Abstract:
A system for synchronizing chaotic transmitters and receivers that is less sensitive to channel effects than other known chaotic communication methods. The system employs duplicate transmitter and receiving modules and in addition to the chaotic output a synchronizing signal which occupies a reduced bandwidth. The small bandwidth affords the system a greater resistance to the affects of frequency dependent channel distortion and noise. The broad band chaotic signal is transmitted and appears to be noise to an unauthorized listener. The receiving unit employs band pass filtering, and when the signal is received the receiver filters the chaotic signal through band pass filters which eliminate channel noise and make gain control easier to implement.

Description:
FIELD OF THE INVENTION 
     The present invention relates generally to synchronizing chaotic systems and more particularly a system which allows the synchronizing of one chaotic system to another chaotic system using only a narrow band signal. 
     DESCRIPTION OF THE RELATED ART 
     A synchronized nonlinear system can be used as 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. 
     It is known to those skilled in the art that a nonlinear dynamical system can be driven (the response) with a signal from another nonlinear dynamical system (the drive). With such a configuration the response system actually consist of duplicates of subsystems of the drive system, which are cascaded and the drive signal, or signals, come from parts of the drive system that are included in the response system. FIG. 1 shows a cascaded chaotic system  100  known in the prior art. Drive system  100  comprises a chaotic drive circuit  140 , housed in transmitter system  139 , and a chaotic response circuit  160 , housed in a receiver system  166 . Chaotic drive circuit comprises subsystems  198  and  199  which are duplicated by subsystem  169  and  170  in the response circuit. A nonlinear function  150  is contained in drive circuit  140  and is used to drive the system into chaotic operation. 
     A chaotic system has extreme sensitivity to initial conditions. The same chaotic system started at infinitesimally different initial conditions may reach significantly different states after a period of time. Lyapunov exponents (also known in the art as “characteristic exponents”) measure this divergence. A system will have a complete set of Lyapunov exponents, each of which is the average rate of convergence (if negative) or divergence (if positive) of nearby orbits in phase space as expressed in terms of appropriate variables and components. 
     Sub or Conditional Lyapunov exponents are characteristic exponents which depend on the signal driving the system. It is also known to those skilled in the art that, if the sub-Lyapunov, or conditional Lyapunov, exponents for the driven response system are all negative, then all signals in the response system will converge over time or synchronize with the corresponding signals in the drive. When the response system is driven with the proper signal from the drive system, the output of the response system is identical to the input signal. When driven with any other signal, the output from the response is different from the input signal. 
     In brief, a dynamical system can be described by the equation 
     
       
         dα/dt=f(α).  (1) 
       
     
     The system is then divided into two subsystems. α=(β,χ); 
     
       
         dβ/dt=g(β, χ) 
       
     
     
       
         dχ/dt=h(β, χ)  (2) 
       
     
     where β=(α 1  . . . α n ), g=(f 1 (α) . . . f n (α)), h=(f n+1 (α) . . . f m (α)), χ=(α n+1 , . . . α m ), where α, β and χ are measurable parameters of a system, for example vectors representing a electromagnetic wave. 
     The division is arbitrary since the reordering of the α i  variables before assigning them to β, χ g and h is allowed. A first response system is created by duplicating a new subsystem χ′ identical to the χ system, and substituting the set of variables β for the corresponding β′ in the function h, and augmenting Eqs. (2) with this new system, giving, 
     
       
         dβ/dt=g(β, χ), 
       
     
     
       
         dχ/dt=h(β, χ)  (3) 
       
     
     
       
         dχ′/dt=h(β, χ′). 
       
     
     If all the sub-Lyapunov exponents of the χ′ system (i.e. as it is driven) are less than zero, then [χ′−χ]→0 as t infinity. The variable β is known as the driving signal. 
     One may also reproduce the β subsystem and drive it with the χ′ variable, giving 
     
       
         dχ/dt=g(β, χ), 
       
     
     
       
         dχ/dt=h(β, χ), 
       
     
     
       
         dχ′/dt=h(β, χ′).  (4) 
       
     
     
       
         dχ′/dt′=g(β″, χ′) 
       
