Cascading synchronized chaotic systems

Abstract

A cascaded synchronized nonlinear system includes a nonlinear transmitter having stable first and second subparts. The first subpart produces a first transmitter signal for driving the second subpart and the second subpart produces a second transmitter signal for driving the first subpart. The nonlinear transmitter transmits the second transmitter signal to a nonlinear cascaded receiver. The receiver, being for producing an output signal in synchronization with the second transmitter signal, includes a first stage (a duplicate of the first subpart) responsive to the second transmitter signal for producing a first receiver signal. The receiver further includes a second stage (a duplicate of the second subpart) responsive to the first receiver signal for producing the output signal. The cascaded synchronized nonlinear system can be used in an information transfer system. The transmitter, responsive to an information signal produces a drive signal for transmission to the receiver. An error detector compares the drive signal and the output signal produced by the receiver to produce an error signal indicative of the information contained in the information signal.

Claims

What is claimed is:

1. A cascaded synchronized nonlinear electrical system comprising:

(a) a nonlinear transmitter having stable first and second subparts, said first subpart producing a first transmitter signal for driving said second subpart, said second subpart producing a second transmitter signal for driving said first subpart, said nonlinear transmitter being for transmission of said second transmitter signal; and

(b) a nonlinear cascaded receiver responsive to said second transmitter signal transmitted by said transmitter, said receiver being for producing an output signal in synchronization with said second transmitter signal, said receiver comprising a first stage responsive to said second transmitter signal for producing a first receiver signal, said first stage being a duplicate of said first subpart, and said receiver further comprising a second stage responsive to said first receiver signal for producing said output signal, said second stage being a duplicate of said second subpart.

2. The system of claim 1 wherein said first stage produces said first receiver signal in synchronization with said first transmitter signal.

3. The system of claim 1 wherein said transmitter comprises a non-Lorenz system.

4. The system of claim 3 wherein said transmitter comprises a non-Chua circuit.

5. The system of claim 1 wherein said transmitter comprises a non-Chua circuit.

6. The system of claim 1, wherein at least part of said first subpart of element (a) is external to said second subpart of element (a), and at least part of said second subpart of element (a) is external to said first subpart of element (a).

7. An electrical information transfer system comprising:

(a) a nonlinear transmitter responsive to an information signal for producing a drive signal, said transmitter having stable first and second subparts, said first subpart producing a first transmitter signal for driving said second subpart, said second subpart producing a second transmitter signal for driving said first subpart, said nonlinear transmitter being responsive to said second transmitter signal for transmission of said drive signal; and

(b) a nonlinear cascaded receiver responsive to said drive signal, said receiver comprising a first stage responsive to said drive signal for producing a first receiver signal, said first stage being a duplicate of said first subpart, and said receiver further comprising a second stage responsive to said first receiver signal for producing an output signal, said second stage being a duplicate of said second subpart; and

(c) an error detector coupled to said receiver being responsive to said drive signal and to said output signal for producing an error signal indicative of the information contained in said information signal.

8. The system of claim 7 wherein said transmitter comprises a non-Lorenz system.

9. The system of claim 8 wherein said transmitter comprises a non-Chua circuit.

10. The system of claim 7 wherein said transmitter comprises a non-Chua circuit.

11. The system of claim 7 wherein said transmitter comprises a modulator responsive to said second transmitter signal and to said information signal for producing said drive signal.

12. The system of claim 11 wherein said modulator is a nonlinear modulator.

13. The system of claim 7 wherein said transmitter further comprises means responsive to said information signal for varying the value of a transmitter parameter.

14. The system of claim 13 further comprising means responsive to said error signal for varying the value of a receiver parameter corresponding to said transmitter parameter so that the variation of said receiver parameter is in synchronization with the variation of said transmitter parameter.

15. A synchronized electrical system, comprising:

a nonlinear dynamic drive system for producing a drive signal and transmitting the drive signal, the drive system having at least two stable subsystems; and

a cascaded nonlinear driven system responsive to said drive system to receive the drive signal and comprising cascade connected subsystems of the driven system, said driven system subsystems being substantially duplicates of said at least two drive system subsystems, said at least two driven subsystems comprising a first subsystem responsive to the drive signal and a second subsystem for producing an output signal in synchronization with the drive signal.

