameter values; and, integrating all of said non-linear differential equations at least one thousand times without inputting values for said coupling constants.
9. The method of claim 1 further including the steps of: displaying said distribution of chaotic systems to show aggregate action after integration on a cathode ray tube (CRT); and, viewing said CRT
display to evaluate if sufficient stability is achieved.
display to evaluate if sufficient stability is achieved.
10. The method of claim 1 further including the steps of using said computer to: calculate the sum of net diffusion for said distribution of said chaotic systems; and, monitor said sum of net diffusion to determine achievement of stability as when said sum of net diffusion is constant.
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|---|---|---|---|
| EP93904505A EP0630500A4 (en) | 1993-01-19 | 1993-01-19 | Method for control of chaotic systems. |
| CA002132194A CA2132194A1 (en) | 1993-01-19 | 1993-01-19 | Method for control of chaotic systems |
| PCT/US1993/000360 WO1994017478A1 (en) | 1993-01-19 | 1993-01-19 | Method for control of chaotic systems |
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| CA002132194A CA2132194A1 (en) | 1993-01-19 | 1993-01-19 | Method for control of chaotic systems |
| PCT/US1993/000360 WO1994017478A1 (en) | 1993-01-19 | 1993-01-19 | Method for control of chaotic systems |
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| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA002132194A Abandoned CA2132194A1 (en) | 1993-01-19 | 1993-01-19 | Method for control of chaotic systems |
Country Status (2)
| Country | Link |
|---|---|
| CA (1) | CA2132194A1 (en) |
| WO (1) | WO1994017478A1 (en) |
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN104303115A (en) * | 2011-11-03 | 2015-01-21 | 牛津布鲁克斯大学 | A method of controlling a dynamic physical system that exhibits a chaotic behaviour |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| JPH10507905A (en) * | 1994-08-17 | 1998-08-04 | ジェネティック セラピー インコーポレイテッド | Retroviral vectors produced in cell lines resistant to lysis by human serum |
| WO2001065755A1 (en) * | 2000-03-03 | 2001-09-07 | Chil Min Kim | Method for synchronizing a plurality of chaotic systems and method for multichannel communication using synchronized chaotic systems |
| CN102594556A (en) * | 2011-12-09 | 2012-07-18 | 北京工业大学 | Identity authentication method based on synchronization of spatiotemporal chaos |
Family Cites Families (7)
| Publication number | Priority date | Publication date | Assignee | Title |
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| US4916635A (en) * | 1988-09-12 | 1990-04-10 | Massachusetts Institute Of Technology | Shaping command inputs to minimize unwanted dynamics |
| JP2760543B2 (en) * | 1989-02-10 | 1998-06-04 | 株式会社東芝 | Multiple feedback circuit |
| JP2730195B2 (en) * | 1989-06-30 | 1998-03-25 | 三菱電機株式会社 | Coupling vibration characteristic analyzer |
| US5191524A (en) * | 1989-09-08 | 1993-03-02 | Pincus Steven M | Approximate entropy |
| US5060947A (en) * | 1990-01-25 | 1991-10-29 | Hall Guy E | Magnetic pendulum random number selector |
| US5134685A (en) * | 1990-02-06 | 1992-07-28 | Westinghouse Electric Corp. | Neural node, a netowrk and a chaotic annealing optimization method for the network |
| US5163016A (en) * | 1990-03-06 | 1992-11-10 | At&T Bell Laboratories | Analytical development and verification of control-intensive systems |
-
1993
- 1993-01-19 CA CA002132194A patent/CA2132194A1/en not_active Abandoned
- 1993-01-19 WO PCT/US1993/000360 patent/WO1994017478A1/en not_active Application Discontinuation
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN104303115A (en) * | 2011-11-03 | 2015-01-21 | 牛津布鲁克斯大学 | A method of controlling a dynamic physical system that exhibits a chaotic behaviour |
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| Publication number | Publication date |
|---|---|
| WO1994017478A1 (en) | 1994-08-04 |
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