# Diffusive Synchronization of Hyperchaotic  Lorenz Systems

Authors: Ruy Barboza
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

The synchronizing properties of two diffusively coupled hyperchaotic Lorenz 4D systems are investigated by calculating the transverse Lyapunov exponents and by observing the phase space trajectories near the synchronization hyperplane. The effect of parameter mismatch is also observed. A simple electrical circuit described by the Lorenz 4D equations is proposed. Some results from laboratory experiments with two coupled circuits are presented.


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## Body

## 1. Introduction

Coupled
oscillators are currently studied in physics, chemistry, biology, neural
networks, and other fields. A large number of coupled oscillators form a
complex system for which the investigation of coherence and synchronization is
important and in the last decade the discovery that chaotic systems can
synchronize added interest to this topic [1–4]. An approach for contributing to
the investigation of large networks is studying the properties of a small
number of coupled oscillators—this is the
approach used in the present work. Chaos synchronization is also of interest in
secure communication systems. For such application, hyperchaos has drawn
attention for providing more complex waveforms than simply chaotic systems,
thus improving the masking process.  This
is because hyperchaos is characterized by at least two positive Lyapunov
exponents, while simple chaos shows a single one. Related to this subject is
the question on how many variables are necessary to be coupled in order to obtain
synchronization. Although chaotic systems can synchronize by a single variable
coupling, it was for some time believed that in the case of hyperchaos the
minimum number of coupling variables had to be equal to the number of positive
Lyapunov exponents [5]. It was later demonstrated in [6] that it is not true,
and some hyperchaotic systems can achieve synchronization by a single variable.
Other systems, however, for example the Rössler equations for hyperchaos, are
unable to synchronize by only one of its variables [6, 7]. Another problem
refers to synchronization under parameter mismatch, since usually perfect
synchronization is achieved only if the coupling systems are identical. In this
work we are concerned with the question of whether a hyperchaotic Lorenz system
can synchronize and with which variables. The effect of mismatch is also
observed. We present some results from numerical and laboratory experiments on
an eight-dimensional dynamical system obtained by diffusively coupling two
hyperchaotic Lorenz systems.


## 2. Equations and Numerical Simulations

## 2.1. Diffusive Coupling

Consider
the four-dimensional systemdx/dt=f(x),
where  x=[x,y,z,w]t. Unidirectional diffusive coupling
involving two such systems is obtained by using the system dx1/dt=f1(x1) to drive the response
system  dx2/dt=f2(x2)+K(x1−x2). In this work we consider K a diagonal matrix, where each element kii is the strength of the coupling related to the
corresponding variable. We also consider f1(x)=f2(x)=f(x), and each entry k11, k22,
k33, or k44 is either zero or equal
to a positive constant k. For
example, the case k11=0, k22=k, k33=0, and  k44=k means simultaneous y-y and w-w coupling. The two coupled systems form a compound
eight-dimensional system. If the systems synchronize, the motion must remain on
the hyperplane x1=x2. For small (x1−x2) we have f(x1)−f(x2)≈(∂f(x1)/∂x1)⋅(x1−x2),
so we can write the variational equation (2.1)dx⊥dt=A(x1)x⊥, where A=∂f(x1)/∂x1−K and x⊥=(x1−x2).
In the case of bidirectional diffusive coupling we have dx1/dt=f(x1)+K(x2−x1) and dx2/dt=f(x2)+K(x1−x2), therefore (2.1) also applies, now
with A=∂f(x1)/∂x1−2K.
We use (2.1) as the locally linear dynamical system associated to a fiducial
trajectory [8] of the coupled system to calculate the transverse Lyapunov
exponents [2, 3, 7, 9], so called because the perturbation x⊥ is
transverse to the synchronization hyperplane. The coupled system will remain
stably synchronized if all transverse Lyapunov exponents (TLEs) are negative.


## 2.2. Complete Replacement

If the drive system 1 transmits the scalar componentx1 and the corresponding variable x2 of the response system 2 is replaced by the
transmitted one x1, this
is called complete replacement [4]. As explained in [4], unidirectional
diffusive coupling and complete replacements are related, since at very high
values of k the variable x1 slaves x2.  Therefore, in our numerical and experimental investigations,
complete replacement of one or more variables corresponds to k→∞.