16. A system as recited in claim 15, wherein said drive system produces a first signal to be synchronized and said driven system reproduces the first signal.

17. The system of claim 15, wherein said driven system comprises a non-Lorenz device.

18. A synchronized electrical system, comprising:

a nonlinear dynamic drive system for producing a dynamic drive signal and having at least a first and a second interdependent and stable subsystem for producing a first signal to be synchronized; and

a nonlinear dynamic driven system coupled to said drive system and receiving the drive signal, said driven system comprising:

a first driven subsystem comprising a duplicate of said first subsystem of said drive system, said first driven subsystem being interdependent to synchronize and reproduce the first signal; and

a second subsystem comprising a duplicate of said second subsystem of said drive system, said second driven subsystem being connected to the first driven subsystem and being responsive thereto to receive the reproduced signal and being interdependent to synchronize and reproduce the drive signal.

19. The system of claim 18, further comprising a comparator for comparing the drive signal and the reproduced drive signal to confirm synchronization.

20. A method of synchronizing, comprising the steps of:

(a) electrically producing a drive value and a first value to be matched using a differential equation model:

(b) broadcasting the drive value to a receiver; and

(c) electrically producing in the receiver an output value and a receiver first value in synchronization with the drive value and the first value, respectively, by using the drive value received and at least a pair of stable differential equation model subsystems which are subsets of the equation model, wherein said pair of differential equation model subsystems includes at least one common variable, said pair of differential equation model subsystems comprising a first subsystem dependent on the drive value for producing the receiver first value and a second subsystem dependent on the receiver first value for producing the output value.

21. A method as recited in claim 20, further comprising comparing the drive value and the reproduced drive value to confirm matching of the first value and the reproduced first value.

22. An electrical information transfer system, comprising:

a transmitter having at least two stable subsystems for producing a dynamic drive signal carrying an information signal; and

a cascaded receiver coupled to said transmitter to receive the drive signal and comprising:

cascade connected subsystems of the receiver, said receiver subsystems being substantially duplicates of said at least two transmitter subsystems for producing a reproduced signal comprising the drive signal with the information signal removed; and

a combining circuit for combining the dynamic drive signal and the reproduced signal to reproduce the information signal.

23. The system of claim 22, wherein said combining circuit comprises a nonlinear device.

24. A system as recited in claim 22, wherein said transmitter comprises a drive system producing a drive signal and an adder combining the drive signal and the information signal to produce the dynamic drive signal carrying the information signal.

25. A system as recited in claim 22, wherein said combining circuit is an error detector.

26. The system of claim 22, wherein said transmitter comprises a drive system for producing a drive signal and a nonlinear combining circuit for combining the drive signal and the information signal to produce the dynamic drive signal.

27. A method of transferring information, comprising the steps of:

(a) electrically producing drive value using a differential equation model;

(b) combining the drive value with an information value to produce a broadcast value;

(c) broadcasting the broadcast value to a receiver;

(d) electrically reproducing the drive value in the receiver from the broadcast value and at least a pair of stable differential equation model subsystems which includes subsets of the equation model, wherein said pair of differential equation model subsystems includes at least one common variable; and

(e) combining the reproduced drive value with the broadcast value to reproduce the information value.

28. A method as recited in claim 27, wherein step (b) comprises adding the drive value and the information value and step (e) comprises subtracting the reproduced drive value from the broadcast value.

29. An electrical information transfer system, comprising:

a transmitter including a parameter and having at least two stable subsystems, said transmitter being for producing a dynamic drive signal carrying an information signal produced by parameter variations; and

a cascaded receiver coupled to said transmitter to receive the drive signal and comprising:

cascade connected subsystems of the receiver, said receiver subsystems being substantially duplicates of said at least two transmitter subsystems for producing a reproduced signal comprising the drive signal with the information signal removed; and

a combining circuit combining the drive signal and the reproduced signal to produce a parameter difference signal indicating variations in the parameter.