     
     The functions h and g may contain some of the same variables. If all the sub-Lyapunov exponents of the χ′, β″ subsystem (i.e. as it is driven) are less than 0, then β″→β as t→infinity. The example of the eqs. (4) is referred to as cascaded synchronization. Synchronization is confirmed by comparing the driving signal β with the signal β″. 
     Generally, since the response system is nonlinear, it will only synchronize to a drive signal with the proper amplitude. If the response system is at some remote location with respect to the drive system, the drive signal will probably be subjected to some unknown attenuation. This attenuation can be problematic to system synchronization. 
     It is also known by those skilled in the art, that it is possible to pass chaotic signals from a drive system through some linear or nonlinear function and use the signals from the response system to invert that function as discussed, for example, in Carroll, et al., “Transforming Signals with Chaotic Synchronization,” Phys. Rev. E. Vol. 54, p. 4676 (1996). 
     The present invention builds on the design of three previous inventions, the synchronizing of chaotic systems, U.S. Pat. No. 5,245,660, the cascading of synchronized chaotic systems, U.S. Pat. No. 5,379,346, and a method for synchronizing nonlinear systems using a filtered signal, U.S. Pat. No. 5,655,022 each herein incorporated by reference. The present invention extends those principles to allow the synchronization of a broad band chaotic receiver to a broad band chaotic transmitter, using only a narrow band chaotic signal. 
     SUMMARY OF THE INVENTION 
     It is therefore an object of the invention to provide systems for producing synchronized signals, and particularly nonlinear dynamical (chaotic) systems. 
     Another object of the invention is to provide a chaotic communications system for encryption utilizing synchronized nonlinear transmitting and receiving circuits using a narrow-band version of the chaotic signal to synchronize the broader band chaotic transmitter and receiver. 
     A further object of the invention is to provide a chaotic communication system which employs a narrow band version of the chaotic signal for synchronizing transmitter and receiver units to facilitate efficient use on existing telephone or FM radio channels. 
     The present invention is an autonomous system design featuring subsystems which are nonlinear and possibly chaotic, but will still synchronize when the drive signal is attenuated or amplified by an unknown amount. The system uses filters to produce a narrow band version of the wideband chaotic signal to synchronize the chaotic transmitter to the chaotic receiver. The small bandwidth affords the system a greater resistance to the effects of noise, specifically the systems resistance to channel distortion and accompanying phase shifts is greatly increased by employing a narrow band. The broad band chaotic signal is transmitted and appears to be noise to an unauthorized listener. The receiving unit employs band pass filtering, and when the signal is received the receiver filters the broadband chaotic signal. The filters produce a narrowband chaotic signal and that narrowband signal is used to synchronize the transmitter and receiver. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING FIGURES 
     FIG. 1 is a block diagram of a cascaded chaotic circuit 
     FIG. 2 is a block diagram of an autonomous cascaded chaotic circuit employing band pass filters 
     FIG. 3 is a block diagram of an autonomous cascaded chaotic circuit employing band stop filters 
     FIGS. 4 and 5 are block diagram of autonomous chaotic circuits employing band stop and band pass filters. 
     FIG. 6 is a schematic diagram of a autonomous cascaded drive circuit. 
     FIG. 7 is a schematic diagram of a cascaded response circuit. 
     FIG. 8 is a schematic diagram of a band-stop filter. 
     FIG. 9 is a graph of the synchronization of the x and x′ terms in an example chaotic circuit. 
     FIG. 10 is a graph of the power spectrum of u as a function of frequency. 
     FIG. 11 is a graph of the power spectrum of u f  as a function of frequency. 
     FIG. 12 is a plot of the largest Lyapunov exponent for the response system as a function of the bandpass filter frequency. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Referring to the remaining figures, wherein like references refer to like components, FIGS. 2,  3 ,  4 , and  5  show a block diagrams of techniques of synchronizing chaotic systems using filters. The filter synchronized chaotic systems each comprise a chaotic drive circuit, and a chaotic response circuit. With reference to FIG. 2 which shows an embodiment comprising a Chaotic drive circuit  301  coupled to band pass filter circuit  400  producing a chaotic signal u f . Chaotic drive circuit  301  is contained in a transmitter system  300  or other means for transmission. Drive circuit  301  is coupled to a filter circuit  400 . Filter circuit may be contained in transmitter system or it may be housed in receiving system  500 . In the embodiment illustrated in FIG. 2 the filter is contained in receiving system  500 . 
