## Abstract

We analyse anticipating synchronization in chaotic systems with time-delay coupling. Two algorithms for extending the prediction horizon are considered. One of them is based on the design of a suitable coupling matrix compensating the phase lag in the time-delay feedback term of the slave system. The second algorithm extends the first by incorporating, in the coupling law, information from many previous states of the master and slave systems. We demonstrate the efficiency of both algorithms with the simple dynamical model of coupled unstable spirals, as well as with the coupled Rössler systems. The maximum prediction time attained for the Rössler system is equal to the characteristic period of chaotic oscillations.

## 1. Introduction

Synchronization of oscillations has been known to scientists since the historical discovery of this phenomenon by Huygens in pendulum clocks. Later, synchronization was observed and shown to play an important role in a large variety of systems in physics, chemistry and biology (Pikovsky *et al.* 2001; Boccaletti *et al.* 2002). Original notions and classical theory of synchronization imply periodicity of oscillators. The discovery of chaotic oscillators has originated a new subfield in nonlinear dynamics referred to as synchronization of chaos. A generic feature of nonlinear systems exhibiting chaotic motions is the extreme sensitivity to initial conditions. This feature, known as the butterfly effect, would seem to defy synchronization among dynamical variables in coupled chaotic systems. Nonetheless, several different regimes of chaos synchronization have been discovered and investigated in the past two decades.

Afraimovich *et al.* (1986) and Pecora & Carroll (1990) introduced the notion of identical chaos synchronization. Here, the coupled chaotic systems are identical, and synchronization appears as a coincidence of corresponding variables of coupled systems **r**_{2}(*t*)=**r**_{1}(*t*), as *t* approaches infinity. The identical chaos synchronization in time-delay systems was considered by Pyragas (1998). Rulkov *et al.* (1995) showed that synchronization may appear in unidirectionally coupled non-identical chaotic systems as a functional relationship **r**_{2}(*t*)=**F**[**r**_{1}(*t*)] between the variables of the drive (master) **r**_{1} and response (slave) **r**_{2} systems. This type of synchronization is called the generalized synchronization. Generalized synchronization can also appear in weakly coupled identical chaotic systems and usually precedes the identical synchronization (Pyragas 1996). A particular case of the generalized synchronization is the projective synchronization, when there exists a scale factor in the amplitude of the master’s state variable and that of the slave’s, i.e. **r**_{2}(*t*)=*α***r**_{1}(*t*) (Mainieri & Rehacek 1999). Rosenblum *et al.* (1996) detected phase synchronization, which means the difference between the phase of the master’s state and that of the slave’s is constant during interaction, whereas their amplitudes remain chaotic and uncorrelated, i.e. *nφ*_{1}−*mφ*_{2}=const. (*n* and *m* are integers). Another type of chaos synchronization introduced by Rosenblum *et al.* (1997) is the lag synchronization, when the state of the slave is retarded with the time length of *τ* compared to that of the master, i.e. **r**_{2}(*t*)=**r**_{1}(*t*−*τ*).

More recently, Voss (2000) discovered so-called anticipating synchronization of chaos, which is most counterintuitive among the types of synchronization listed above. Here, unidirectionally coupled systems synchronize in such a way that the slave system predicts the behaviour of the master system, i.e. **r**_{2}(*t*)=**r**_{1}(*t*+*τ*). Voss considered two coupling schemes. The first scheme implies the presence of an internal delay in the master system and the coupling is introduced by complete replacement of the variables
1.1a
and
1.1b
Here, **r**_{1}(*t*) and **r**_{2}(*t*) are the dynamic vector variables of the master and slave system, respectively, **f** is a nonlinear vector function, *τ* is a delay time and *β* is a scalar parameter. It is easy to see that the anticipatory synchronization manifold **r**_{2}(*t*)=**r**_{1}(*t*+*τ*) is a solution of equations (1.1). The stability of this manifold is defined by the equation for the difference Δ(*t*)=**r**_{1}(*t*+*τ*)−**r**_{2}(*t*), which is simply derived from equations (1.1),
1.2
For *β*>0, the anticipatory synchronization manifold is globally stable. Thus, the slave anticipates by an amount *τ* the output of the master.

