We study the stability of linear fractional order maps. We show that in the stable region, the evolution is described by Mittag-Leffler functions and a well defined effective Lyapunov exponent can be obtained in these cases. For one-dimensional systems, this exponent can be related to the corresponding fractional differential equation. A fractional equivalent of map $f(x)=ax$ is stable for $a_c(α)<a<1$ where $α$ is a fractional order parameter and $a_c(α)\approx -α$. For coupled linear fractional maps, we can obtain `normal modes' and reduce the evolution to effectively one-dimensional system. If the eigenvalues are real the stability of the coupled system is dictated by the stability of effectively one-dimensional normal modes. For complex eigenvalues, we obtain a much richer picture. However, in the stable region, the evolution of modulus is dictated by Mittag-Leffler function and the effective Lyapunov exponent is determined by modulus of eigenvalues. We extend these studies to synchronized fixed points of fractional nonlinear maps.
In this work, we introduce a new three-dimensional chaotic differential dynamical system. We find equilibrium points of this system and provide the stability conditions for various fractional orders. Numerical simulations will be used to investigate the chaos in the proposed system. A simple linear control will be used to control the chaotic oscillations. Further, we propose an optimal control which is based on the fractional order of the system and use it to synchronize new chaotic system.
Synchronization of fractional-order chaotic systems is a hot topic in the field of nonlinear study. The co-coupled synchronization between two fractional-order chaotic systems with different initial conditions is investigated in this paper. Based on Lyapunov stability principle and Gerschgorin theorem, the co-coupled synchronization theorem of fractional-order chaotic systems is deduced, and the range of coupling coefficients is confirmed for synchronization of fractional-order unified chaotic systems. By building up the synchronization simulation model on Simulink, the co-coupled synchronizat…
The chaotic dynamics of fractional order systems begin to attract much attentions in recent years. In this brief report, we study the master-slave synchronization of fractional order chaotic systems. It is shown that fractional order chaotic systems can also be synchronized.
The chaotic dynamics of fractional (non-integer) order systems have begun to attract much attention in recent years. In this paper, we study the projective synchronization in two coupled fractional order chaotic oscillators. It is shown that projective synchronization can also exist in coupled fractional order chaotic systems. A simple feedback control method for controlling the scaling factor onto a desired value is also presented.
The aim of this study is to prove analytically that synchronization of a piece-wise continuous class of systems of fractional order can be achieved. Based on our knowledge, there are no numerical methods to integrate differential equations with discontinuous right hand side of fractional order which model these systems. Therefore, via Filippov's regularization [1] and Cellina's Theorem [2,3], we prove that the initial value problem can be converted into a continuous problem of fractional-order, to which numerical methods for fractional orders apply. In this way, the synchronization problem tra…
This paper is devoted to the problem of synchronization between fractional-order chaotic systems with Gaussian fluctuation by the method of fractional-order sliding mode control. A fractional integral (FI) sliding surface is proposed for synchronizing the uncertain fractional-order system, and then the sliding mode control technique is carried out to realize the synchronization of the given systems. One theorem about sliding mode controller is presented to prove the proposed controller can make the system synchronize. As a case study, the presented method is applied to the fractional-order Che…
In this paper, an adaptive fractional controller has been designed to control chaotic systems. In fact, this controller is a fractional PID controller, which the coefficients will be tuned according to a proper adaptation mechanism. The adaptation law will be constructed from a sliding surface via gradient method. The adaptive fractional controller is implemented on a gyro system to signify the performance of the proposed technique.
In this paper, fractional order Coullet system is studied. An active control technique is applied to control this chaotic system. This type of controller is also applied to synchronize chaotic fractional-order systems in master-slave structure. The synchronization procedure is shown via simulation. The boundary of stability is obtained by both of theoretical analysis and simulation result. The numerical simulations show the effectiveness of the proposed controller.
