WO1994017478A1

WO1994017478A1 - Method for control of chaotic systems

Info

Publication number: WO1994017478A1
Application number: PCT/US1993/000360
Authority: WO
Inventors: Robert R. Klevecz; James L. Bolen, Jr.
Original assignee: City Of Hope
Priority date: 1993-01-19
Filing date: 1993-01-19
Publication date: 1994-08-04
Also published as: CA2132194A1

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

METHOD FOR CONTROL OF CHAOTIC SYSTEMS

Technical Field

The present invention relates to methods for effecting concordant action of a collection of interacting, i.e., coupled, chaotic systems organized in a distribution. More specifically, the present invention relates to methods for determining operating parameters in conjunction with selected initial conditions for controlling interacting chaotic systems organized in a distribution so as to effect concordant action.

Background Of the Invention

Chaotic systems are ubiquitous in the natural world. They are found on all scales of time and space, from: astronomical, e.g., asteroid distribution and motion (G.J. Sussman and J. Wisdom, Science 257 (1992) 56) ; through the terrestrial, e.g. mechanical pendulums; to the molecular, e.g. turbulent flow in stirred fluids (H.L. Swinney, and J.P. Gollub, "The Transition To Turbulence," Physics Today 31, No. 8 (August 1978) 41), including levels of chemical constituents seen in the Belousov-Zhabotinski reaction (I.e. Roux, "Experimental Studies of Bifurcations Leading To Chaos In The Belousov- Zhabotinski Reaction", Physica 7D (1983) 57); to the atomic, e.g., pinning site frequency and distribution in high-T_c (YBCO) superconducting films (M. Hawley, I.D. Raistrick, J.G. Berry, R.J. Houlton, "Growth Mechanism of Sputtered Films of YBa_z Cu₃0₇ Studied By Scanning Tunneling Microscopy," Reports (29 March 1991) 1587), and radiation emitters such as lasers and masers (R.A. Elliott, R.K. DeFreez, T.L. Paoli, R.O. Burnham, and W. Streifer, IEEE J. Quantum Electron. QE-21, 598 (1985) ; and even to the subatomic, e.g., chaotic quantum mechanical systems (M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics. (1990) Springer-Verlag, New York) .

For more than a century chaotic behavior in physical systems has been a recognized phenomenon. Two characteristics of chaotic behavior are that it is deterministic, i.e., for precisely selected initial conditions the resulting chaotic behavior can be predicted, but this deterministic behavior is non- periodic. Chaotic behavior was first recognized as such when simple mechanical systems were shown to have very complicated motions. Not only is such behavior exceedingly sensitive to precise values for starting or initial conditions, but chaotic behavior never settles into predictable final states via recognizable patterns.

Chaotic behavior is pervasive, as outlined above, and can be found even in the presence of deterministic periodic behavior. For example, laser light output intensities are now recognized as having an intrinsically unstable component with chaotic intensity fluctuations. This chaotic behavior exists in spite of the fact that laser outputs are coherent with exceedingly narrow band widths. Specifically, individual lasers 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. Burnham, and W. Streifer, IEEE J. Quantum Electron QE-21, 598 (1985) ; and S.S. Ward and H.G. Winful, Appl. Phys. Lett. 52, 1774 (1988)). Such behavior is unavoidably detrimental in applications where temporal stability is important. 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 situation where multiple chaotic systems are: (i) spatially arranged in distributions; and, (ii) interact with each other by means such as evanescent or diffusive coupling. As used here the term distribution is intended to be synonymous with other terms for groupings and arrangements such as array and structure. The aggregate output action from such distributions can exhibit some level of concordance or can be chaotic as if individual systems were independently functioning without interaction or coupling. Most probable aggregate actions are either spatiotemporal chaos, which often results from excessive interaction, or asynchronous chaos, which often results from insufficient interaction.

Summary of the Invention

Principal objects of this invention are to provide a method for determining how to achieve concordant aggregate action from distributions of interacting chaotic systems and to then use such control for achieving specific levels of concordant action. Distributions of interacting chaotic systems here include spatially and temporally periodic substructures, that may exist in a homogeneous media, where normal aggregate behavior is chaotic, irregular or turbulent. Concordant action 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 chaotic systems in the distribution, and by also using predetermined dimensional arrangements for the chaotic systems in distributions. Resulting distributions can be skinny bars, i.e., such skinny bars can be considered one-dimensional; or they can be flat plates, i.e., two- dimensional; or cubes, i.e., three-dimensional. Achieved concordant output actions for interacting chaotic systems controlled using the method of the invention are stable and resistant to perturbations and noise.

