Creating New Chaotic Signals with Reservoir Computers

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Creating New Chaotic Signals with Reservoir

Computers

Thomas L. Carrolla,

aCode 6392, US Naval Research Lab, Washington DC 20375 USA.

Abstract

While there have been many publications on potential applications of chaos to

ﬁelds such as communications, radar, sonar, random signal generation, channel

equalization and others, designing continuous chaotic systems is still an unsolved

problem. There are a number of well known chaotic systems used for applica-

tions, but if any application is to become widely used, some way of generating

many diﬀerent chaotic signals is necessary. This work shows that one may use a

reservoir computer to create a set of chaotic signals that are correlated but eas-

ily distinguishable from one chaotic signal with desirable properties. The ability

to distinguish the new signals is demonstrated with a simple communications

example.

Keywords: Chaos; reservoir computer; chaotic communications

1. Introduction

While much has been published on the topic of using chaos for communications[1,

2, 3, 4, 5, 6, 7, 8, 9, 10] or radar [11, 12, 13, 14, 15, 16, 17, 18, 19] , one problem

that has not been much addressed is the design and implementation of chaotic

systems for these applications. There have been a number of chaotic systems

proposed for these uses, such as the Lorenz [20] or R¨ossler [21] systems, the

Chua system [22], the 19 Sprott systems [23] and others, but more alternatives

are needed for actual applications. There are no rules for designing chaotic sig-

Email address: thomas.carroll@nrl.navy.mil (Thomas L. Carroll )

Preprint submitted to Chaos, Solitons & Fractals October 13, 2022

arXiv:2210.06250v1 [eess.SP] 9 Sep 2022

nals to have desired properties; Sprott does give a list of chaotic systems, but

these were found by a systematic search. Adding to the design complexity, in

some cases one may want to use the self synchronizing property of chaos, but

the chaotic system must be designed speciﬁcally to allow this possibility. One

may also create diﬀerent signals from a known chaotic system by changing a

parameter, but there may be limits on how far the parameter may be changed

without encountering a bifurcation.

In this work we propose a method to create a large number of chaotic signals

from a particular chaotic system that has desirable properties. We use a chaotic

signal from a desirable system such as a Lorenz system to drive a reservoir

computer. A reservoir computer is a high dimensional dynamical system that

may be created by connecting a number of nonlinear nodes in a recursive network

[24, 25] . Usually the output signals from this network are combined to ﬁt a

training signal; for our purposes, we instead make random combinations of

signals from the reservoir to create a new set of signals. These signals are

nonlinear functions of the original driving signal; while they are still correlated

with the driving signal, they can still be distinguished from the driving signal

and each other by training a second reservoir computer on the new signals. We

show that these signals may be used to communicate using chaos shift keying

(CSK), in which diﬀerent communications symbols are represented by signals

from diﬀerent chaotic systems.

One feature of reservoir computers is that because the training only takes

place on the output, they may be constructed from analog systems. Reservoir

computers that are all or part analog include photonic systems [26, 27, 28,

29, 30, 31], analog electronic circuits [32], mechanical systems [33] and ﬁeld

programmable gate arrays [34]. Many other examples are included in the review

paper [35]. Building reproducible analog chaotic circuits that operate at high

frequencies or high powers is diﬃcult, so one could envision driving an analog

reservoir computer with a digital signal to produce a number of analog chaotic

signals.

In this work, a reservoir computer will be used to create a new set of chaotic

signals from an input signal. A second reservoir computer will be trained on

each of these new signals and the training coeﬃcients for each new signal will be

stored. To transmit information, for each data interval, one of these new signals

will be transmitted. The job of the receiver is to determine which of these signals

was sent for each data interval. Two types of receiver are studied; one where

the receiver is synchronized to the original chaotic signal in the transmitter and

one in which it is not synchronized.

2. Reservoir Computers

The reservoir computer we use in this work, often known as the leaky hy-

perbolic tangent reservoir computer [24], is common in the literature. It is

described by

R(n+ 1) = (1 −α)R(n) + αtanh AR +Wins(n)+1(1)

where Ris a vector of reservoir variables, Ais the adjacency matrix that de-

scribes how the diﬀerent nodes are connected, sis the input signal and Win is

the vector of input coeﬃcients. The individual components of R(n) are ri(n),

where iis the index of a particular node. The reservoir computer has Mnodes,

so the dimensions of Rand Win are M×1 while Ais M×M.