In the second scheme proposed by Voss, the master system does not possess an internal delay; however, the delay is introduced in the coupling term of the slave system
1.3a
and
1.3b
Here, *k* is a scalar parameter defining the coupling strength and **K** is a coupling matrix. Again, one can check that the anticipatory synchronization manifold **r**_{2}(*t*)=**r**_{1}(*t*+*τ*) is a solution of equations (1.3). However, the proof of stability of this manifold is a more complicated task as for the first scheme. While in the scheme of complete replacement, the anticipation time *τ* coincides with the internal delay time of the master system and can be arbitrarily large, the delay coupling scheme requires some constraints on the anticipation time *τ* and coupling **K** for the synchronization solution to be stable. Despite this fact, the delay coupling scheme is more interesting and promising for applications, since it does not require the presence of delay in the master system, and the anticipation time *τ* can be varied without altering the master’s dynamics.

Anticipating synchronization of chaos has been studied numerically for a variety of systems by Masoller (2001), Calvo *et al.* (2004) and Kostur *et al.* (2005), justified experimentally in electronic circuits by Voss (2002) and in chaotic semiconductor lasers by Sivaprakasam *et al.* (2001) and Tang & Liu (2003). It has also been observed in excitable systems driven by random forcing (Ciszak*et al.* 2003, 2004).

Implementation of anticipating synchronization as a strategy for real-time forecasting of a given dynamics requires the design of coupling schemes with a possibly large anticipation time. The analysis performed by Voss (2000) shows that schemes (1.3) with a diagonal matrix **K** are ineffective. The maximum stable anticipation time is much shorter than the characteristic time scales of the system’s dynamics. To enlarge the prediction time, Voss (2001) proposed to extend equation (1.3b) with a chain of *N* unidirectionally coupled slave systems
1.4
Formally, the prediction time of this scheme is *N* times larger when compared to the scheme (1.3b). However, Mendoza *et al.* (2004) showed that the chain (1.4) is unstable to propagating perturbations and this convective-like instability limits the number of slaves in the chain that can operate in a stable regime.

In our recent publication (Pyragas & Pyragienė 2008), we addressed the question of whether the prediction time can be considerably enlarged with the single-slave system (1.3b) via a suitable choice of the coupling matrix **K**. For typical low-dimensional chaotic systems, we gave a positive answer. We proposed an algorithm of design **K** based on a phase-lag compensating coupling (PLCC) and showed that the prediction time can be enlarged several times in comparison to the diagonal coupling usually used in the literature. Here, we further develop this idea by introducing an extended PLCC (EPLCC) and show that the prediction horizon can be prolonged to the characteristic period of chaotic oscillations.

## 2. Phase-lag compensating coupling

We demonstrate a heuristic idea of the PLCC algorithm with the Rössler (1976) system, which is given by a three-dimensional vector variable **r**=[*x*,*y*,*z*] and vector field
2.1
In the following, we set *a*=0.15, *b*=0.2 and *c*=10 and suppose that both **r** and **f** are the vector columns. The phase portrait of the Rössler system for the given values of the parameters is shown in figure 1. Although the Rössler system has two fixed points, the strange attractor has originated from one of them **r**_{0}=[(*c*−*s*)/2,(*s*−*c*)/2*a*,(*c*−*s*)/2*a*] located close to the origin, where *s*=(*c*^{2}−4*ab*)^{1/2}. The fixed point is a saddle focus with an unstable two-dimensional manifold (an unstable spiral) almost coinciding with the (*x*,*y*)-plane and a stable one-dimensional manifold almost coinciding with the *z*-axis. The phase point of the system spends most time in the (*x*,*y*)-plane moving along the unstable spiral according to approximate equations and . Whenever *x* approaches a value *x*≈*c*, the *z* variable comes into play. The phase point leaves, for a short time, the (*x*,*y*)-plane and then returns to the origin via a stable *z*-axis manifold.

Taking into account such a topology of the strange attractor, we choose the coupling matrix as **K**=**Q**, where
2.2
is a 3×3 matrix that projects the vector field onto the unstable (*x*,*y*)-plane and rotates this projection by the angle *α*=*ωτ*. Here, *ω* is a frequency of the unstable spiral, for which the Rössler system is ≈1. The main advantage of such a choice consists of phase-lag compensation of the time-delay feedback term in equation (1.3b). When the system moves along the unstable spiral in the (*x*,*y*)-plane, the vector **Qr**_{2}(*t*−*τ*) is in phase with the vector **r**_{2}(*t*) (cf. figure 2), and thus the term **Kr**_{2}(*t*−*τ*) provides a correct negative feedback. We refer to this coupling law as a PLCC. In figure 3*a*,*b*, we compare the effect of the PLCC with the usual diagonal coupling, when **K**=diag[1,1,1]. The time of reliable prediction for the PLCC is *τ*≈3.8. It exceeds four times the maximum prediction time for the diagonal coupling. The characteristic period of chaotic oscillations for the Rössler system is *T*≈6.3. Thus, the PLCC algorithm allows us to make prediction for more than a half of this period.