30. A system as recited in claim 29, wherein the receiver includes a fixed parameter.

31. A system as recited in claim 30, wherein said combining circuit comprises an error detector.

32. A method of information transfer, comprising the steps of:

(a) electrically producing a drive value using a differential equation model with a varied equation parameter varied responsive to an information value;

(b) broadcasting the drive value to a receiver;

(c) electrically reproducing the drive value in the receiver from the broadcast drive value and at least a pair of stable differential equation model subsystems which are subsets of the equation model, wherein said pair of differential equation model subsystems includes at least one common variable with the parameter fixed; and

(d) combining the reproduced drive value with the broadcast drive value to produce a parameter change value corresponding to the information value.

33. A method as recited in claim 32, wherein step (a) comprises changing the parameter between two values and the parameter change value changes between a pair of values.

34. An electrical information transfer system, comprising:

a transmitter including a parameter having at least two stable subsystems for producing a dynamic drive signal carrying an information signal produced by parameter variation; and

a cascaded receiver coupled to said transmitter to receive the drive signal and comprising:

cascade connected subsystems of the receiver, including the parameter said receiver subsystems being substantially duplicates of said at least two transmitter subsystems and being for producing a reproduced signal comprising the drive signal with the information signal removed;

a combining circuit combining the drive signal and the reproduced signal to produce a parameter error signal; and

a parameter change circuit changing the parameter of the receiver to match the parameter variation of said transmitter to reproduce the information signal responsive to the parameter error signal.

35. A system as recited in claim 34, wherein said combining circuit comprises an error detection circuit and said parameter change circuit performs stepwise matching of the parameter variation.

36. A method of transferring information, comprising the steps of:

(a) electrically producing a drive value using a source differential equation model with a varied equation parameter varied responsive to an information value;

(b) broadcasting the drive value to a receiver;

(c) electrically reproducing the drive value in the receiver from the broadcast value and at least a pair of stable differential equation model subsystems which are subsets of the equation model, wherein said pair of differential equation model subsystems includes at least one common variable and which includes the parameter;

(d) combining the reproduced drive value with the broadcast value to produce a parameter error; and

(e) changing the parameter of the differential equation subsystems responsive to the parameter error.

Description

CROSS REFERENCE TO RELATED PATENTS AND APPLICATIONS

This application is related to commonly assigned U.S. patent application Ser. No. 07/656,330 filed Feb. 19, 1991 (now U.S. Pat. No. 5,245,660) by the same inventors and having Navy Case no. 72,593. This application is also related to commonly assigned U.S. application Ser. No. 07/880,789 filed May 11, 1992 by the same inventors and having Navy case no. 73,912. Both U.S. Pat. No. 5,245,660 and U.S. application Ser. No. 07/880,789 are incorporated herein by reference.

1. Field Of The Invention

The present invention relates generally to physical systems with dynamical characteristics and, in particular, to systems for producing synchronized signals which are capable of information transfer.

2. Background of the Invention

The design of most man-made mechanical and electrical systems assumes that the systems exhibit linear behavior (stationary) or simple nonlinear behavior (cyclic). In recent years there has been an increasing understanding of a more complex form of behavior, known as chaos, which is now recognized to be generic to most nonlinear systems. Systems evolving chaotically (chaotic systems) display a sensitivity to initial conditions, such that two substantially identical chaotic systems started with slightly different initial conditions (state variable values) will quickly evolve to values which are vastly different and become totally uncorrelated, even though the overall patterns of behavior will remain the same. This makes chaotic systems nonperiodic (there are no cycles of repetition whatsoever), unpredictable over long times, and thus such systems are impossible to synchronize by conventional methods. Y. S. Tang et al., "Synchronization and Chaos," IEEE Transactions of Circuits and Systems, Vol. CAS-30, No. 9, pp. 620-626 (September 1983) discusses the relationship between synchronization and chaotic systems in which selected parameters are outside some range required for synchronization.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide systems for producing synchronized signals, and particularly nonlinear dynamical systems.

It is another object of the present invention to provide communication systems for encryption utilizing synchronized nonlinear sending and receiving circuits.

