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Abstract
q-deformations of functions and distributions have been used in the literature to explain several experimental observations. In this work, we study the dynamics of the Tinkerbell map under q-deformations. The system exhibits a rich variety of dynamical behavior as q varies, including occurrences of interior crises, paired cascades, simultaneous occurrence of Neimark-Sacker and reverse Neimark-Sacker bifurcations, and co-existence of attractors and multistability. Numerical analysis reveals the existence of 3 limit cycles occurring simultaneously in a certain parameter regime. An appropriate choice of initial conditions enables one to choose a desired attractor for the system among other co-existing ones, thus switching the system between different dynamical states. We demonstrate the possibility of secure encryption and decryption of messages with the q-deformed Tinkerbell map. The system's sensitivity to the initial conditions and to the deformation parameter makes the cryptic message secure, and decrypting the original message difficult. We propose the use of the q-deformed map as a novel method for transmission of messages securely.
... In simpler terms, the dynamical analysis of the q-deformed Hénon system exhibits greater complexity than those of the classical Hénon system. Furthermore, the effects of q-deformations on the Tinkerbell system are discussed in [30]. Lately, there has been an increasing focus on investigating q-deformation within chaotic systems [30][31][32], capturing considerable attention in research and sparking heightened interest within the scientific community. ...
... Furthermore, the effects of q-deformations on the Tinkerbell system are discussed in [30]. Lately, there has been an increasing focus on investigating q-deformation within chaotic systems [30][31][32], capturing considerable attention in research and sparking heightened interest within the scientific community. ...
... The master system for Eq. (15) is described by (30) and the slave system is expressed as ...
Variable-order differential operators can be useful for modeling chaotical systems and nonlinear. This paper introduces a novel discrete fractional-order q-deformed Stefanski map and investigates its dynamical properties, stability, synchronization, and cryptographic applications. The study begins the incorporation of q-deformation and fractional-order dynamics. The proposed system’s complex behavior is analyzed using the Lyapunov exponent, entropy analysis, bifurcation diagrams, phase plots, time history plots, and the 0-1 test, revealing various attractors, including chaotic, quasi-periodic, and non-chaotic states. These findings highlight the map’s enhanced complexity and potential for diverse applications. A detailed chaos analysis is also performed, showcasing the strengthened unpredictability and sensitivity of the q-deformed Stefanski map. Stability and synchronization are explored using adaptive control techniques, demonstrating their effectiveness in regulating chaotic behavior. Furthermore, the study uses the chaotic properties of the system to develop a robust image encryption scheme. The encryption framework integrates permutation and diffusion mechanisms, ensuring enhanced security and resistance against statistical and differential attacks. Comparative analysis confirms the superiority of the proposed encryption approach over traditional methods.
... On other words, the dynamics of q-deformed Hénon map are richer than those of classical Hénon map. The influence of q-deformations on the Tinkerbell map has been discussed in [20]. Also, the fractional chaotic maps with q deformation has been investigated in [29]. ...
... Taking the parameter h 1 as bifurcation parameter and slightly perturb it around h * 1 where |h * 1 | 1, then we can write system (20) as ...
... On other words, the dynamics of q-deformed Hénon map are richer than those of classical Hénon map. The influence of q-deformations on the Tinkerbell map has been discussed in [20]. Also, the fractional chaotic maps with q deformation has been investigated in [29]. ...
... Taking the parameter h 1 as bifurcation parameter and slightly perturb it around h * 1 where |h * 1 | 1, then we can write system (20) as ...
The aim of this work is to analytically investigate the nonlinear dynamic behaviors of a proposed reduced Lorenz system based on q-deformations. The effects of varying the new q-deformation parameter on the dynamical behaviors of the system along with the induced bifurcations of fixed points are explored. In particular, the codimension-one bifurcation analysis is carried out at interior fixed point of the q-deformed system. Explicit conditions for the existence of pitchfork and Neimark–Sacker bifurcations are obtained. Numerical simulations are performed to confirm stability and bifurcation analysis in addition to investigate the effects of variations in system parameters. The changes in system dynamics are explored via the bifurcation diagrams, phase portraits and time series diagrams. Moreover, the quantification of system complex behaviors is depicted through the maximal Lyapunov exponent plots. A cascaded version of the model is proposed to boost its complex dynamics. Then, a chaos-based image encryption scheme, relying on a proposed key-distribution algorithm, is introduced as an application. Finally, several aspects of security analysis are examined for the encryption system to prove its efficiency and reliability against possible attacks.