We stress that the PLCC enables forecasting of the global dynamics of the system, although the coupling matrix (2.2) takes into account only the local properties of the phase space. Indeed, the phase-lag compensation via rotation of the vector field is strongly valid only in the vicinity of the fixed point. It is notable that a similar rotation feedback gain has been recently used by Fiedler et al. (2007) in a problem of the delayed feedback control (DFC; Pyragas 1992) to overcome the so-called odd-number limitation.

The success of the PLCC algorithm can be explained by a simple analytical model. The instability of the Rössler attractor is defined by the unstable spiral lying in the (*x*,*y*)-plane, where the phase point of the system spends most time. Thus, one can expect that the main properties of anticipating synchronization of the Rössler systems can be derived from local motion on the unstable spirals, i.e. by considering the simplified problem of anticipating synchronization of the mere spirals. Specifically, assume that the dynamical system under consideration is described by two linear equations and that define an unstable spiral with the positive increment *γ* and frequency *ω*. For the complex variable *Z*=*x*+*iy*, this system can be presented by a single equation . Then, equations for anticipating synchronization of two spirals take the form
2.3a
and
2.3b
where *k*e^{iα} is a complex coupling coefficient. By a suitable choice of the phase *α*, we can model both the PLCC (*α*=*ωτ*) and diagonal coupling (*α*=0). The solution of the master system is *Z*_{1}(*t*)=*Z*_{0}e^{(γ+iω)t} and the deviation *δZ*_{2}(*t*)=*Z*_{2}(*t*)−*Z*_{0}e^{(γ+iω)(t+τ)} from the anticipated state satisfies . Substituting *δZ*_{2}(*t*)=*C*e^{(λ+iω)t}, we obtain the characteristic equation
2.4
that defines the eigenvalues λ of the synchronized state in a rotating frame with the frequency *ω*; the real parts of λ represent the transversal Lyapunov exponents of the anticipation manifold. For the PLCC, *α*=*ωτ* and the solution of equation (2.4) is
2.5
where *W*(*z*) is the Lambert *W* function, satisfying the definition *W*(*z*)e^{W(z)}=*z* (Corless *et al.* 1996). The dependence of the leading transversal Lyapunov exponent Re λ on the coupling strength *k* is shown in figure 4*a*; it is determined by the principal branch of the Lambert function. The synchronized state is stable in some interval of the coupling strength *k*_{1}<*k*<*k*_{2}, where Reλ(*k*)<0. The lower bound of stability is determined by the spiral’s increment, *k*_{1}=*γ*, and is independent of *τ*. The dependence of the upper bound on *τ*, *k*_{2}=*k*_{2}(*τ*), is defined by the parametric equation presented in the caption of figure 5. The leading transversal Lyapunov exponent reaches the minimal value Re λ(*k*_{0})=*γ*−1/*τ* at *k*_{0}=e^{γτ−1}/*τ*. Thus, the necessary condition for stability of the synchronized state is Re λ(*k*_{0})<0 or
2.6
where *τ*_{H} is the prediction horizon. As expected from a general theory, the limit for the prediction horizon is determined by the inverse of the largest Lyapunov exponent of the system, which for the spiral is 1/*γ*. In figure 5, we compare stability regions in the (*τ*,*k*)-plane for the PLCC (*α*=*ωτ*) and diagonal coupling (*α*=0). We see that the region of the PLCC is considerably larger than that of the diagonal coupling. Note that these regions are in approximate quantitative agreement with the corresponding regions of the Rössler system (cf. Pyragas & Pyragienė 2008). Thus, the characteristic parameters of anticipating chaotic synchronization can be estimated analytically from the simple linear equations (2.3) that model local dynamics of the chaotic system close to the fixed point.