It is a further object of the present invention to provide improved control devices which rely on wide-frequency band synchronized signals.

It is an object of the present invention to provide synchronizable systems in which only a single drive signal is transmitted between systems, multiple synchronized signals can be produced and in which the drive signal can be reproduced to confirm synchronization.

It is an object of the present invention to provide synchronizable systems in which the total dimension of the synchronized systems or the number of elements can be the same.

It is also an object of the present invention to provide synchronizable chaotic systems using cascaded subsystems.

It is a further object of the present invention to transmit information using cascaded synchronizable chaotic systems.

It is still another object of the present invention to transmit information using cascaded synchronizable chaotic systems using parameter changes to transmit information.

The above objects can be accomplished by two types of systems: single stage systems and cascaded systems. The difference in single stage and cascaded systems lies primarily on the receiver side. In single stage systems only a single subsystem of the transmitter is reproduced. In cascaded systems the receiver is cascaded in that the receiver includes two or more cascade connected subsystems of the transmitter. In both types of systems the synchronizable chaotic transmitter and receiver systems can be at a great distance from each other.

In single stage systems transmitter and receiver are synchronized by transmitting a drive signal produced by the chaotic system in the transmitter, to a receiver which is a duplicate of part of the transmitter.

To transmit information the transmitter transmits information by adding an information signal to a primary signal and transmitting the modified primary signal along with the drive signal to the receiver. The receiver determines the information signal by using the drive signal.

The above-described system can be improved to remove the need for transmitting different drive and synchronized signals by cascade connecting subsystems of the chaotic system in the receiver.

A cascaded synchronized nonlinear system includes a nonlinear transmitter having stable first and second subparts. The first subpart produces a first transmitter signal for driving the second subpart and the second subpart produces a second transmitter signal for driving the first subpart. The nonlinear transmitter transmits the second transmitter signal to a nonlinear cascaded receiver. The receiver, being for producing an output signal in synchronization with the second transmitter signal, comprises a first stage (a duplicate of the first subpart) responsive to the second transmitter signal for producing a first receiver signal. The receiver further comprises a second stage (a duplicate of the second subpart) responsive to the first receiver signal for producing the output signal.

The cascaded synchronized nonlinear system can be used in an information transfer system. The transmitter, responsive to an information signal produces a drive signal for transmission to the receiver. An error detector compares the drive signal and the output signal produced by the receiver to produce an error signal indicative of the information contained in the information signal.

These and other objects, features and advantages of the present invention are described in or apparent from the following detailed description of preferred embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

The preferred embodiments will be described with reference to the drawings, in which like elements have been denoted throughout by like reference numerals, and wherein:

FIG. 1 is a general block diagram of a nonlinear dynamical physical system according to the present invention;

FIG. 2 is a schematic circuit diagram of a synchronized chaotic circuit system constructed in accordance with the present invention;

FIG. 3 is a first embodiment of an encrypted communication system constructed in accordance with the present invention which uses the system of FIG. 1;

FIG. 4 is a second embodiment of an encrypted communication system constructed in accordance with the present invention which uses the system of FIG. 1; and

FIG. 5 is a third embodiment of a communication system constructed in accordance with the present invention which the system of FIG. 1;

FIG. 6 is a general block diagram of an embodiment according to the present invention;

FIG. 7 is a general block diagram of an application of the present invention to encryption;

FIG. 8 is a general block diagram of another application of the present invention to encryption.

FIGS. 9A and 9B illustrate a cascaded synchronized system with two stages;

FIG. 10 depicts a cascaded system with three stages allowing multiple signals to be synchronized and synchronization to be verified;

FIG. 11 depicts an embodiment of a two stage cascaded synchronization system;

FIGS. 12-15 illustrate the circuit details of the FIG. 11 embodiment;

FIGS. 16 and 17 are flow charts of two software versions of the FIG. 11 embodiment;

FIG. 18 illustrates a two stage cascaded information transfer system;

FIG. 19 illustrates a two stage cascaded information transfer system using parameter variation;

FIG. 20 illustrates the circuit details of the error detector 960 of FIG. 18;