Dynamics, bifurcations, strange attractors, and related nonlinear phenomena have been investigated in numerous one and two-dimensional maps. The maps could be smooth, non-differentiable, or even discontinuous. Various deformations of these systems could yield valuable insight into the dynamics of these maps. The quantum deformation or q-deformation of nonlinear maps has been investigated recently in this context. It leads to large excursions in the phase space, and the nature of attractors is very different from those usually observed. We investigate the q-deformed Lozi map. The q-deformation of either variable or both variables is studied from the viewpoint of bifurcations, possible multistability, and the nature of attractors. A rich structure of basins of coexisting attractors for various strengths of q-deformation is investigated. The initial conditions that lead to unbounded trajectories in the absence of q-deformation may get confined for slight q-deformation. Chaotic synchronization of two coupled q-deformed Lozi maps is observed, and an analytic criterion has been given for the same.
The concept of q-deformation has been expanded in a variety of situations including q-deformed nonlinear maps. Coupled q-deformed logistic maps with positive and negative q-diffusive coupling are investigated. The transition to state of synchronized fixed point in the thermodynamic limit for random initial conditions is studied as dynamic phase transitions. Though rare for one-dimensional maps, q-deformed maps show multistability. For weak coupling, the synchronized state may not be observed in the thermodynamic limit with random initial conditions even if it is linearly stable. Thus multistability persists. However, for large lattices with random initial conditions, five well-defined critical points are observed where the transition to a synchronized state occurs. Two of these show continuous phase transition. One of them is in the directed percolation universality class. In another case, the order parameter shows power-law decay superposed with oscillations on a logarithmic scale at the critical point and does not match with known universality classes.
In 1991, T. Puu proposed a cubic investment map in order to analyze chaos in business cycles [34]. In this study, we propose a q-deformation of the aggregate variable income difference in order to analyze the dynamics of the q-deformed version considering that the original model is a particular case of the q-deformation case with q=1.
A chaotic image encryption algorithm based on a deformation of the logistic map is proposed. The use of a deformed chaotic map has several advantages. In particular, we are able to not only enlarge the key space size but also describe a parametric region for the selection of a key avoiding non-chaotic regions, which is a fundamental requirement for real applications. Moreover, a security analysis of the proposed algorithm shows its efficiency and robustness in resisting attacks.
We analyze the q-deformed logistic map, where the q-deformation follows the scheme inspired in the Tsallis q-exponential function. We compute the topological entropy of the dynamical system, obtaining the parametric region in which the topological entropy is positive and hence the region in which chaos in the sense of Li and Yorke exists. In addition, it is shown the existence of the so-called Parrondo's paradox where two simple maps are combined to give a complicated dynamical behavior.
Periodic outbreaks of the larch budmoth Zeiraphera diniana population (and the massive forest defoliation they engender) have been recorded in the Alps over the centuries and are known for their remarkable regularity. But these have been conspicuously absent since 1981. On the other hand, budmoth outbreaks have been historically unknown in the larches of the Carpathian Tatra mountains. To resolve this puzzle, we propose here a model which includes the influence of climate and explains both the 8-9 year periodicity in the budmoth cycle and the variations from this, as well as the absence of cycles. We successfully capture the observed trend of relative frequencies of outbreaks, reproducing the dominant periodicities seen. We contend that the apparent collapse of the cycle in 1981 is due to changing climatic conditions following a tipping point and propose the recurrence of the cycle with a changed periodicity of 40 years-the next outbreak could occur in 2021. Our model also predicts longer cycles.
This paper proposes a novel pseudo-random bit generation algorithm combining Chebyshev polynomial and Tinkerbell map. We calculated the key space of the proposed scheme. The output zero-one bits are statistically tested with three packages: NIST, DIEHARD and ENT. The experimental results show that the data are uniformly distributed with suffcient enough statistical properties to disturb brute-force attacks.
The chaotic map based pseudo-random number generators with good statistical properties have been widely used in modern cryptography algorithms. This paper proposes a Tinker bell map as a novel pseudo-random number generator. We evaluated the proposed approach with various statistical packages: NIST, DIEHARD and ENT. The results of the analysis demonstrate that the new derivative bit stream scheme is very suitable for embedding in critical cryptographic applications.
In this paper, the dynamical behaviors of the Tinkerbell map are investigated in detail. Conditions for the existence of fold bifurcation, flip bifurcation and Hopf bifurcation are derived, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations include bifurcation diagrams in two- and three-dimensional spaces, phase portraits, and the maximum Lyapunov exponent and fractal dimension, as well as the distribution of dynamics in the parameter plane, which exhibit new and interesting dynamical behaviors. More specifically, this paper reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2, 7, 8, 9, 10, 13, 17, 19, 23, 26 and so on, and suddenly appearing or disappearing chaos, convergence of an invariant circle to a period-one orbit, symmetry-breaking of periodic orbits, interlocking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2, 10, 13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on, all in the Tinkerbell map. In particular, it is found that there is no obvious road from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invariant circle and then to transient chaos as the parameters are varied. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the Tinkerbell map is obtained.