To use the method of the invention, dynamics for chaotic systems in the distribution need to be described using non-linear differential equations with boundary conditions inserted prior to integration of the equations. These boundary conditions are determined from both physical characteristics for the distribution, i.e. , size, shape and number of chaotic systems, and for achievable values for chaotic system operating parameters including coupling constants. Before integration is begun, a subgroup of interacting chaotic systems from within the distribution is selected. For those selected chaotic systems in the subgroup a set of operating parameter values defining an initial seed condition is input to their corresponding non-linear differential equations. Initial seed conditions define a set of operating parameter values that initiate a concordant output from chaotic systems in the subgroup. A variety of initial seed conditions is appropriate for the present invention as discussed 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 measures are used to evaluate aggregate output as determined by integration of the non-linear differential equations. These stability measures include viewing aggregate output performance on are appropriate device such as a cathode ray tube (CRT) or by use of mathematical tests as discussed below. When stability is achieved integration is terminated, and identified optimum operating values for achieving determined concordant action are used to operate chaotic systems in selected distributions.

Brief Description of Drawings

The various objectives, advantages and novel features of the present invention will become more readily apprehended from the following detailed description when taken in conjunction with the appended drawings, in which:

Fig. 1 shows in field space, using gray tones, an initial seed condition in the form of a minimum spiral seed (MSS) for a 2 x 2 array of interacting chaotic systems as a substructure in a larger planar distribution of chaotic systems;

Fig. 2 shows instantaneous phase seed conditions in x, y phase space associated with the initial seed conditions depicted in Figure 1;

Fig. 3 shows in field space, using gray tones, a concordant aggregate output in the form of an asynchronous periodic spiral (ASPS) from a two dimensional distribution of interacting chaotic systems, that can result 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 output in the form of periodic banding structure (PBS) for a two dimensional distribution of interacting chaotic systems resulting from initial seed conditions depicted in Figures 1 and 2 with high value coupling constants;

Fig. 5 shows instantaneous phase values in x, y phase space for the PBS aggregate output depicted in Fig. 4; and,

Fig. 6 is a schematic for a two-dimensional array of lasers.

Detailed Description of the Invention

The present invention provides a method for determining how to control interacting chaotic systems organized in distributions so they aggregately act in concordant fashion. All types of distributions, namely one, two and three spatial dimensions, involving action of one, two, three and even more variables for individual chaotic systems in the distributions, are within the scope of this invention. Each chaotic system organized in a distribution for the present invention, though, must be capable of interacting with at least one other chaotic system in the distribution. Using distribution parameters, e.g., spacing between chaotic systems, for controlling coupling interactions 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 organized 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 non-linear differential equations that can be of the form:

dv_i/dt=R_Vι { v₁ , v₂ V ^v n ) ' +D v τ^v? V l-

n=l to ... infinity where V, are local state variables, D_v are coupling constants, and R_v (V_lf V₂, ...V_n) are functions describing kinetics for the i th chaotic system. It is noted that it may not be necessary to describe the action of each chaotic system in the distribution with a corresponding individual non-linear differential equation if parameter values for initial seed (antipodal phase apposition) conditions capable of leading to concordant aggregate action are used with the non-linear differential equations for those chaotic systems in a selected subgroup within the distribution. Occurrence of this situation, however, will be dependent on characteristics of those chaotic systems in a selected distribution.

The next step of this invention is to select both a finite number of chaotic systems in a subgroup of the distribution and select a set of initial seed parameters for the chaotic systems in the subgroup at the time of initiation. Selected initial seed parameters are then input to their respective non-linear differential equations for the chosen chaotic systems and the entire field of all non-linear differential equations describing the distribution are used for calculating operating values. The calculations, involving integrations, can be made on a computer using known programming techniques. As the integrations proceed as an iteration process, the optimum coupling constants are determined. Determined coupling constants may be a function of distribution spacing.

An example of an appropriate initial seed can include minimum spiral seed (MSS) conditions as shown in Figures 1 and 2. Here the selected array is a 30 x 30 two-dimensional distribution, and the achieved concordant output is an asynchronous periodic spiral (ASPS) as shown in Figure 3, which shows ASPS in field space. The coupling constant for this example is 0.2 To achieve this ASPS result a central 2 x 2 subgroup of chaotic systems are set to have initial seed conditions shown in field space in Figure 1, and instantaneous phase seed conditions shown in x, y phase space as depicted in Figure 2. Specifically the selected instantaneous values for phase for those chaotic systems in the subgroup are shown as dots in relation to steady state 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 values are such that in clockwise or counter clockwise directions, they approximate values, i.e. state space variables of the non-linear dynamic system, at 90 degree phase differences. Now the resulting output, given proper coupling between chaotic systems, can be ASPS in character, which has the general property of having a constant aggregate output and is a function of the number of units in the distribution. This constant output is additionally resistant to perturbation from both external and intrinsic (or deterministic) noise which is damped. Such a result is realized because stable 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 constants, can produce a periodic banding structure (PBS) aggregate output. A property of PBS is a synchronous periodic amplitude output as shown in the field space depiction set out in Figure 4. Instantaneous phase values for this PBS output are shown by the four separate values 100, 112, 114 and 116 forming an isochron as set out in the x, y phase space depiction in Figure 5.