FIGS. 21A and B are flowcharts of a software version of FIG. 20;

FIG. 22 illustrates a second parameter variation system;

FIGS. 23-26 depict the circuit details of the FIG. 22 embodiment; and

FIGS. 27A and B depict the FIG. 22 embodiment implemented in software.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

All physical systems can be described by state variables. For example, a billiard game can be described by the position and the velocity of a ball at any instant of time; and an electronic circuit can be described by all of its currents and voltages at a particular time. This invention is a tangible system which can be of any form. The state variables and associated signals can be, as further examples, pressure or other force, temperature, concentration, population, or electro-magnetic field components. The evolution of a physical system depends on the dynamical relations between the state variables, which are usually expressed as functional relations between the rates of change of the variables. Thus, most but not all physical systems are describable in terms of ordinary differential equations (ODEs). Mathematical models of chaotic systems often involve two types of systems: flows and iterative maps. The former evolve as solutions of differential equations, and the latter evolve in discrete steps, such as by difference equations. For example, seasonal measurements of populations can be modeled as iterative maps. Cf., Eckmann et al., Rev. Mod. Phys., Vol. 57, pp. 617-618, 619 (1985). Some iterative maps could be considered as numerical solutions to differential equations. Solution or approximate solution of these equations, such as approximate, numerical, or analytical solution, provides information about the qualitative and quantitative behavior of the system defined by the equations.

As used herein, the synchronization of two or more evolving state variables of a physical system means the process by which the variables converge toward the same or linearly related but changing set of values. Thus, if one synchronized variable changes by a certain amount, the change of the other synchronized variable will also approach a linear function of the same amount. Graphically, the plot of the synchronized variables against each other as they evolve over time would approach a straight line.

Referring to FIG. 1, an n-dimensional autonomous nonlinear dynamical primary system 9 can be arbitrarily divided, as shown, into first and second parts or subsystems 12 and 14, each of which subsystems is also a nonlinear dynamical system. Primary system 9 and more specifically, subsystem 14, has output signal S.sub.o.

The following discussion involves mathematical modeling of the system 10 in terms of solutions to differential equations and provides theoretical support for the invention. However, it is not necessary in practicing this invention that system 10 be susceptible to such modeling. For example, as stated earlier, some iterated maps cannot be modeled as solutions to differential equations, and yet this invention encompasses systems evolving according to iterated maps. As a further example, it is impractical to accurately model an ideal gas by individually considering the position and momentum of every molecule because of the vast number of molecules and variables involved.

This discussion about mathematical modeling is in two parts to correspond to two sources of difficulty in synchronizing signals: instability within a single system (chaos) and instability between two systems (structural instability). It is understood that both discussions apply to this invention and neither part should be read separately as limiting the practice of this invention.

A system with extreme sensitivity to initial conditions is considered chaotic. The same chaotic system started at infinitesimally different initial conditions may reach significantly different states after a period of time. As known to persons skilled in the art and discussed further, for example, in Wolf et al., Determining Lyapunov Exponents from a Time Series, Physica, Vol. 16D, p. 285 et seq. (1985), Lyapunov exponents (also known in the art as "characteristic exponents") measure this divergence. A system will have a complete set (or spectrum) of Lyapunov exponents, each of which is the average exponential rate of convergence (if negative) or divergence (if positive) of nearby orbits in phase space as expressed in terms of appropriate variables and components. If all the Lyapunov exponents are negative, then the same system started with slightly different initial conditions will converge (exponentially) over time to the same values, which values may vary over time. On the other hand, if at least one of the Lyapunov exponents is positive, then the same system started with slightly different initial conditions will not converge, and the system behaves chaotically. It is also known by persons skilled in the art that "in almost all real systems there exist ranges of parameters or initial conditions for which the system turns out to be a system with chaos . . . " Chernikov et al., Chaos: How Regular Can It Be?, 27 Phys. Today 27, 29 (November 1988).

Primary system 9 can be described by the ODE

du(t)/dt=f{u(t)} or u=f(u) (1)

where u(t) are the n-dimensional state variables.