     Receiving system  500  is coupled to transmitter system  300 , and may be located remote from transmitter system  300 . Receiving system  500 , comprises filter circuit  400 . Preferably filter circuit  400  is a band pass or band stop type filter, however any filter may be used. Receiving system  500  further comprises a response circuit,  501  which comprises subsystems which duplicate those which are contained in drive circuit  301  and a filter circuit  401  which is coupled to the output of response circuit  501  and shares a linear relationship to the filter circuit  400 . Filter circuit  400  is coupled to a difference circuit  510  which combines the signal received from the drive circuit with a version of the signal produced by duplicate subsystems contained in the response circuit. Response system  500  further comprises a filter circuit  401  which is coupled to the combined output of the duplicate subsystems contained in the response circuit. Filter circuit is coupled to difference circuit  510 , which combines output of the filter circuit  401  and the output of filter circuit  400 . Response circuit  500  further contains gain elements, b 1 , b 2  and b 3 , the input of each coupled to difference circuit  510  and the output of b 1 , b 2  and b 3  each coupled to one of the duplicate subsystems of response circuit. 
     Drive circuit  301  comprises subsystems A, B, and C,  373 ,  383 , and  393  each of which and coupled in a cascaded configuration and together produce output signal x, y, and z, at least one of which has nonlinear or chaotic properties. Signals x, y, and z are multiplied by linear constants k 1 , k 2 , and k 3  and are combined to form a linear combination these signals u which may be expressed u=k 1 x+k 2 y+k 3 z. Cascaded drive circuit Sol formed by subsystem A  373 , subsystem B  383 , subsystem C  393 , k 1 , k 2 , and k 3  and the linear combination output it produces are known in the art. Signal u is a chaotic broadband signal. 
     The orientation of subsystems  373 ,  383 , and  393 , each being driven by the other 2 subsystems eliminates the need for an independent driving chaotic signal, thus providing autonomous operation. No outside periodic driving source is required to drive the chaotic drive circuit. 
     In the present embodiment, the broadband chaotic signal U is then transmitted by the transmitter system  300  and received at a location remote from the transmitter by the receiver system  500 . The receiver system  500 , receives the broadband chaotic signal U and filters U producing a filtered chaotic signal U f . U f  features a narrow or reduced bandwidth in comparison to U. A benefit of this feature is that the information signal is contained within a broader band which increases reduces the chance of unauthorized decryption of the information signal. In other embodiments discussed supra, the signal is filtered prior to transmission, thus a narrowband signal is transmitted, offering benefits such as lower transmitter power requirements. However, in either configuration, the chaotic signal is synchronized using a chaotic signal using a narrow band. 
     Filter circuit  400  is preferably of the band pass or band stop type, however other filter circuit or a combinations of filter circuits may be used. 
     The response circuit  501 , comprises subsystems A′  373 ′, B′  383 ′ and C′  393 ′ which are duplicates of the subsystems contained in drive circuit  301  which combine to produce a linear combination of signals v. Using the sections of the response system identical to those in drive system  301 , response system  501  creates v where v=k 1 x′+k 2 y′+k 3 z′. The signal v is then passed through filter  401 , which is identical or shares a linear relationship to the filter  400 , and is used to generate the chaotic signal U f . Filter  401  filters signal V to generates a filtered signal V f . 
     In the configuration shown in FIG. 2 a band pass filter is used. Response system  501  then creates the signal w by taking the difference between v f  and u f , thus w=u f −v f . Signal w is then multiplied by constant values, b 1 , b 2 , and b 3  creating  3  different signals (signals reflecting a different gain), b 1  w, b 2  w and b 3  w. Signal b 1  w is then fed into A′, the part of the response system that produced x′, b 2  w is fed into B′ the part of the response system that produced y′ and b 3  w is coupled to the part of the response system that produced z′. 
     The values for k 1 , k 2 , k 3 , b 1 , b 2  and b 3  are selected to add stability to the system. If the response system is stable, it will synchronize. These values for k 1 , k 2 , k 3 , b 1 , b 2  and b 3  may be selected by combining any standard algorithm for computing Lyapunov exponents with a numerical minimization both known in the art. Routines such as those found in J. P. Eckmann and D. Ruelle, “Ergodic Theory of Chaos and Strange Attractors”Review of Modern Physics, vol 57 pp. 617-656 (1985) may be used for computing the Lyapunov exponents and routines such as those found W. H. press et al, “Numerical Recipes”, (Cambridge, N.Y. 1990) may be used for the numerical minimizations. In the band pass embodiment one may use a regular bandpass filter as illustrated in FIG. 2 or employ a band stop filter and subtract the band stop filter output from u as shown in FIG.  3 . 
     In a numerical example of the band pass embodiment the synchronization of the well known Lorenz equations are illustrated. The Lorenz equations are: 
     