Other initial seed conditions that can be used include: (i) random uniformly distributed state variables, whose local state variable, V-, values are centered about a steady state; and, (ii) random uniformly distributed state variables in which transients are destroyed by running the non-linear differential equations through a large number of iterations, e.g., 10⁶ iterations (100 iterations/unit time) , before initiating coupling.

As an example for using the present invention, a two dimensional distribution of lasers is described below. Positioning of 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 identify an exclusive application of the invention. Other distributions of chaotic systems capable of interaction can also be used with the present invention. For example, even arbitrarily discrete portions of continuous chaotic systems can be used with the present invention. However, for such applications where a continuous chaotic system is divided there must also be a means for controlling magnitudes of coupling between subdivided portions. This is a necessary condition for using the present invention.

Groups of lasers, e.g., two or more semiconductor lasers, may be arranged spatially so their evanescent fields overlap and effect coupling between adjacent lasers. Such a spatial arrangement is shown in Fig. 6, and this spatial arrangement or distribution is generally designated by reference number 10. The lasers 12 are positioned with respect to each other in the distribution 10 so evanescent fields from each laser 12 overlaps with that of at least one adjacent laser 12. Each laser 12 in the distribution 10 is operated in the same longitudinal and transverse mode, e.g., TEM (0,0), TEM (1,0) or TEM

(1,1).

In spite of coupling between lasers 12 effected by overlapping evanescent fields, the aggregate radiation output from the spatial arrangement 10 without additional control will, except for a few unique cases, be chaotic, i.e., irregular and unordered even as if, for large enough arrays, each of the lasers 12 were emitting chaotic intensity radiation patterns. Such chaotic aggregate radiation output patterns will occur even when individual lasers 12 are operated in the fundamental TEM (0,0) mode.

The distribution 10 of lasers 12, with each operated in the same longitudinal and transverse mode, provides an example of where the present invention can be applied for determining the magnitude of coupling between lasers 12 required for effecting a non-chaotic aggregate system output.

For each laser 12 in the spatial distribution 10 the time varying electric field for the guided mode is describable as E_j(t)e^",t)0t, where the complex amplitude (E,(t) ) varies slowly compared to the optical frequency S)₀. This function is identified as being for the jth laser 12. Assuming nearest-neighbor coupling, there is an evolution of the mode amplitude (E_j) and the population (N ) in the jth laser that is described by the following equations respectively: dE_j

JG(N_j) --^ {l-ia) E_j (2) dt

+i (£'^₁+£₇..₁)

^ =p-_2 -G(N.) |£',l² , (2) dt τ„ ^{J J}

where G is gain, τ (~lps) is photon lifetime, τ_s ( 2nanosecond (ns) ) is lifetime of the active population, P is pump rate, and K is coupling strength between adjacent lasers. The parameter α is known as the line width enhancement 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 as G(N_j)=G(N_th)+g(Nj - N_th) , where N_th is the carrier density at threshold, G(N_th)=l/τ_p, and g= dG/θN is differential gain. It should be noted that if the population N. is 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 to transform Eqs. (1) and (2) into dimensionless form for the normalized magnitude (X_j) and phase (ø.) of the electric field, and normalized excess carrier density Z, in the jth laser 12. These equations are:

έ^aZ_j-η [ (x_j+1/X_j) cos (ø^-ø_J+1)

TZ_j=p-Z_j- ( l +2Z_j) X^ _f j=l _f 2 , . . . , N, (5)

with X=X_N+1=0. Here the overdots signify derivatives with respect to a reduced time t/τ and we define the following variables and parameters:

P~9N_thτ_p (P/P_ch-l ) ~~ =Kτ_Dι T=τ_s/τp"

Equations (3)-(5) represent an oscillator assembly, which in the absence of coupling (77=0) would evolve toward a steady state with Z-=0, X, = Vp, and arbitrary phases _j.

For nonzero coupling, the output of spatial distribution 10 of lasers 12 can organize itself into a macroscopically coherent structure with well-defined phase relationships. Prior to the present invention it was believed that the self-organizing principle underlying such collective behavior resulted only from forced synchronization or mutual entrainment. But with the present invention it is determinable that selected ranges of values for coupling are required to achieve the collective behavior.