Defined in terms of the state variables v and w for subsystems 12 and 14, respectively, where u=(v,w), the ODEs for subsystems 12 and 14 are, respectively:

v=g(v,w)

w=h(v,w) (2)

where v and w are m and n-m dimensional, respectively, that is, where v=(u.sub.1, . . . , u.sub.m), g=(f.sub.1 (u) . . . , f.sub.m (u)), w=(u.sub.m+1, . . . , u.sub.n) and h=(f.sub.m+1 (u), . . . , f.sub.n (u)).

The division of primary system 9 into subsystems 12 and 14 is truly arbitrary since the reordering of the u.sub.i variables before assigning them to v, w, g and h is allowed.

If a new subsystem 16 identical to subsystem 14 is added to primary system 9, thereby forming system 10, then substituting the variables v for the corresponding variables in the function h augments equations (2) for the new three-subsystem system 10 as follows:

v=g(v,w)

w=h(v,w)

w'=h(v,w'). (3)

Subsystem 16 has output signal S.sub.o '.

The w and w' subsystems (subsystems 14 and 16) will only synchronize if .DELTA.w.fwdarw.O as T.fwdarw..infin., where .DELTA.w=w'-w.

The rate of change of .DELTA.w (for small .DELTA.w) is:

.DELTA.w=d.DELTA.w/dt=h(v,w')-h(v,w)=D.sub.W h(v,w).DELTA.w+W;(4)

where D.sub.W h(v,w) is the Jacobian of the w subsystem vector field with respect to w only, that is: an (n-m).times.(n-m) linear operator (matrix)

(D.sub.w h).sub.ij =.differential.h.sub.i /.differential.w.sub.j(5)

for (m+1).ltoreq.i.ltoreq.n and 1.ltoreq.j.ltoreq.(n-m), and where W is a nonlinear operator. When Equation 4 is divided by .vertline..DELTA.w(0).vertline., and .xi.=.DELTA.w(t)/.DELTA.w(0), an equation for the rate of change (the growth or shrinkage) of the unit displacement (n-m) dimensional vector, .xi., is obtained. In the infinitesimal limit, the nonlinear operator vanishes and this leads to the variational equation for the subsystem

d.xi./dt=D.sub.W h(v(t), w(t)).xi.. (6)

The behavior of this equation or its matrix version, using the usual fundamental matrix approach, depends on the Lyapunov exponents of the w subsystem. These are hereinafter referred to as sub-Lyapunov exponents to distinguish them from the full Lyapunov spectrum of the (v,w)=(u) system. Since the w subsystem 14 is driven by the v subsystem 12, the sub-Lyapunov exponents of the w subsystem 14 are dependent on the m dimensional v variable. If at least one of the sub-Lyapunov exponents is positive, the unit displacement vector .xi. will grow without bounds and synchronization will not take place. Accordingly, the sub-systems 14 and 16 (w and w') will synchronize only if the sub-Lyapunov exponents are all negative. This principle provides a criterion in terms of computable quantities (the sub-Lyapunov exponents) that is used to design synchronizing systems in accordance with the present invention.

The v=(v.sub.1, . . . , v.sub.m) components (subsystem 12) can be viewed more broadly as driving variables and the w'=(w'.sub.m+1, . . . , w'.sub.n) components (subsystem 16) as responding variables. The drive system 9 (v,w) can be viewed as generating at least one drive signal S.sub.d, in the formula v(t), which is applied to the response systems w and w' (subsystems 14 and 16, respectively) to synchronize the drive system and the response system outputs. This is the approach taken in accordance with the present invention to provide synchronized nonlinear dynamical systems.

In practicing this invention, the above discussion applies to identical subsystems 14 and 16. This might be achievable, for example, in digital systems. In such systems 10, the signals S.sub.o and S.sub.o ' may each be chaotic because the system 9 might be chaotic. They may differ because of different initial conditions in subsystems 14 and 16. However, they will approach each other (.DELTA.w.fwdarw.0) because systems 14 and 16 have all negative sub-Lyapunov exponents when considered as driven by the same at least one drive signal S.sub.d. As used herein, a system, e.g. system 14 or 16, is "stable" with respect to a drive signal if the system has all negative sub-Lyapunov exponents with respect to that drive signal.