       
         dx/dt=16(y−x)  (5) 
       
     
     
       
         dy/dt=−xz+45.92x−y  (6) 
       
     
     
       
         dz/dt=xy−4z  (7) 
       
     
     The signal u is formed as: 
     
       
         u=k 1 x+k 2 y+k 3 z  (8) 
       
     
     Next the equations are numerically integrated with a 4 th  order Runge-Kutta numerical integration routine as known in the art and discussed in (W. H. Press et al, “Numerical Recipes”, (Cambridge, N.Y., 1990)). FIG. 10 shows the power spectrum of signal u. The signal u is then filtered through a band pass filter producing a filtered signal u f . The filter variables are h 1  and h 2 : expressed as follows: 
     
       
         dh/dt=2h 1 /R 1 −(1/2R 2 )(1/R 3 −1/R 1 )h 2 −(1/R 1 )(du/dt)  (9) 
       
     
     
       
         dh 2 /dt=h 1   (10) 
       
     
     
       
         u f =h 2   (11) 
       
     
     Variables h 1  and h 2  represent a second order Butterworth band-pass filter. The resonant frequency of the filter is given by (fr)2=(R 1 +R 2 )/(2πCR 1 R 2 R 3 ). The gain Ar=−R 2 /(2R 1 ), and the Q factor is Q=πR 2 C. For equation 9, C=1, Q=20, Ar=−1 and R 1 , R 2 , and R 3  are set to select fr. The value for fr=5.4. 
     FIG. 11 shows a graph of the power vs. frequency of the filtered signal u f  when fr=5.44. The reader should note that filtered signal u f  illustrated in FIG. 11, reflects a reduced frequency band in comparison to the unfiltered graph of power vs frequency of u as illustrated in FIG.  10 . 
     The response system is described by the equations: 
     
       
         dx′/dt=16(y′−x′)+b 1 w  (12) 
       
     
     
       
         dy′/dt=−x′z′+45.92x′−y′+b 2 w  (13) 
       
     
     
       
         dz′/dt=x′y′−4z′+b 3 w  (14) 
       
     
     
       
         v=k 1 x′+k 2 y′+k 3 z′  (15) 
       
     
     
       
         dh′ 1 /dt=−2h′ 1 /R 1 −(1/2R 2 )(1/R 3 −1/R 1 )h′ 2 −(1/R 1 )(dv/dt)  (16) 
       
     
     
       
         dh′ 2 /dt=h′ 1   (17) 
       
     
     
       
         v f =h′ 2   (18) 
       
     
     
       
         w=u f −v f   (19) 
       
     
     The values for the k 1 , k 2 , k 3 , b 1 , b 2 , and b 3  for which the response system is stable, are determined by minimizing the largest Lyapunov exponent for the response system using conventional minimization routines. In the present example k 1 =273.0212, k 2 =23.26557, k 3 =16.24705, b 1 =18.93643, b 2 =20.51921, and b 3 =3.04397, thus the largest exponent for the response system is −4.9523, and the response synchronizes to the drive. FIG. 9 shows a plot of the synchronization of the x′ signal form the response system to the x signal from the drive system. The other terms, y and z, will synchronize when the x term synchronizes. There are a large number a k-b sets and a large number of fr&#39;s that will give synchronization. 
     FIG. 12 is a plot of the largest Lyapunov exponent for the response system as a function of the bandpass filter frequency. λ max  represent the maximum value of the Lyapunov exponent. The k&#39;s and b&#39;s are the same as the values listed above. The largest Lyapunov exponent for the response is negative over a broad range of fr, so many different filter frequencies are possible for a given set of k&#39;s and b&#39;s. Thus one may transmit one signal u to multiple chaotic response systems, each of which uses a bandpass filter with a different center frequency fr. If the transmission channel contains frequency dependent noise, one can compare the different response systems to improve the overall signal to noise ratio. 
     In an alternative embodiment for synchronization, one will keep only narrow band information from the drive signal u and keep only broadband information from the response signal v. As shown in FIG. 4, one way to do this is to pass u through the band stop filter and subtract the band stop output from u to produce u f . The signal v from the response system is passed through a bandstop filter to produce v f . The signal created is of the form w=u f +v f . When the drive and response systems are synchronized, w=u=v. One of the response variables may be replaced with its equivalent, reconstructed from w: for example x′ in the response system may be replaced with xd=(w−k2y′−k3z′)/k1. Similar substitutions may be possible for other variables. The constants k 1 , k 2  and k 3  may be chosen so that the response system will synchronize. 
     As a specific example FIGS. 6 and 7 show an electrical drive and response circuit constructed in accordance with the system disclosed in FIG.  3 . The drive circuit of FIG. 3 was described by: 
      dx 1 /dt=10 4 (0.05x 1 +0.5x 2 +x 3 )  (20) 
     