In this paper, a sliding mode controller is designed to synchronize a chaotic fractional-order system. To construct a corrective control input, a saturation function sat(.), with a modified sliding surface is proposed. Finally, Chaos in the Duffing-Holmes system with fractional orders is investigated, and a numerical simulation (synchronizing fractional-order Duffing-Holmes _ Duffing-Holmes system) are presented to show the effectiveness of the proposed controller.
In this paper, synchronization of fractional order Coullet system with precise and also unknown parameters are studied. The proposed method which is based on the adaptive backstepping, has been developed to synchronize two chaotic systems with the same or partially different attractor. Sufficient conditions for the synchronization are analytically obtained. There after an adaptive control law is derived to make the states of two slightly mismatched chaotic Coullet systems synchronized. The stability analysis is then proved using the Lyapunov stability theorem. It is the privilege of the approa…
In the present Letter we show that the concept of the generalized synchronization regime in discrete maps needs refining in the same way as it has been done for the flow systems [PRE, 84 (2011) 037201]. We have shown that\alkor{, in the general case, when the relationship between state vectors of the interacting chaotic maps are considered,} the prehistory must be taken into account. We extend the phase tube approach to the systems with a discrete time coupled both unidirectionally and mutually and analyze the essence of the generalized synchronization by means of this technique. Obtained resu…
Synchronization of identical chaotic systems subjected to common noise has been the subject of recent research. Studies on several chaotic systems have shown that, the synchronization is actually induced by the non-zero mean of the noise, and symmetric noise with zero-mean cannot lead to synchronization. Here it is presented that synchronization can be achieved by {\sl zero-mean} noise in some chaotic maps with large convergence regions.
We study the synchronization of time-delayed nonidentical maps subject to unidirectional (master-slave) coupling. The individual dynamics of the maps have a delay n1, and the coupling acts with a delay n2. We show analytically that, suitably tuning the slave map parameters, two distinct synchronization regimes can occur. In one regime the lag time between the slave and the master maps is given by the delay of the coupling, n2, while in the other regime is given by the difference between the delays, n1-n2. We analyze the effect of the coupling strength on the different synchronization regimes i…
We prove a sufficient condition for synchronization for coupled one-dimensional maps and estimate the size of the window of parameters where synchronization takes place. It is shown that coupled systems on graphs with positive eigenvalues (EVs) of the normalized graph Laplacian concentrated around 1 are more amenable for synchronization. In the light of this condition, we review spectral properties of Cayley, quasirandom, power-law graphs, and expanders and relate them to synchronization of the corresponding networks. The analysis of synchronization on these graphs is illustrated with numerica…
Unidirectionally coupled area-preserving maps with a mixed phase space may show identical synchronization in the sticky neighborhood of the regular islands. We use this fact to devise numerical procedures to control (delay and expedite) the process of synchronization in two standard maps coupled under the Pecora-Carroll coupling scheme. The delay method is based on controlled kicking of trajectories away from synchronization traps for as long as necessary. The method to expedite the process is achieved by a parameter perturbation technique which rapidly drives the chaotic trajectories to synch…
We study synchronization of low-dimensional ($d=2,3,4$) chaotic piecewise linear maps. For Bernoulli maps we find Lyapunov exponents and locate the synchronization transition, that numerically is found to be discontinuous (despite continuously vanishing Lyapunov exponent(s)). For tent maps, a limit of stability of the synchronized state is used to locate the synchronization transition that numerically is found to be continuous. For nonidentical tent maps at the partial synchronization transition, the probability distribution of the synchronization error is shown to develop highly singular beha…
We investigate the phenomenon of chaos synchronization in systems subject to coexisting autonomous and external global fields by employing a simple model of coupled maps. Two states of chaos synchronization are found: (i) complete synchronization, where the maps synchronize among themselves and to the external field, and (ii) generalized or internal synchronization, where the maps synchronize among themselves but not to the external global field. We show that the stability conditions for both states can be achieved for a system of minimum size of two maps. We consider local maps possessing rob…
We introduce a new method for determining the global stability of synchronization in systems of coupled identical maps. The method is based on the study of invariant measures. Besides the simplest non-trivial example, namely two symmetrically coupled tent maps, we also treat the case of two asymmetrically coupled tent maps as well as a globally coupled network. Our main result is the identification of the precise value of the coupling parameter where the synchronizing and desynchronizing transitions take place.
