WO1994017478A1 - Procede pour le controle de systemes chaotiques - Google Patents

Procede pour le controle de systemes chaotiques Download PDF

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Publication number
WO1994017478A1
WO1994017478A1 PCT/US1993/000360 US9300360W WO9417478A1 WO 1994017478 A1 WO1994017478 A1 WO 1994017478A1 US 9300360 W US9300360 W US 9300360W WO 9417478 A1 WO9417478 A1 WO 9417478A1
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WIPO (PCT)
Prior art keywords
chaotic systems
distribution
chaotic
action
systems
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Application number
PCT/US1993/000360
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English (en)
Inventor
Robert R. Klevecz
James L. Bolen, Jr.
Original Assignee
City Of Hope
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
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Publication date
Application filed by City Of Hope filed Critical City Of Hope
Priority to EP93904505A priority Critical patent/EP0630500A4/fr
Priority to CA002132194A priority patent/CA2132194A1/fr
Priority to PCT/US1993/000360 priority patent/WO1994017478A1/fr
Publication of WO1994017478A1 publication Critical patent/WO1994017478A1/fr

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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]

Definitions

  • 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.
  • 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 (March 1978) 41), including levels of chemical constituents seen in the Belousov-Zhabotinski reaction (I.e.
  • 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.
  • 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.
  • individual lasers can exhibit undamped chaotic intensity output spiking behavior on the order of 100 picoseconds (ps) intervals.
  • Chaotic behavior can be spatiotemporally extended by arranging individual chaotic systems in one, two and three spatial dimensional distributions.
  • multiple chaotic systems are: (i) spatially arranged in distributions; and, (ii) interact with each other by means such as evanescent or diffusive coupling.
  • 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.
  • 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.
  • 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.
  • 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.
  • a subgroup of interacting chaotic systems from within the distribution is selected.
  • 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.
  • 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.
  • CRT cathode ray tube
  • 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;
  • MSS minimum spiral seed
  • 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;
  • ASS asynchronous periodic spiral
  • 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;
  • PBS periodic banding structure
  • Fig. 5 shows instantaneous phase values in x, y phase space for the PBS aggregate output depicted in Fig. 4;
  • Fig. 6 is a schematic for a two-dimensional array of lasers.
  • 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.
  • n l to ... infinity
  • V are local state variables
  • D v are coupling constants
  • R v (V lf V 2 , ...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.
  • MSS minimum spiral seed
  • the selected array is a 30 x 30 two-dimensional distribution
  • 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
  • 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.
  • 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.
  • 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.
  • 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.
  • PBS periodic banding structure
  • initial seed conditions 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., 10 6 iterations (100 iterations/unit time) , before initiating coupling.
  • 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 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
  • 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.
  • 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
  • G gain
  • ⁇ ( ⁇ lps) photon lifetime
  • ⁇ s ( 2nanosecond (ns) ) lifetime of the active population
  • P pump rate
  • K coupling strength between adjacent lasers.
  • the parameter ⁇ is known as the line width enhancement factor in semiconductor lasers and is a measure of carrier-density-dependent refractive index.
  • 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.
  • the output of spatial distribution 10 of lasers 12 can organize itself into a macroscopically coherent structure with well-defined phase relationships.
  • 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.
  • the first represents a quiescent state in which amplitudes X- and carrier Z, are constant in time while phases ⁇ . evolve linearly in time at the same rate (possibly zero) for all lasers.
  • 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.
  • 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.
  • a distribution 10 of lasers 12 that is larger than a 2 X 2 distribution is considered.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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:
  • 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.
  • 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.

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Abstract

L'invention concerne un procédé conçu pour déterminer l'importance des liaisons entre des systèmes chaotiques interactifs disposés dans une population qui produit une action collective concordante. Le procédé utilise un sous-groupe de systèmes chaotiques dans la population qui sont actionnés avec un jeu de valeurs d'état d'origine initial selectionné. Les équations différentielles non linéaires décrivant les actions des systèmes chaotiques sont intégrées pour déterminer les valeurs nécessaires pour la constante de liaison.
PCT/US1993/000360 1993-01-19 1993-01-19 Procede pour le controle de systemes chaotiques WO1994017478A1 (fr)

Priority Applications (3)

Application Number Priority Date Filing Date Title
EP93904505A EP0630500A4 (fr) 1993-01-19 1993-01-19 Procede pour le controle de systemes chaotiques.
CA002132194A CA2132194A1 (fr) 1993-01-19 1993-01-19 Methode pour controler les systemes chaotiques
PCT/US1993/000360 WO1994017478A1 (fr) 1993-01-19 1993-01-19 Procede pour le controle de systemes chaotiques

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CA002132194A CA2132194A1 (fr) 1993-01-19 1993-01-19 Methode pour controler les systemes chaotiques
PCT/US1993/000360 WO1994017478A1 (fr) 1993-01-19 1993-01-19 Procede pour le controle de systemes chaotiques

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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US5952225A (en) * 1994-08-17 1999-09-14 Genetic Therapy, Inc. Retroviral vectors produced by producer cell lines resistant to lysis by human serum
US7010126B1 (en) * 2000-03-03 2006-03-07 Paichai Hakdang Method for synchronizing a plurality of chaotic systems and method for multichannel communication using synchronized chaotic systems
CN102594556A (zh) * 2011-12-09 2012-07-18 北京工业大学 基于时空混沌同步的身份认证方法

Families Citing this family (1)

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GB201119036D0 (en) * 2011-11-03 2011-12-14 Univ Oxford Brookes A method of controlling a dynamic physical system

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US5045713A (en) * 1989-02-10 1991-09-03 Kabushiki Kaisha Toshiba Multi-feedback circuit apparatus
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
US5163015A (en) * 1989-06-30 1992-11-10 Mitsubishi Denki Kabushiki Kaisha Apparatus for and method of analyzing coupling characteristics
US5163016A (en) * 1990-03-06 1992-11-10 At&T Bell Laboratories Analytical development and verification of control-intensive systems
US5191524A (en) * 1989-09-08 1993-03-02 Pincus Steven M Approximate entropy

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US4916635A (en) * 1988-09-12 1990-04-10 Massachusetts Institute Of Technology Shaping command inputs to minimize unwanted dynamics
US5045713A (en) * 1989-02-10 1991-09-03 Kabushiki Kaisha Toshiba Multi-feedback circuit apparatus
US5163015A (en) * 1989-06-30 1992-11-10 Mitsubishi Denki Kabushiki Kaisha Apparatus for and method of analyzing coupling characteristics
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

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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US5952225A (en) * 1994-08-17 1999-09-14 Genetic Therapy, Inc. Retroviral vectors produced by producer cell lines resistant to lysis by human serum
US7010126B1 (en) * 2000-03-03 2006-03-07 Paichai Hakdang Method for synchronizing a plurality of chaotic systems and method for multichannel communication using synchronized chaotic systems
CN102594556A (zh) * 2011-12-09 2012-07-18 北京工业大学 基于时空混沌同步的身份认证方法

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