       
         dx 2 /dt−10 4 (−x 1 −0.11x 2 )  (21) 
       
     
     
       
         dx 3 /dt=−10 4 (x 3 +g(x 1 ))  (22) 
       
     
     g(x)=0 if x&lt;3, 15(x−3) otherwise 
     
       
         u=(k 1 x 1 +k 2 x 2 +k 3 x 3 )/2  (23) 
       
     
     The values of k are k 1 =−1.9, k 2 =1.1 and k 3 =1. 
     Referring now to FIG. 6 which illustrates an example drive circuit. Drive circuit  301 , comprises differential amplifiers  330 ,  332 , and  335 , which are coupled to resistors  301 ,  302 ,  303 ,  307 ,  311 ,  312  and capacitors  321 ,  322  and  323  form integrating circuits which correspond to subsystems  373 ,  383 , and  393  as defined in FIGS. 2,  3 ,  4  and  5 . Drive circuit  301  also comprises differential amplifier  331 , which with resistors  304 ,  306 ,  305 ,  314 ,  316 ,  315 ,  317 ,  318  and  319  form a summer circuit which receives the output signals from subsystems  373 ,  383  and  393 , are (x1, x2 and x3 in equations 20-23) x, y, and z using the combined signals to drive each subsystem with the output from the remaining two subsystems. The summer circuits constructed from  333  and  336  combine the x, y. and z signals to create output signal u, used as the chaotic drive signal. Signals x, y, and z are characteristic voltages of the drive circuit  301 . Drive circuit  301  further comprises a nonlinear function constructed from differential amplifier  334 , diode  341  and resistors  308  and  309 . This nonlinear function is used to drive subsystem  393  causing the z signal to have a chaotic response resulting in the driving of signal U into chaos. In an experimental implementation of the chaotic system  300  which has been successfully tested, amplifiers  330 - 336  are operational amplifiers of type  741  or comparable. 
     The values for the example drive circuit are shown in table 1. 
     
       
         
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
             
             
               
                 301 = 200 kΩ 
                 305 = 10 kΩ 
                 309 = 68 kΩ 
                 313 = 5 MΩ 
                 317 = 100 kΩ 
                 322 = 1000 pf 
               
               
                 302 = 100 kΩ 
                 306 = 10 kΩ 
                 310 = 150 kΩ 
                 314 = 200 kΩ 
                 318 = 125 kΩ 
                 323 = 1000 pf 
               
               
                 303 = 2 MΩ 
                 307 = 100 kΩ 
                 311 = 100 kΩ 
                 315 = 182 kΩ 
                 319 = 100 kΩ 
                 341 = MV210 
               
               
                   
                   
                   
                   
                   
                 1 diode 
               
               
                 304 = 75 kΩ 
                 308 = 10 kΩ 
                 312 = 100 kΩ 
                 316 = 100 kΩ 
                 321 = 100 pf 
               
               
                   
               
             
          
         
       
     
     Drive circuit  301  and response circuit  501  are subdivided into 3 subsystems  373 ,  383 , and  393 , however, this is not necessary and the division of each circuit into subparts in order to determine the proper configuration for a synchronized response circuit is made in accordance with the analysis described herein. 
     In operation drive circuit  300 , produces a broad band chaotic output signal u, by driving each integrating circuit with the signals x, y, and z (referred to as x 1 , x 2  and x 3  in equations 20-23) in a continuous feedback configuration. The integrating circuit which produces the z signal driven with a nonlinear function to produce a stable chaotic signal u. 
     Referring again to FIG. 3 the chaotic signal u produced by drive circuit  301  passes through filter  400 . In this embodiment a band stop filter  400  is used, however other filters, such band pass type filters may be used. In the band stop configuration one may subtract the bands passed from the band stopped to produce the narrowband signal. 
     FIG. 8, shows a schematic of an example filter circuit  400 . The filter circuit shown is a Wien-Robinson bandstop filter. The device is configured to subtract the filter&#39;s output from the complete signal to achieve a narrow band chaotic signal. Filter  400  is described by the equations: 
     
       
         dx 4 /dt=−(1/RC)(3x 4 /(1+a 1 )+x 5 +b 1 u/(1+a 1 )−[RCb 1 /(1+a 1 )]d 2 u/dt 2   (24) 
       