We study synchronization of random one-dimensional linear maps for which the Lyapunov exponent can be calculated exactly. Certain aspects of the dynamics of these maps are explained using their relation with a random walk. We confirm that the Lyapunov exponent changes sign at the complete synchronization transition. We also consider partial synchronization of nonidentical systems. It turns out that the way partial synchronization manifests depends on the type of differences (in Lyapunov exponent or in contraction points) between the systems. The crossover from partial synchronization to comple…
In this paper two important aspects related to Caputo fractional-order discrete variant of a class of maps defined on the complex plane, are analytically and numerically revealed: attractors symmetry-broken induced by the fractional-order and the sensible problem of determining the right bifurcation diagram of discrete systems of fractional-order. It is proved that maps of integer order with dihedral symmetry or cycle symmetry loose their symmetry once they are transformed in fractional-order maps. Also, it is conjectured that, contrarily to integer-order maps, determining the bifurcation diag…
The problem of synchronization of coupled Hamiltonian systems exhibits interesting features due to the non-uniform or mixed nature (regular and chaotic) of the phase space. We study these features by investigating the synchronization of unidirectionally coupled area-preserving maps coupled by the Pecora-Carroll method. We find that coupled standard maps show complete synchronization for values of the nonlinearity parameter at which regular structures are still present in phase space. The distribution of synchronization times has a power law tail indicating long synchronization times for at lea…
The phenomenology of a system of two coupled quadratic maps is studied both analytically and numerically. Conditions for synchronization are given and the bifurcations of periodic orbits from this regime are identified. In addition, we show that an arbitrarily large number of distinct stable periodic orbits may be obtained when the maps parameter is at the Feigenbaum period-doubling accumulation point. An estimate is given for the coupling strength needed to obtain any given number of stable orbits.
The properties of functional relation between a non-invertible chaotic drive and a response map in the regime of generalized synchronization of chaos are studied. It is shown that despite a very fuzzy image of the relation between the current states of the maps, the functional relation becomes apparent when a sufficient interval of driving trajectory is taken into account. This paper develops a theoretical framework of such functional relation and illustrates the main theoretical conclusions using numerical simulations.
The phenomena of synchronization and nontrivial collective behavior are studied in a model of coupled chaotic maps with random global coupling. The mean field of the system is coupled to a fraction of elements randomly chosen at any given time. It is shown that the reinjection of the mean field to a fraction of randomly selected elements can induce synchronization and nontrivial collective behavior in the system. The regions where these collective states emerge on the space of parameters of the system are calculated.
Through numerical simulations we analyze the synchronization time and the Lyapunov dimension of a coupled map lattice consisting of a chain of chaotic logistic maps exhibiting power law interactions. From the observed behaviors we find a lower bound for the size $N$ of the lattice, independent of the range and strength of the interaction, which imposes a practical lower bound in numerical simulations for the system to be considered in the thermodynamic limit. We also observe the existence of a strong correlation between the averaged synchronization time and the Lyapunov dimension. This is an i…
We study the phase synchronization and cluster formation in coupled maps on different networks. We identify two different mechanisms of cluster formation; (a) {\it Self-organized} phase synchronization which leads to clusters with dominant intra-cluster couplings and (b) {\it driven} phase synchronization which leads to clusters with dominant inter-cluster couplings. In the novel driven synchronization the nodes of one cluster are driven by those of the others. We also discuss the dynamical origin of these two mechanisms for small networks with two and three nodes.