A recent proposal of q-deformation scheme for the nonlinear maps has stimulated new directions in the studies of various nonlinear dynamical systems. Such studies are advantageous in the analytical modeling of several nonlinear physical phenomena which cannot be perfectly represented by the standard or canonical models. This paper attempts to numerically analyze the behavior of q-deformed version of Henon map, which is one of the prototypical models exhibiting strange chaotic attractor. We particularly investigate the effect of q-deformation on the various kinds of long-term asymptotic dynamics of the canonical Henon map and also characterize the complete deformation parameter space into different regions corresponding to the periodic and strange chaotic motions. We also identify the route to chaos and new structures in the chaotic strange attractor of q-deformed Henon map.
The dynamics of an optical ring phase conjugated resonator with Tinkerbell chaos is presented. It is shown that a non-linear behavior takes places when a chaos generating element is introduced on the resonator. The matrix formalism is used in order to carry out the analysis of the resonator in his full-round trip when all the optical resonator elements are involved. Assuming ray optics inside the cavity with parameters y(z) and θ{z) for the effective distance to the optical axis and the angle to the same axis respectively, the expressions for the n-th trip are obtained. The matrix of an optical chaos generating element [a, b, c, e] able to produce Tinkerbell chaos in an ideal ring phase conjugation resonator are obtained for the first time to our knowledge.
Signals generated by chaotic systems represent a potentially rich class of signals both for detecting and characterizing physical phenomena and in synthesizing new classes of signals for communications, remote sensing, and a variety of other signal processing applications. Since classical techniques for signal analysis do not exploit the particular structure of chaotic signals there is both a significant challenge and an opportunity in exploring new classes of algorithms matched to chaotic signals. The authors outline a variety of signal processing issues associated with the analysis and synthesis of chaotic signals. In addition two examples are described in detail, illustrating some possible ways in which the characteristics of chaotic signals and systems can be exploited. One example is a binary signaling scheme using chaotic signals. The second example is the use of synchronized chaotic systems for signal masking and recovery.
The asymptotic angular stability of a dynamical system may be quanti-fied by its rotation number or its winding number. These two quantities are shown to result from different assumptions, made about the flow generating the Poincareap which results from the sequence of homeomorphisms in S l . An ergodic theorem of existence a.s. of the rotation number for non-linear systems is given. The advantages and disadvantages of both the rotation and winding numbers are discussed. Numerical calculations of the distribution of rotation number and winding number arising from different initial conditions are presented for three different chaotic maps.
With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS
q
k [1 –
i=1
W
p
i
q
]/(q-1), whereq characterizes the generalization andp
i are the probabilities associated withW (microscopic) configurations (W). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq1 limit.
Shadowing is a method of backward error analysis that plays a important role in hyperbolic dynamics. In this paper, the shadowing by containment framework is revisited, including a new shadowing theorem. This new theorem has several advantages with respect to existing shadowing theorems: It does not require injectivity or differentiability, and its hypothesis can be easily verified using interval arithmetic. As an application of this new theorem, shadowing by containment is shown to be applicable to infinite length orbits and is used to provide a computer assisted proof of the presence of chaos in the well-known noninjective Tinkerbell map.
We analyze the effects of a strategy of constant effort harvesting in the global dynamics of a one-dimensional discrete population model that includes density-independent survivorship of adults and overcompensating density dependence. We discuss the phenomenon of bubbling (which indicates that harvesting can magnify fluctuations in population abundance) and the hydra effect, which means that the stock size gets larger as harvesting rate increases. Moreover, we show that the system displays chaotic behaviour under the combination of high per capita recruitment and small survivorship rates.
The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos.
Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value μ2 of the map at which there is chaos. We show that often virtually all (i.e. all but finitely many) "regular" periodic orbits at μ2 are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired — connected to exactly one other cascade, or solitary — connected to exactly one regular periodic orbit at μ2. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F(μ2, ⋅). Examples discussed include the forced-damped pendulum and the double-well Duffing equation.
We demonstrated experimentally that the momentum distribution of cold atoms in dissipative optical lattices is a Tsallis distribution. The parameters of the distribution can be continuously varied by changing the parameters of the optical potential. In particular, by changing the depth of the optical lattice, it is possible to change the momentum distribution from Gaussian, at deep potentials, to a power-law tail distribution at shallow optical potentials.