When working with small laser arrays, e.g. two lasers, there are two levels of synchronization involved. The first represents a quiescent state in which amplitudes X- and carrier Z, are constant in time while phases ø. evolve linearly in time at the same rate (possibly zero) for all lasers. For weakly coupled lasers (η≤lO^'5) , a stable phase-locked or quiescent state is achievable in which amplitude distributions across the distribution 10 are nearly uniform. This uniform phase- locked state, however, is not always stable. Above a critical coupling strength the quiescent state loses stability through a supercritical Hopf bifurcation. It has been thought delayed response of carriers leads to phase lags between oscillators and destroys phase locking. Dynamical variables—amplitudes, phases, and carrier densities—all pulsate in time. These pulsations occur in different elements of a distribution and may or may not be in time step with each other.

Typical behavior for small distributions of coupled interacting lasers, e.g., 1 x 3, is chaotic for most pump currents and coupling constants above ry=10^"3-⁵. This chaotic aggregate output continues for larger distributions of lasers in the absence of preferred initial seed conditions, coupling strengths and spatial distributions.

Now a specific example for controlling and eliminating such chaotic behavior in small, e.g., 2 x 2, or large, e.g., 160 x 160, distributions is given. For semiconductor lasers spatially arranged in a distribution it is possible through selection of initial conditions to obtain an ASPS output across the entire distribution. Other output patterns, to include PBS, are achievable through use of the invention, but here achievement of ASPS will be considered.

A distribution 10 of lasers 12 that is larger than a 2 X 2 distribution is considered. Within this larger distribution 10 a 2 X 2 subgroup of four lasers 12 is arbitrarily selected. Lasers 12 in the distribution 10, including the four lasers 12 in the identified subgroup, are selected to have a 1 ns natural frequency. As an initial seed condition, the four lasers 12 in the selected 2 X 2 subgroup are positioned and operated so they are coupled by their evanescent fields and form a closed path with maximum intensity occurring at approximately 250 ps intervals around the orthogonal neighborhood in a clockwise or counter clockwise fashion. Adjusting positioning and operation parameters for the four lasers 12 in the distribution 10 to function in such a manner provides MSS initial seed conditions to the distribution 10. However, just selecting a 2 X 2 subgroup within the distribution 10 and operating the four lasers 12 in the subgroup under MSS initial seed conditions does not necessarily, even probably, result in concordant aggregate output from the distribution 10 much less an ASPS aggregate output.

A set of non-linear differential equations, as described above, for each of the lasers 12 in the distribution 10 is created. Before calculations using these non-linear differential equations can be made, boundary conditions as dictated by physical parameters for lasers 12 and distribution 10 must be put in the differential equations. Specifically these boundary conditions are: (i) the size, shape and number of chaotic systems, i.e., lasers 12, in distribution 10; (ii) identification of which lasers 12 interact with each laser 12 in distribution 10; (iii) value ranges for each physical parameter, including coupling constants, that are achievable for lasers 12 in distribution 10; and, (iv) specific values for each physical parameter required to establish MSS initial seed conditions in the selected 2 X 2 subgroup.

With this description of lasers 12 in the defined distribution 10, the non-linear differential equations are integrated at multiple points in time and over a range of coupling constants to determine aggregate characteristics for the distribution 10. Resulting aggregate characteristics can range from unordered chaos to some form of concordant action or stability. As part of the method of this invention stability measures are used to evaluate performance of initial seed conditions in combination with laser 12 operating parameters including coupling constant values. Such stability measures can include:

(i) Displaying results of integrating the non- linear differential equations on a cathode ray tube (CRT) so aggregate output action can be directly viewed to determine if stability has been achieved, (ii) Mapping fⁿ (U_j) vs fⁿ (U_j+1) where n is the n^th crossing of a Poincare section, U is a selected local state variable (V_{) representative of aggregate output from distribution 10. Then dispersion is measured as standard deviations from the map fⁿ(U_j) = fⁿ (U_j+1) . Stability is achieved at minimum dispersion, (iii) Calculating the sum of absolute net diffusion for the system and monitoring these values until they reach a constant value characteristic of the dynamic system and its parameters. When stability is achieved integration of the non-linear differential equations is terminated. If stability is not achieved, the range of values for operating parameters and coupling constants must be changed or the initial seed conditions must be changed. Possibly changing both is required.

After it is determined that stability is achieved, values for a distribution arrangement, including shape and size, are identifiable from the integration results along with operating parameters and coupling constant values. Accordingly a distribution of chaotic systems in conformity with these results can be built and operated, beginning with the identified initial seed conditions for those chaotic systems in an identified subgroup, to produce the selected concordant output. The above discussion and related illustrations of the present invention are directed primarily to preferred embodiments and practices. However, it is believed numerous changes and modifications in actual implementation of described concepts will be apparent to those skilled in the art, and it is contemplated that such changes and modifications may be made without departing from the scope of the invention as defined by the following claims.

Claims

WE CLAIM:

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.

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.

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.

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 state 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.

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.