     
     
       
         dx 5 /dt=x 4 /(RC)  (25) 
       
     
     
       
         u f =u+x 5   (26) 
       
     
     The gain of the bandstop part of filter  400  is A 0 =−b1/(1+a 1 ), the Q factor is (1+a 1 )/3, and the center frequency is fr=1/(2πRC). The gain is set to −1 and the Q to 7. The capacitor C is 0.01 μf, the value of C 444 , C 445  or C 446 . The variable b1=R 421 /R 420  and a1=R 421 /R 422 . R, the value of R 423  and R 424  is chosen to set fr to the peak frequency in the circuit of equation 24. For a center frequency of 1145 Hz, R=14,179 ohms. 
     The circuit example shown in FIG. 8 features and input comprising an integrator circuit with resistor  420 ,  421 , and  422  and capacitor  444  coupled to the negative terminal of amplifier  450 . Resistor  421 , and capacitor  444  are also coupled to the output of amplifier  450 . The output of amplifier  450  is also coupled to the negative input terminal of amplifier  451  via resistor  425 . The output of amplifier  450  is coupled to the positive terminal of amplifier  451  via capacitor  445  and resistor  423  through a common node. Capacitor  446  is also coupled to the same node. Resistor  424  is coupled to capacitor  446  through yet another node. The output of amplifier  451  shares a common node with resistor  422 ,  426  and  427 . Amplifier  451  through resistor  427  drives the negative terminal of amplifier  452 . The output terminal of amplifier  452  is coupled to resistors  428  and  429 , with resistor  428  coupled in a feedback configuration to the negative terminal of amplifier  452  in a common node with resistor  427  and driving the negative terminal of amplifier  453  by way of resistor  429 . Resistors  430  and  432  a coupled in a common node with the positive terminal of amplifier  453 . The output of amplifier  453  is fed back into the negative terminal of amplifier  453  via resistor  431 . The output of the filter  400  is taken at the output of amplifier  453  and the input of the filter  400  is a common node formed by resistor  420  and resistor  430 . 
     Referring again to FIG.  3  and to FIG. 8, in this example embodiment filters  400  and  401  are identical the only difference being the input signal. In FIG. 3 the drive signal u is coupled to filter  400  which filters signal u to produces the signal u f . Filtered signal u f  features chaotic characteristics, however signal u f  features a reduced bandwidth in comparison to signal u. Filter  401  is coupled to the response system and produces a filtered signal v f  as its output. The signal v f  is used to synchronize the chaotic drive with the chaotic response. For the configuration disclosed in FIG. 3, using dual BUTTERWORTH filters, when w=0 the system is synchronized. The values for the example band stop filter circuit are shown in table 2. 
     
       
         
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
             
             
               
                 420 = 1 MΩ 
                 424 = 14,179 Ω 
                 428 = 100 kΩ 
                 432 = 100 kΩ 
               
               
                 421 = 1 MΩ 
                 425 = 100 kΩ 
                 429 = 100 kΩ 
                 444 = 100 pf 
               
               
                 422 = 50 kΩ 
                 426 = 200 kΩ 
                 430 = 100 kΩ 
                 445 = 1000 pf 
               
               
                 423 = 14,179 Ω 
                 427 = 12,500 Ω 
                 431 = 100 kΩ 
                 446 = 1000 pf 
               
               
                   
               
             
          
         
       
     