We study the synchronization of two chaotic maps with unidirectional (master-slave) coupling. Both maps have an intrinsic delay $n_1$, and coupling acts with a delay $n_2$. Depending on the sign of the difference $n_1-n_2$, the slave map can synchronize to a future or a past state of the master system. The stability properties of the synchronized state are studied analytically, and we find that they are independent of the coupling delay $n_2$. These results are compared with numerical simulations of a delayed map that arises from discretization of the Ikeda delay-differential equation. We show…
The phenomenon of synchronization occurring in a locally coupled map lattice subject to an external drive is compared to the synchronization process in an autonomous coupled map system with similar local couplings plus a global interaction. It is shown that chaotic synchronized states in both systems are equivalent, but the collective states arising after the chaotic synchronized state becomes unstable can be different in these two systems. It is found that the external drive induces chaotic synchronization as well as synchronization of unstable periodic orbits of the local dynamics in the dri…
Coupled map lattices (CMLs) are prototypical dynamical systems on networks/graphs. They exhibit complex patterns generated via the interplay of diffusive/Laplacian coupling and nonlinear reactions modelled by a single iterated map at each node; the maps are often taken as unimodal, e.g., logistic or tent maps. In this letter, we propose a class of higher-order coupled dynamical systems involving the hypergraph Laplacian, which we call coupled hypergraph maps (CHMs). By combining linearized (in-)stability analysis of synchronized states, hypergraph spectral theory, and numerical methods, we det…
We consider regular lattices of coupled chaotic maps. Depending on lattice size, there may exist a window in parameter space where complete synchronization is eventually attained after a transient regime. Close outside this window, an intermittent transition to synchronization occurs. While asymptotic transversal Lyapunov exponents allow to determine the synchronization threshold, the distribution of finite-time Lyapunov exponents, in the vicinity of the critical frontier, is expected to provide relevant information on phenomena such as intermittency. In this work we scrutinize the distributio…
Synchronization among globally coupled, chaotic map lattices can be related to stable periodic windows in isolated chaotic maps. This relation provides a simple predictive tool for the understanding of complicated behavior in coupled physical, chemical and biological systems.
We study the fractional maps of complex order, $α_0e^{i r π/2}$ for $0<α_0<1$ and $0\le r<1$ in 1 and 2 dimensions. In two dimensions, we study H{é}non and Lozi map and in $1d$, we study logistic, tent, Gauss, circle, and Bernoulli maps. The generalization in $2d$ can be done in two different ways which are not equivalent for fractional-order and lead to different bifurcation diagrams. We observed that the smooth maps such as logistic, Gauss, and H{é}non maps do not show chaos while discontinuous maps such as Lozi, Bernoulli, and circle maps show chaos. The tent map is continuous but not diffe…
In this paper, we show that the stability analysis in the paper A note on stability of fractional logistic maps, Appl. Math. Lett. 125 (2022) 107787 is incorrect and repeat a proof of a theorem on the convergence of a convolution of the product of a converging series with a converging sequence. We also mention that the paper commented on should have been edited more carefully.