## 2.3. Coupled Hyperchaotic Lorenz Systems

The preceding coupling scheme will now be applied to systems described by the
following Lorenz equations linearly extended to four dimensions:(2.2)dxdt=σ(y−x),dydt=x(r−z)−y+w,dzdt=xy−bz,dwdt=−γx, which shows hyperchaos and was
theoretically analyzed in [10]. In the case of unidirectional coupling the
eight-dimensional system is given by (2.3)dx1dt=σ(y1−x1),dy1dt=x1(r−z1)−y1+w1,dz1dt=x1y1−bz1,dw1dt=−γx1,dx2dt=σ(y2−x2)+k11(x1−x2),dy2dt=x2(r−z2)−y2+w2+k22(y1−y2),dz2dt=x2y2−bz2+k33(z1−z2),dw2dt=−γx2+k44(w1−w2). For this system
the matrix A is (2.4)A=[−σ−k11σ00z1−r−1−k22−x11y1x1−b−k330−γ00−k44] for
unidirectional coupling. In the case of bidirectional coupling, the only
modification is replacing kii by 2kii along the diagonal. The synchronizing
properties of system (2.2) will now be numerically investigated by calculating
the TLE of (2.3) as a function of the coupling strength k. Parameter mismatch is also examined. In the following, only
unidirectional coupling is considered.


## 2.3.1. Parametersσ=10,b=8/3,r=30,γ=10

We first illustrate the synchronizing
properties of (2.2) for the above parameter values.  The values σ=10 and b=8/3 are the
classical, or most popular, used in studies of the original Lorenz 3D
system.  For the hyperchaotic Lorenz 4D
system the extra parameter γ=10 is
included. In Figure 1, the two largest TLE for single-variable coupling are
plotted as a function of k, for r=30. The synchronization thresholds are k=8.1 and k=1.8 for x-x and y-y couplings, respectively. The systems
will never get synchronized if w alone is the coupling variable. On the other hand, z-z coupling provides a small window of stable synchronization.Transversal Lyapunov exponents (TLEs) forσ=10,b=8/3,r=30,γ=10.
(a)(b)(c)(d)


## 2.3.2. Parametersσ=4,b=0.3,r=30,γ=1.6

The above values of the parameters are of
interest in this work because small values ofσ  and b are easier to
realize with practical component values in the circuit model presented in Section
3. (At this point it is worth remembering the observation by Sparrow [11] that
small b  leads to very complex
behavior of the Lorenz equations.) Therefore, the system properties for such
small parameter values will be examined with more detail in the following. In
Figure 2, the Lyapunov spectrum for these parameter values is plotted as a
function of r, showing a broad range
of hyperchaotic behavior. Also shown is the one-dimensional bifurcation diagram
along the same r range.(a) Lyapunov spectrumΛk(r); (b) bifurcation diagram obtained from the crossings through the Poincaré surface dz/dt=0, corresponding to maxima of z(t). Parameter values: σ=4,b=0.3,r=30,γ=1.6.
(a)(b)The two largest TLEs for single-variable
coupling, plotted as a function ofk,
are shown in Figures 3(a)–3(c). The synchronization thresholds are k=3.2 and k=1.3 for x-x and y-y couplings, respectively. For z-z coupling, the system shows a window
of stable synchronization ranging from k=0.7
to k=9.0. The system does not
synchronize in the case of w-w coupling.
However, the nonsynchronizing variables z and w, when working together in the z-z plus w-w double-coupling scheme, provide stable synchronization above k=0.34. In the case of all-variable
coupling, stable synchronization is achieved above k=0.21.Transversal Lyapunov exponents (TLEs) forσ=4,b=0.3,r=30,γ=1.6.
(a) and (b) One-variable coupling; (c) two- and four-variable couplings.
(a)(b)(c)