In most physical systems, subsystems 14 and 16 are not identical. For example, two electrical components with the same specifications typically do not have identical characteristics. The following explanation based on mathematical modeling shows that the signals S.sub.o and S.sub.o ' will nevertheless be synchronized if both subsystems 14 and 16 have negative sub-Lyapunov exponents. According to this mathematical model, the synchronization is affected by differences in parameters between the w and w' systems which are found in real-life applications. Let .mu. be a vector of the parameters of the w subsystem (subsystem 14) and .mu.' of the w' subsystem (subsystem 16), so that h=h(v,w,.mu.), for example. If the w subsystem were one-dimensional, then for small .DELTA.w and small .DELTA..mu.=.mu.'-.mu.:

.DELTA.w.apprxeq.h.sub.W .DELTA.w+h.sub..mu. .DELTA..mu. (7)

where h.sub.W and h.sub..mu. are the partial derivatives of h with respect to w and .mu., respectively. Roughly, if h.sub.W and h.sub..mu. are nearly constant in time, the solution of this equation will follow the formula ##EQU1## If h.sub.W <0, the difference between w and w' will level off at some constant value and the systems will be synchronized. Although this is a simple one dimensional approximation, it turns out to be the case for all systems that have been investigated numerically, even when the differences in parameters are rather large (.about.10-20%). This is also the case in the exemplary electronic synchronizing circuit described in more detail hereinbelow. Furthermore, it can be established on a mathematical basis that the small changes in parameters only lead to proportionally small degradations of synchronization, which approach a constant value. See Pecora et al., "Driving systems with chaotic signals", Physical Review A, Vol. 44, No. 4, Aug. 15, 1991, pages 2374-2383.

Since m-dimensional variable v may be dependent on (n-m)-dimensional variable w, there may be feedback from subsystem 14 to subsystem 12. As shown in FIG. 6, a response part 15 of subsystem 14 may produce a feedback signal S.sub.F responsive to the m-dimensional driving variable v, and a drive part 17 of subsystem 12 may respond to the feedback signal S.sub.F to produce the at least one drive signal S.sub.d.

As shown in FIG. 6, subsystems 14 and 16 need not be driven by the same at least one drive signal S.sub.d but could be driven by at least one input signals S.sub.I and S.sub.I ' responsive to the at least one drive signal S.sub.d. System 10 could have primary and secondary means 18 and 19, respectively, coupled to subsystem 12 and responsive to the at least one drive signal S.sub.d for generating input signals S.sub.I and S.sub.I ', respectively If these primary and secondary means 18 and 19, respectively are linearly responsive to the at least one drive signal S.sub.d, then the above mathematical analysis would continue to apply since linear transformations do not affect the signs of the sub-Lyapunov exponents.

In accordance with the present invention, in order to develop electrical circuits, for example, which have chaotic dynamics, but which will synchronize, a nonlinear dynamical circuit (the driver subsystem) is duplicated (to form a response subsystem). A selected portion of the response circuit is removed, and all broken connections are connected to voltages produced at their counterparts in the drive circuit. These driving voltages constitute the at least one drive signal S.sub.d shown in FIG. 1, and advantageously are connected to the response circuit via a buffer amplifier to ensure that the drive circuit is not affected by the connection to the response circuit, i.e., it remains autonomous.

As a specific example, FIG. 2 shows an electrical circuit system 20 constructed in accordance with the present invention which has two synchronized nonlinear dynamical subsystems, a drive circuit 22 and a response circuit 16. Circuits 22 and 16 correspond to the u and w' systems, respectively, discussed above.