We investigate the parametric evolution of riddled basins related to synchronization of chaos in two coupled piecewise-linear Lorenz maps. Riddling means that the basin of the synchronized attractor is shown to be riddled with holes belonging to another basin in an arbitrarily fine scale, which has serious consequences on the predictability of the final state for such a coupled system. We found that there are wide parameter intervals for which two piecewise-linear Lorenz maps exhibit riddled basins (globally or locally), which indicates that there are riddled basins in coupled Lorenz equations…
Covariant Lyapunov vectors for scale-free networks of Henon maps are highly localized. We revealed two mechanisms of the localization related to full and phase cluster synchronization of network nodes. In both cases the localization nodes remain unaltered in course of the dynamics, i.e., the localization is nonwandering. Moreover this is predictable: the localization nodes are found to have specific dynamical and topological properties and they can be found without computing of the covariant vectors. This is an example of explicit relations between the system topology, its phase space dynamics…
We study the synchronization of a linear array of globally coupled identical logistic maps. We consider a time-delayed coupling that takes into account the finite velocity of propagation of the interactions. We find globally synchronized states in which the elements of the array evolve along a periodic orbit of the uncoupled map, while the spatial correlation along the array is such that an individual map sees all other maps in his present, current, state. For values of the nonlinear parameter such that the uncoupled maps are chaotic, time-delayed mutual coupling suppress the chaotic behavior …
The synchronization behavior of delay coupled chaotic smooth unimodal maps over a ring network with stochastic switching of links at every time step is reported in this paper. It is observed that spatiotemporal synchronization never appears for nearest neighbor connections; however, stochastic switching of connections with homogeneous delay $(τ)$ is capable of synchronizing the network to homogeneous steady state or periodic orbit or synchronized chaotically oscillating state depending on the delay parameter, stochasticity parameter and map parameters. Linear stability analysis of the synchron…
We investigate the dynamics of an array of logistic maps coupled with random delay times. We report that for adequate coupling strength the array is able to synchronize, in spite of the random delays. Specifically, we find that the synchronized state is a homogeneous steady-state, where the chaotic dynamics of the individual maps is suppressed. This differs drastically from the synchronization with instantaneous and fixed-delay coupling, as in those cases the dynamics is chaotic. Also in contrast with the instantaneous and fixed-delay cases, the synchronization does not dependent on the connec…
Several coupled maps models are sketched and reviewed in this short communication. First, a discrete logistic type model that was proposed for the symbiotic interaction of two species. Second, a model of many of these symbiotic species evolving together under the coupling action of a global mean field. Third, a similar model with many of these species interacting in a symbiotic way with the nearest neighboring species in different kind of networks. The phenomena of bistability and synchronization is discussed in these models. Finally, some new results concerning the last model embedded in diff…
Local Lyapunov analysis via micro-ensembles: finite-time Lyapunov exponent estimation and KNN-based predictive comparison in complex-valued BAM neural networks
A new type of nonlinear time series analysis is introduced, based on phases, which are defined as polar angles in spaces spanned by a finite number of delayed coordinates. A canonical choice of the polar axis and a related implicit estimation scheme for the potentially underlying auto-regressive circle map (next phase map) guarantee the invertibility of reconstructed phase space trajectories to the original coordinates. The resulting Fourier approximated, Invertibility enforcing Phase Space map (FIPS map) is well suited to detect conditional asymptotic stability of coupled phases. This rather …
We study the synchronization of a coupled map lattice consisting of a one-dimensional chain of logistic maps. We consider global coupling with a time-delay that takes into account the finite velocity of propagation of interactions. We recently showed that clustering occurs for weak coupling, while for strong coupling the array synchronizes into a global state where each element sees all other elements in its current, present state [Physica A {\bf 325} (2003) 186, Phys. Rev. E {\bf 67} (2003) 056219]. In this paper we study the effects of in-homogeneities, both in the individual maps, which are…
In this paper we extend the notion of an $α$-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling factorial-law memory which is asymptotically power-law memory. We introduce the fractional difference Universal, Standard, and Logistic $α$-Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory.