The PLCC algorithm can be generalized for any Rössler-type chaotic system. Suppose that a strange attractor of a three-dimensional dynamical system
2.7
has originated from a saddle-focus fixed point **r**_{0}, such that **f**(**r**_{0})=0. Generally, the unstable and stable manifolds of the fixed point may have arbitrary orientations in the phase space. To design the coupling for this general case, we first shift coordinates to the fixed point and rewrite the governing equation in the form
2.8
where **J**=∂**f**/∂**r**|_{r=r0} is the Jacobian matrix, **N**(**R**)=**f**(**r**_{0}+**R**)−**JR** is a nonlinear function and **R**=**r**−**r**_{0}. Then, equation (1.3b) for the slave system can be written as
2.9
According to our assumptions, the Jacobian **J** has a pair of complex conjugate eigenvalues λ_{1,2}=*γ*±*iω*, with *γ*>0, corresponding to a two-dimensional unstable spiral manifold, and a real negative eigenvalue λ_{3}, representing the stable one-dimensional manifold. By a suitable change of variables, the Jacobian **J** can be transformed to Jordan normal form, i.e.
2.10
where **E** is the matrix of eigenvectors of **J**. This transformation orients the unstable two-dimensional manifold towards the (*x*,*y*)-plane and the stable manifold towards the *z*-axis. After such a transformation, we can apply the above theoretical arguments and use the coupling law **K**=**Q**. This means that, in the original (non-transformed) variables, the coupling matrix has to be constructed as
2.11

Equation (2.11) gives a general algorithm of coupling design for typical chaotic systems. Application of the general coupling law (2.11) to the Rössler system does not advance significantly the forecasting algorithm in comparison to the above-considered heuristic approach. This is because the Jacobian of the Rössler system is close to Jordan normal form and the matrix **EQE**^{−1} does not differ significantly from the matrix **Q**. The coupling law (2.11) not only works well for mono-scroll chaotic attractors, but it also extends considerably the prediction time of double-scroll chaotic systems, such as the Chua circuit or the Lorenz system (cf. Pyragas & Pyragienė 2008).

## 3. Extended phase-lag compensating coupling

To further improve the PLCC algorithm, we employ an idea from DFC theory (cf. Pyragas 2006). The aim of this theory is the stabilization of unstable periodic orbits embedded in chaotic attractors. In the original DFC algorithm (cf. Pyragas 1992), the coupling law is analogous to that of the anticipating synchronization. Gauthier *et al.* (1994) proposed an extension of the original DFC algorithm, which incorporates, in the feedback loop, information from many previous states of the system in the form closely related to the amplitude of light reflected from a Fabry–Perot interferometer. This modification, known as an extended DFC (EDFC), improves the algorithm considerably. The EDFC makes the stabilization of high-periodic orbits with large positive Lyapunov exponents possible (cf. Pyragas 1995).

Here, we use the idea of extended time-delay feedback to improve the performance of the PLCC algorithm. We refer to this modification as an EPLCC. The PLCC uses the single difference between the current state of the master system and the time-delayed state of the response system, whereas the EPLCC extends this perturbation to an infinite series of corresponding differences that are time-delayed by integer multiples of *τ*. We first demonstrate the idea for two coupled unstable spirals
3.1a
and
3.1b
The EPLCC perturbation *S*(*t*) is governed by
3.2
where *R* is a memory parameter that regulates the weight of the time-delayed information. We require −1<*R*<1 to guarantee the convergence of the infinite sum. The case when *R*=0 corresponds to the PLCC considered in the previous section (cf. equation (2.3b)). Note that *S*(*t*) vanishes for any value of *R* when the system moves along the synchronization manifold, since the equality *Z*_{1}(*t*−*nτ*)=*Z*_{2}(*t*−(*n*+1)*τ*) holds on the manifold for all *n*. The phase lag of each term in the sum is compensated by the multiplier e^{i(n+1)α}, and the amplitude is weighted by the multiplier *R*^{n}, such that the contribution of terms exponentially decreases with the increase of time delay.

At first sight, the perturbation (3.2) seems to be very complex. Fortunately, the infinite sum can be rewritten in an equivalent, recursive form
3.3
and implemented experimentally with the single delay line (cf. Gauthier *et al.* 1994). Now, the stability of the synchronized state is defined by variational equations
3.4a
and
3.4b
where Δ(*t*)=*Z*_{1}(*t*)−*Z*_{2}(*t*−*τ*) describes the deviation from the synchronization manifold. This leads to the characteristic equation
3.5
For *R*=0, it coincides with equation (2.4) analysed in the previous section. Here, as well as in the previous section, we set *α*=*ωτ* and analyse the simplified version of equation (3.5)
3.6

Unfortunately, the solution of equation (3.6) for *R*≠0 cannot be expressed through the Lambert function. Numerical solutions of this equation for different values of *R* are presented in figure 4. We see that the negative values of *R* improve the stability properties of the synchronization manifold.