## 2.3.3. Parameter Mismatch

A qualitative method of investigating the
hyperchaos synchronization phenomena is by observing the projections of the
eight-dimensional attractor onto the planes(x1,x2), (y1,y2), (z1,z2), and (w1,w2). In these planes the
straight lines x1=x2, y1=y2, z1=z2, and w1=w2 correspond to the
synchronization hyperplane. In the following we show only the (z1,z2) plane, since the components z1 and z2 seem to be the most difficult to synchronize. For all plots we used x-x coupling. Figure 4 refers to
identical parameters (i.e., without mismatch), illustrating the inability of
the systems to synchronize if k is
less than the threshold value obtained from Figure 3(a), while perfect
synchronization is achieved above the threshold: the same alignment along the
diagonal is observed in all the four projection planes.Synchronization for identical parameters:k below and above the threshold
k=3.2(σ=4,b=0.3,r=30,γ=1.6).
(a)(b)In Figures5 and 6, we observe the effect of mismatch on synchronization (for x-x coupling). In these examples we
applied the same mismatch to all parameters, that is,   Δσ/σ=Δb/b=Δr/r=Δγ/γ. In Figure 5, where k=4.0, we see
that for k values just above the
threshold, some good degree of synchronization is obtained for 1% mismatch;
however, for 5% large deviations from
the diagonal are observed. Figure 6 shows the effect of mismatch for 1% and 5%
in the case of k=10.Effect of mismatch on synchronization fork=4(σ=4,b=0.3,r=30,γ=1.6).
(a)(b)Effect of mismatch on synchronization fork=10(σ=4,b=0.3,r=30,γ=1.6).
(a)(b)


## 3. Experiments with a Simple Electrical Circuit

## 3.1. Circuit Description and Equations

The Lorenz system, being one of the most
important paradigms of chaos, has inspired many attempts to make a physical
system representing its equations, mainly in the form of an electrical circuit.
Some authors have proposed replacing the cross-products of variables by
discontinuities (switching circuits) as in [12, 13], or by continuous piecewise
linear resistors [14], thus resulting in very simple and practical circuits,
although not truly described by the Lorenz equations. More accurate realization,
though more complex, is by the analog computer approach using smooth
cross-product functions, as in [15], which employs 10 integrated circuits (2
multipliers and 8 op amps) and 23
passive components, therefore a total of 33 circuit components for the Lorenz
3D circuit. In the present work we are proposing a simpler easy-to-build
circuit with smooth functions, aiming at encouraging more experimental
approaches on hyperchaos investigation, even by those researchers not trained
in electronics. As stated in [3], to facilitate experiments with coupled
chaotic oscillators the circuit is required to exhibit chaos in a large range
of parameters in order that the coupling will not destroy the attractor, and it
is needed to be simple enough so that several practically identical oscillators
can be easily constructed. The simplest possible smooth Lorenz 3D circuit
appeared in [16], using only 2 integrated circuits (2 multipliers) and 7
passive components (a total of 9 components, thus about 70% smaller than the
circuit by Cuomo et al. in [15]), as
shown in Figure 7(a). The good performance of this circuit is illustrated in
Figure 7(b), which shows the experimental attractor and examples of single-cusp
and double-cusp Lorenz maps (displayed in-line by the circuit, via a
Poincaré-section circuitry). In the present work we extended to 4D that
simplest circuit by adding 2 op amps and 6 passive components, obtaining the
hyperchaotic Lorenz circuit shown in Figure 8(a), redrawn in Figure 8(b) using
circuit theoretic symbols. The following equations describe the circuit: (3.1)C1dv1dt=i−v1R1,Ldidt=−v1v210−R2i+v3,C2dv2dt=R2iv110R4−v2+ER3,C3dv3dt=−v1R1,  where we used the multipliers
transfer function W=0.1(X1−X2)(Y1−Y2)+Z, where Z=v1+v3 on the input of the first multiplier (on the left
side in Figure 8(a)), and Z=v2 on the input of the second
one. In deriving (3.1) we assumed R8=2R9 and R5=R6=R7≫R1. Now, by defining the new
variables and parameters (3.2)x=v110LR1R4C2,y=R1i10LR1R4C2,z=v210R1R2+r,w=v310R1R2LR1R4C2,σ=LR1R2C1,b=LR2R3C2,r=R1E10R2,γ=LR22C3, we obtain (2.2). Therefore, the proposed circuit realizes, exactly,
the Lorenz hyperchaotic system given by (2.2), obviously with some usual
practical restrictions imposed by parasitic effects and finite bandwidth,
slew-rate, excursion range, and so forth.(a)  Simplest circuit for the 3D standard Lorenz equations. (b)  Experimental results from the circuit of Figure7(a): butterfly attractor on
the plane v1×v2; Lorenz maps showing single and double cusps v2(tn)×v2(tn+1).
The straight line in each picture is given by v2(tn+1)=v2(tn). L=10mH,C1=22nF,C2=1nF,R1=1.5kΩ,R2=100Ω,R3=160kΩ,R4=1kΩ.
(a)(b)(a)  Simplest circuit for the 4D hyperchaotic Lorenz system of (2.2). (b) Equivalent Lorenz hyperchaotic circuit.
(a)(b)