In the training stage, the reservoir computer is driven with the input signal

s(n) to produce the reservoir computer output signals ri(n). In all the examples

in this paper, the input signal is normalized to have a mean of zero and a

standard deviation of 1. The reservoir output matrix Ω1is constructed from

the reservoir signals as

Ω1=







r1(1) · · · rM(1)

r1(2) rM(2)

r1(N)· · · rM(N)







(2)

2.1. Creating New Signals

In normal use the a linear combination of the columns of the matrix Ω1would

be used to ﬁt a training signal. Instead, to create Nsnew signals, we create a

M×Nsrandom matrix of coeﬃcients Ψ. The elements of Ψ are drawn from a

uniform random distribution between -1 and 1. To insure that the columns of

Ψ are not too similar to each other, then are then made orthonormal to each

other by a Gram-Schmidt or other method to yield ΨO. We produce Nsnew

signals as

Θ=Ω1ΨO.(3)

The new signals are Θj(n), n = 1 . . . N, j = 1 . . . Ns, where Nis the number

of points in the reservoir time series.

2.2. Distinguishing the New Signals

A second reservoir computer may be used to distinguish the diﬀerent chaotic

signals. The reservoir computer is driven with one of the signals from the matrix

of signals Θ:

Rj(n+ 1) = (1 −α)Rj(n) + αtanh ARj+WinΘj(n)+1(4)

Before driving, each signal in Θ is normalized by subtracting the mean and

dividing by the standard deviation.

The output signals from the reservoir computer driven with each of the new

signals are each arranged in a matrix

Ωj=







r1j(1) · · · rMj (1) 1

r1j(2) rMj (2) 1

r1j(N)· · · rMj (N) 1







(5)

where the ﬁrst index of rindicates the node number. The last column of Ωjis

set to 1.0 to ﬁt any constant oﬀset.

The reservoir computer is trained on a particular chaotic signal, the reservoir

is trained by predicting that signal one time step into the future. The training

signal is gj(n)=Θj(n+ 1). For each of the Nsnew signals, the matrix Ωjis

used to ﬁt the training signal as

gj≈hj= ΩjWout

j(6)

where the ﬁt is done using ridge regression to prevent overﬁtting. The ﬁt coeﬃ-

cients are in the vector Wout. The training error ∆RC

jis the standard deviation

of gj−hj, normalized by the standard deviation of g.

For signal identiﬁcation, the reservoir computer of eq. (1) is driven with

a signal ˜sfrom the same dynamical system with diﬀerent initial conditions.

The output signals ˜

Rare arranged in a matrix ˜

Ω1and a set of new signals is

created as ˜

Θ = ˜

Ω1ΨO. The testing signals are ˜gj(n) = ˜

Θj(n+ 1), which may be

approximated as ˜

hj=˜

ΩjWout

j. The testing error ∆tx

jis the standard deviation

of ˜gj−˜

hj.

3. Communications: Synchronous and non-Synchronous

The diﬀerent signals Θj, j = 1 . . . Nsmay be used as communications sym-

bols, an encoding commonly known as chaos shift keying (CSK). Typically CSK

would proceed by sending signals from diﬀerent chaotic systems (also known as

attractor shift keying) or by sending signals from one chaotic system with dif-

ferent parameters. The version of CSK described here is equivalent to sending

diﬀerent components from a chaotic system.

The communications system may be divided into signal encoding and signal

decoding. The encoding is implemented by switching between diﬀerent com-

ponents of Θj, while the decoding uses a reservoir computer and the training

coeﬃcients Wout

jfrom eq. (6) to determine which component was transmitted.

If there are Nspossible components of Θ that can be transmitted, then the

number of bits of information in each data interval is log2(Ns). The detection

may be done coherently (using a synchronized receiver) or non-coherently (using

an asynchronous receiver).

The data signal consists of a series of discreet values I(k), k = 1 . . . Nd,

where I(k) is an integer in the range 1 to Ns. In the k’th time slot, the signal

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摘要：

CreatingNewChaoticSignalswithReservoirComputersThomasL.Carrolla,aCode6392,USNavalResearchLab,WashingtonDC20375USA.AbstractWhiletherehavebeenmanypublicationsonpotentialapplicationsofchaostoeldssuchascommunications,radar,sonar,randomsignalgeneration,channelequalizationandothers,designingcontinuouscha...

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