Drive circuit 22 comprises a hysteretic circuit formed by a differential amplifier 30, resistors 42, 44, 46, 48, 50 and 52; potentiometer 74; capacitor 76; and diodes 82 and 84 connected as shown; and an unstable oscillator circuit formed by differential amplifiers 32, 34, 36, 38 and 40; resistors 58, 60, 62, 64, 66, 68, 70 and 72; and capacitors 78 and 80 connected as shown. In an experimental implementation of circuit system 20 which has been successfully tested, amplifiers 30-40 were MM741 operational amplifiers, and diodes 82 and 84 were 1N4739A diodes. Component values for the resistors and capacitors which were used are set forth in the following table:

    ______________________________________
    Resistor 42 = 10 K.OMEGA.
                     Resistor 64 = 150 K.OMEGA.
    Resistor 46 = 10 K.OMEGA.
                     Resistor 66 = 150 K.OMEGA.
    Resistor 48 = 20 K.OMEGA.
                     Resistor 68 = 330 K.OMEGA.
    Resistor 50 = 100 K.OMEGA.
                     Resistor 70 = 100 K.OMEGA.
    Resistor 52 = 50 K.OMEGA.
                     Resistor 72 = 100 K.OMEGA.
    Resistor 54 = 3 K.OMEGA.
                     Potentiometer 74 = 10 k.OMEGA.
    Resistor 56 = 20 K.OMEGA.
                     Capacitor 76 = 0.01 .mu.F
    Resistor 58 = 100 K.OMEGA.
                     Capacitor 78 = 0.01 .mu.F
    Resistor 60 = 100 K.OMEGA.
                     Capacitor 80 = 0.001 .mu.F
    Resistor 62 = 220 K.OMEGA.
    ______________________________________

    ______________________________________
    Resistor R1 = 100 k.OMEGA.
                       Resistor R11 = 221 k.OMEGA.
    Resistor R2 = 100 k.OMEGA.
                       Resistor R12 = R.sub.v
    Resistor R3 = 100 k.OMEGA.
                       Resistor R13 = 100 k.OMEGA.
    Resistor R4 = 100 k.OMEGA.
                       Resistor R14 = 200 k.OMEGA.
    Resistor R5 = 68.2 k.OMEGA.
                       Resistor R15 = 100 k.OMEGA.
    Resistor R6 = 100 k.OMEGA.
                       Resistor R16 = 100 k.OMEGA.
    Resistor R7 = 100 k.OMEGA.
                       Resistor R17 = 100 k.OMEGA.
    Resistor R8 = 68.2 k.OMEGA.
                       Resistor R18 = 100 .OMEGA.
    Resistor R9 = 1 M.OMEGA.
    Resistor R10 = 100 k.OMEGA.
    Capacitor C1 = 910 pf
                       Capacitor C3 = 910 pf
    Capacitor C2 = 910 pf
                       Capacitor C4 = 910 pf.
    ______________________________________

    ______________________________________
    Resistors R21 = 27.4 k.OMEGA.
                      Resistors R29 = 50.1 .OMEGA.
    Resistors R22 = 27.4 k.OMEGA.
                      Resistors R30 = 50.1 .OMEGA.
    Resistors R23 = 49.9 k.OMEGA.
                      Resistors R31 = 50.1 .OMEGA.
    Resistors R24 = 49.9 k.OMEGA.
                      Resistors R32 = 50.1 .OMEGA.
    Resistors R25 = 200 k.OMEGA.
                      Resistors R33 = 20 k.OMEGA.
    Resistors R26 = 200 k.OMEGA.
                      Resistors R34 = 178 k.OMEGA.
    Resistors R27 = 825 k.OMEGA.
                      Resistors R35 = 156.2 k.OMEGA.
    Resistors R28 = 825 k.OMEGA.
                      Resistors R36 = 100 k.OMEGA.
    ______________________________________

• Inventors
  Pecora, Louis M.; Carroll, Thomas L.;
• Assignee
  The United States of America as represented by the Secretary of the Navy (Washington, DC)
• Published
  Jan-3-1995
• Current US Classes:
  331/78
  380/263
  380/46
  708/250
• Application #
  129495
• International Classes
  H03B 029/00; G06F 001/02
• Field of Search
  331/78 364/717 380/9 380/46 380/48 380/49 380/50
• Examiner
  Gregory; Bernarr E.
• Agent
  Kalish; Daniel, McDonnell; Thomas E.
• US Patent References:
  5007087
  5245660
  5291555