Although an analytical solution of equation (3.6) is not available, the boundaries of stability can be obtained analytically. The lower bound of stability *k*_{1}=*γ*(1−*R*) is deduced from equation (3.6) by substituting λ=0. The upper bound of stability *k*_{2} is defined by a Hopf bifurcation. Substituting λ=i*Ω* in equation (3.6) and denoting *Ωτ*=*β*, we obtain the dependence *k*_{2} on *τ* in a parametric form
3.7
where the parameter *β* is varied in the interval . These boundaries are depicted in figure 5 for *R*=−0.3. We see that the EPLCC extends the stability region of the synchronization manifold in the (*τ*,*k*)-plane. The maximum prediction time can be obtained from equation (3.7) in the limit . The necessary condition for stability of the synchronized state is
3.8
Comparing this with equation (2.6) derived for the PLCC, we see that the EPLCC extends the prediction horizon by a factor (1−*R*). This factor is greater than 1 only for negative values of the parameter *R*. When , this factor reaches a maximal value equal to 2. Thus, for coupled spirals, the maximum prediction horizon of the EPLCC exceeds two times the maximum prediction horizon of the PLCC. Note that, in the case of negative values of *R*, the feedback law (3.2) describes a series with alternating signs. Only such a series advances the stability of anticipating synchronization.

We now consider an application of the EPLCC algorithm for chaotic systems in general. We extend equations (1.3) for the EPLCC as follows:
3.9a
and
3.9b
where the perturbation **S**(*t*) is governed by the recursive equation
3.10
and the coupling matrix **K** is defined by equation (2.11). For *R*=0, these equations describe the PLCC considered previously. Note that the anticipatory synchronization manifold **r**_{2}(*t*)=**r**_{1}(*t*+*τ*) is a solution of equations (3.9) and (3.10) for any *R*.

The linear stability of the anticipatory synchronization manifold is determined by variational equations
3.11a
and
3.11b
where Δ(*t*)=**r**_{1}(*t*)−**r**_{2}(*t*−*τ*) is a transversal deviation from the synchronization manifold. The growth rates of this deviation define the transversal Lyapunov exponents. A necessary condition for the synchronized regime to be stable is that the maximum transversal Lyapunov exponent λ_{⊥} is negative.

The performance of the EPLCC is demonstrated for the Rössler system in figures 3*c* and 6. In figure 6, we plot the dependence of λ_{⊥} on the delay time *τ*. This characteristic period is obtained by numerical integration of the variational equations (3.11) together with equation (3.9a). The transversal Lyapunov exponent has a minimum at the delay time *τ*≈6.3, coinciding with the characteristic period of chaotic oscillations *T*=2*π*/*ω*, where *ω* is the frequency of the relevant saddle-fixed point of the strange attractor. Thus, the EPLCC provides an optimal prediction for time intervals close to the characteristic period of the strange attractor. In figure 3*c*, we show the dynamics of the drive (3.9a) and response (3.9b) Rössler systems in the case of the EPLCC (3.10), with *R*=−0.45. In comparison to the PLCC (cf. figure 3*b*), the EPLCC prolongs the maximum prediction horizon by almost twice.

## 4. Conclusions

We have considered two algorithms that extend the anticipation horizon of chaos synchronization schemes with a time-delay coupling. The algorithms are referred to as PLCC and EPLCC. The PLCC uses the topological properties of a typical chaotic attractor originated from a saddle-focus fixed point. The main idea is to design the coupling matrix in such a way that it compensates the phase lag in the time-delayed feedback term of the slave system. The PLCC extends, by several times, the maximum prediction horizon when compared to the diagonal coupling usually used in the literature. The second, EPLCC, algorithm is even more advantageous. It uses information from many previous states of the master and slave systems. The idea of the EPLCC comes from DFC theory, where a similar feedback law has shown its merits. The EPLCC modification exceeds about twice the maximum prediction of the PLCC. The maximum prediction time attained with the EPLLC is equal to the characteristic period of chaotic oscillations.

The PLCC and EPLLC algorithms can be used as a strategy for real-time forecasting of chaotic dynamics in many technical applications. They can be implemented using an analogue technique, and are especially advantageous for forecasting the dynamics of fast chaotic systems. We hope that our findings will stimulate the search for appropriate coupling laws in other problems of anticipating synchronization, e.g. to enhance the predictability of chaotic systems with unknown dynamical models (Ciszak *et al.* 2005) or excitable systems driven by random forcing (Ciszak *et al.*2003, 2004).

## Footnotes

One contribution of 13 to a Theme Issue ‘Delayed complex systems’.

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