     Referring again to FIG. 3, the filtered signal is received by response circuit  501 . Response circuit, comprises a section identical to drive circuit  300  producing signal V as it output. V then passe through filter  401  to produce signal V f  Filtered V f  is subtracted from the signal received from drive circuit  301  and coupled back into response circuit  501 . 
     FIG. 7 is a schematic diagram of response circuit  501 . The input of response circuit  501  is coupled to a difference circuit comprising of amplifier  522  coupled to resistors  533 ,  534 ,  535  and  536 . The V f  input signal is coupled to amplifier  522  via resistor  534 . The u f  input is coupled to amplifier  522  via resistor  533 . The output of amplifier  522  is coupled to input of integrator circuits formed by amplifiers  523 ,  525 , and  527 . Amplifiers  523 ,  525 , and  527  form subsystems  373 ′,  383 ′, and  393 ′ corresponding duplicates to subsystems  373 ,  383 , and  393  contained in drive circuit  301 . Summer circuits are formed by amplifier  524  via resistors  541 ,  542  and  543 , amplifier  528 , resistor 553-555 and amplifier  529  and resistors  556 - 558  are likewise identical to the summer circuits contained in drive circuit  301 . 
     Response circuit  501  may be described by the following equations: 
     
       
         dx′ 1 /dt=−10 4 (0.05x′ 1 +0.5x′ 2   +x′   3 )+2b 1 w  (27) 
       
     
     
       
         dx′ 1 /dt=−10 4 (−x′ 1 0.11x′ 2 )+2b 2 w  (28) 
       
     
     
       
         dx′ 3 /dt=−10 4 (x′ 3 +g(x′ 1 ))+2b 3 w  (29) 
       
     
     
       
         v=(k 1 x′ 1 +k 2 x′ 2 +k 3 x′ 3 )/2  (30) 
       
     
     
       
         dx′ 4 /dt=(1/RC)(3x′ 4 /(1+a 1 )+x′ 5 +b 1 v/(1+a 1 ))−[RCb 1 /(1+a 1 )]d 2 v/dt 2   (31) 
       
     
     
       
         dx′ 5 /dt=x′ 4 /RC  (32) 
       
     
      v f =v+x′ 5   (33) 
     
       
         w=u−v  (34) 
       
     
     Where the b values are b 1 =1, b 2 =1, b 3 =1. The values for the example response circuit are shown in table 3. 
     
       
         
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
             
             
               
                 533 = 100 kΩ 
                 538 = 200 kΩ 
                 543 = 10 kΩ 
                 548 = 68 kΩ 
                 553 = 182 kΩ 
                 558 = 100 kΩ 
               
               
                 534 = 100 kΩ 
                 539 = 2 MΩ 
                 544 = 50 kΩ 
                 549 = 150 kΩ 
                 554 = 200 kΩ 
                 527 = 1000 pf 
               
               
                 535 = 100 kΩ 
                 540 = 100 Ω 
                 545 = 100 kΩ 
                 550 = 50 kΩ 
                 555 = 100 kΩ 
                 528 = 1000 pf 
               
               
                 536 = 100 kΩ 
                 541 = 75 kΩ 
                 546 = 5 MΩ 
                 551 = 100 kΩ 
                 556 = 100 kΩ 
                 529 = 1000 pf 
               
               
                 537 = 50 kΩ 
                 542 = 10 kΩ 
                 547 = 10 kΩ 
                 552 = 100 kΩ 
                 557 = 125 kΩ 
                 530 = MV210 
               
               
                   
                   
                   
                   
                   
                 1 diode 
               
               
                   
               
             
          
         
       
     
     Referring again to FIGS. 3,  6 ,  7 , and  8  drive system  301  is preferably be housed in a transmitter  300 , the chaotic signal U created by drive system  301  is transmitted to receiver  500 , which comprises, filter  400 , response circuit  501  and filter  401 . Receiver  500  receives the transmitted signal u and uses filter  400  to filter signal u producing signal u f . Signal u f  is a narrow band chaotic signal and is used to synchronize drive circuit  301  with response circuit  501 . Response circuit  501  produces a chaotic signal v, which is filtered by filter  401  to produce signal v f . Signal v f  is combined with signal u f  to produce signal W. Signal W is used to drive response circuit  501 . Response circuit  501  compares narrow band signal u f  to narrow band signal v f  to determine synchronization. A plot of the synchronization of an example chaotic system as described in FIG. 3 is shown in FIG.  9 . The drive circuit  301  is as described in FIG.  6  and the example response circuit  501  is as described in FIG.  7 . Filters,  400  and  401  are as described in FIG.  8 . 
     It should therefore readily be understood that many modifications and variations of the present invention are possible within the purview of the claimed invention. It is therefore to be understood that, within the scope of the appended claims, the invention may be practiced otherwise than as specifically described.