A network of delay-coupled logistic maps exhibits two different synchronization regimes, depending on the distribution of the coupling delay times. When the delays are homogeneous throughout the network, the network synchronizes to a time-dependent state [Atay et al., Phys. Rev. Lett. 92, 144101 (2004)], which may be periodic or chaotic depending on the delay; when the delays are sufficiently heterogeneous, the synchronization proceeds to a steady-state, which is unstable for the uncoupled map [Masoller and Marti, Phys. Rev. Lett. 94, 134102 (2005)]. Here we characterize the transition from ti…
Dynamical behaviour of discrete dynamical systems has been investigated extensively in the past few decades. However, in several applications, long term memory plays an important role in the evolution of dynamical variables. The definition of discrete maps has recently been extended to fractional maps to model such situations. We extend this definition to a spatiotemporal system. We define a coupled map lattice on different topologies, namely, one-dimensional coupled map lattice, globally coupled system and small-world network. The spatiotemporal patterns in the fractional system are more orde…
We study the influence of network topology and connectivity on the synchronization properties of chaotic logistic maps, interacting with random delay times. Four different types of topologies are investigated: two regular (a ring-type and a ring-type with a central node) and two random (free-scale Barabasi-Albert and small-world Newman-Watts). The influence of the network connectivity is studied by varying the average number of links per node, while keeping constant the total input that each map receives from its neighbors. For weak coupling the array does not synchronize regardless the topolo…
Starting from kicked equations of motion with derivatives of non-integer orders, we obtain "fractional" discrete maps. These maps are generalizations of well-known universal, standard, dissipative, kicked damped rotator maps. The main property of the suggested fractional maps is a long-term memory. The memory effects in the fractional discrete maps mean that their present state evolution depends on all past states with special forms of weights. These forms are represented by combinations of power-law functions.
This article is a short review of the recent results on properties of nonlinear fractional maps which are maps with power- or asymptotically power-law memory. These maps demonstrate the new type of attractors - cascade of bifurcations type trajectories, power-law convergence/divergence of trajectories, period doubling bifurcations with changes in the memory parameter, intersection of trajectories, and overlapping of attractors. In the limit of small time steps these maps converge to nonlinear fractional differential equations.
Coupled map lattices are paradigmatic models of many collective phenomena. However, quite different patterns can emerge depending on the updating scheme. While in early versions, maps were updated synchronously, there has been in recent years a concern to consider more realistic updating schemes where elements do not change all at once. Asynchronous updating schemes and the inclusion of time delays are seen as more realistic than the traditional parallel dynamics, and, in diverse works, either one or the other has been implemented separately. But are they actually distinct cases? For coupled m…
Caputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. The conditions of the asymptotic stability of the fixed points for maps of the orders $0< α<1$ that are derived in this paper are narrower than the conditions of stability for the discrete convolution equations in general and wider than the well-known conditions of stability for the fractional difference maps. The derived stability conditions for the fractional standard and log…
This paper is devoted to the memory of Professor George M. Zaslavsky passed away on November 25, 2008. In the field of discrete maps, George M. Zaslavsky introduced a dissipative standard map which is called now the Zaslavsky map. G. Zaslavsky initialized many fundamental concepts and ideas in the fractional dynamics and kinetics. In this paper, starting from kicked damped equations with derivatives of non-integer orders we derive a fractional generalization of discrete maps. These fractional maps are generalizations of the Zaslavsky map and the Henon map. The main property of the fractional d…
We investigate the processes of synchronization and phase ordering in a system of globally coupled maps possessing bistable, chaotic local dynamics. The stability boundaries of the synchronized states are determined on the space of parameters of the system. The collective properties of the system are characterized by means of the persistence probability of equivalent spin variables that define two phases, and by a magnetization-like order parameter that measures the phase-ordering behavior. As a consequence of the global interaction, the persistence probability saturates for all values of the …
The possibility of realization of the phenomena of complex analytic dynamics for the realistic physical models are investigated. Observation of the Mandelbrot and Julia sets in the parameter and phase spaces both for the discrete maps and non-autonomous continuous systems is carried out. For these purposes, the method, based on consideration of coupled systems, demonstrating period-doubling cascade is suggested. Novel mechanism of synchronization loss in coupled systems corresponded to the dynamical behavior intrinsic to the complex analytic maps is offered.
In this paper the author compares behaviors of systems which can be described by fractional differential and fractional difference equations using the fractional and fractional difference Caputo Standard $α$-Families of Maps as examples. The author shows that properties of fractional difference maps (systems with falling factorial-law memory) are similar to the properties of fractional maps (systems with power-law memory). The similarities (types of attractors, power-law convergence of trajectories, existence of cascade of bifurcations and intermittent cascade of bifurcations type trajectories…