## 3.2. Some Experimental Results

For our experiments, two circuits following the
schematic diagram of Figure8(a) were constructed, each one using two AD633
multipliers and two LM351 op amps, all powered with ±15 V, and the following
passive components: L=11mH, C1=22nF, C2=1nF, C3=1μF, R1=1.5kΩ, R2=80Ω, R3=470kΩ, R4=1kΩ, R5=R6=R7=100kΩ, R8=2kΩ, R9=1kΩ. Using (3.2), the corresponding parameters of (2.2) are σ=4.2, b=0.3, γ=1.7. We worked with E=15V, giving r=28. (Note: we verified that with an independent DC power source
it is possible to use E values up to
30 V, or r=56, without waveform
clipping.) The six projections of the experimental hyperchaotic attractor are
shown in Figure 9(b); the calculated attractor is shown in Figure 9(a). For the
experiments on synchronization, bidirectional diffusive coupling can be
obtained simply by connecting a resistor R linking the capacitor C1 of the first circuit with the capacitor C1′
of the second circuit, since in this work we have tested only x-x, or v1-v1,
coupling. For unidirectional coupling a voltage follower is added in series
with the resistor R, as sketched in
Figure 10. The coupling strength is given by the relation k=σR1/R, as can be easily verified by adding
the term (v1′−v1)/R to the first of (3.1). Note that each variable of the second
circuit is represented by the same symbol as the corresponding one of the first
circuit, but with an uppercase prime. The four projections of the trajectories
near the synchronization hyperplane are shown in Figures 11(a) and 11(b) for several
values of the coupling resistor R.
The waveforms v2(t) and v2′(t) are
shown in Figure 12. All these results refer to unidirectional coupling through
the variables v1 and v1′.(a) Projections of a calculated hyperchaotic attractor described by (2.2)
for σ=4.2,b=0.3,γ=1.7, and r=28. From left to right, top: x×z; x×y; y×z; bottom: w×z;x×w;y×w. (b) Experimental hyperchaotic Lorenz attractor generated by the circuit of
Figure 8, as projected on the oscilloscope screen. From left to right, top: v1×v2;v1×i;i×v2; bottom: v3×v2;v1×v3;i×v3. The same parameter values as those
of Figure 9(a)—see text.
(a)(b)Coupling schemes: (a) bidirectional; (b) unidirectional.
(a)(b)(a) Experimental trajectories near the synchronization hyperplane for
several values of the unidirectional coupling resistor. A: 0Ω; B: 100 Ω; C: 220 Ω. (b) Continuation of Figure 11(a). D: 470 Ω; E: 1.0 kΩ; F: 2.2 kΩ.
(a)(b)Experimental waveformsv2 and v2′ corresponding to the first column of
Figure 11.
(a)(b)(c)(d)(e)(f)


## 4. Conclusion

In this work the synchronizing properties of diffusively coupled hyperchaotic Lorenz
4D systems, described by (2.2), have been studied both numerically and
experimentally. The numerical investigation was realized by calculating the transverse Lyapunov exponents as a function of the coupling strength k, and also by visually inspecting the
phase space trajectories near the synchronization hyperplane. We concluded that
using a single coupling variable, either x or y (but neither z nor w), guarantees
stable synchronization. Although z and w are not good choices for the
single-variable scheme, double coupling with both z and w easily provides
synchronization.  We also verified that a
small degree of parameter mismatch seems tolerable. For the laboratory work a
very simple electrical circuit described by the Lorenz 4D system was proposed and
described here for the first time. The experiments confirmed the qualitative
behavior predicted by the numerical approach.



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*Source: 174546-2009-04-23.xml*