The study of animal foraging behaviour is of practical ecological importance, and exemplifies the wider scientific problem of optimizing search strategies. Lévy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails, such that clusters of short steps are connected by rare long steps. Lévy flights display fractal properties, have no typical scale, and occur in physical and chemical systems. An attempt to demonstrate their existence in a natural biological system presented evidence that wandering albatrosses perform Lévy flights when searching for prey on the ocean surface. This well known finding was followed by similar inferences about the search strategies of deer and bumblebees. These pioneering studies have triggered much theoretical work in physics (for example, refs 11, 12), as well as empirical ecological analyses regarding reindeer, microzooplankton, grey seals, spider monkeys and fishing boats. Here we analyse a new, high-resolution data set of wandering albatross flights, and find no evidence for Lévy flight behaviour. Instead we find that flight times are gamma distributed, with an exponential decay for the longest flights. We re-analyse the original albatross data using additional information, and conclude that the extremely long flights, essential for demonstrating Lévy flight behaviour, were spurious. Furthermore, we propose a widely applicable method to test for power-law distributions using likelihood and Akaike weights. We apply this to the four original deer and bumblebee data sets, finding that none exhibits evidence of Lévy flights, and that the original graphical approach is insufficient. Such a graphical approach has been adopted to conclude Lévy flight movement for other organisms, and to propose Lévy flight analysis as a potential real-time ecosystem monitoring tool. Our results question the strength of the empirical evidence for biological Lévy flights.
Nonequilibrium systems with large-scale fluctuations of a suitable system parameter are often effectively described by a superposition of two statistics, a superstatistics. Here we illustrate this concept by analysing experimental data of fluctuations in atmospheric wind velocity differences at Florence airport. Comment: 9 pages, 4 figures. New version to appear in Europhysics News (2005)
A scheme is proposed to classify the basins for attractors of dynamical systems in arbitrary dimensions. There are four basic classes depending on their size and extent, and each class can be further quantified to facilitate comparisons. The calculation uses a Monte Carlo method and is applied to numerous common dissipative chaotic maps and flows in various dimensions.
This issue is a collection of contributions on recent developments and achievements of cryptography and communications using chaos. The various contributions report important and promising results such as synchronization of networks and data transmissions; image cipher; optical and TDMA communications, quantum keys etc. Various experiments and applications such as FPGA, smartphone cipher, semiconductor lasers etc, are also included.
It is shown that recourse to Tsallis' generalized entropy makes it
possible to find sensible distribution functions for stellar polytropes,
while that of Boltzmann yields unphysical distributions. Additionally,
some constraints are imposed on Tsallis' entropy.
A scheme of q-deformation of nonlinear maps is introduced. As a specific
example, a q-deformation procedure related to the Tsallis q-exponential
function is applied to the logistic map. Compared to the canonical
logistic map, the resulting family of q-logistic maps is shown to have a
wider spectrum of interesting behaviours, including the co-existence of
attractors—a phenomenon rare in one-dimensional maps.
Nowadays, a variety of controllers used in process industries are still of the proportional–integral–derivative (PID) types. PID controllers have the advantage of simple structure, good stability, and high reliability. A relevant issue for PID controllers design is the accurate and efficient tuning of parameters. In this context, several approaches have been reported in the literature for tuning the parameters of PID controllers using evolutionary algorithms, mainly for single-input single-output systems. The systematic design of multi-loop (or decentralized) PID control for multivariable processes to meet certain objectives simultaneously is still a challenging task. This paper proposes a new chaotic firefly algorithm approach based on Tinkerbell map (CFA) to tune multi-loop PID multivariable controllers. The firefly algorithm is a metaheuristic algorithm based on the idealized behavior of the flashing characteristics of fireflies. To validate the performance of the proposed PID control design, a multi-loop multivariable PID structure for a binary distillation column plant (Wood and Berry column model) and an industrial-scale polymerization reactor are taken. Simulation results indicate that a suitable set of PID parameters can be calculated by the proposed CFA. Besides, some comparison results of a genetic algorithm, a particle swarm optimization approach, traditional firefly algorithm, modified firefly algorithm, and the proposed CFA to tune multi-loop PID controllers are presented and discussed.
Secure communication via chaotic synchronization is experimentally demonstrated using Chua’s circuit. In our experiment the energy lost in the information-bearing signal is approximately 0.6 dBV. The reduction in chaotic signal after the recovery process is between –40 and –50 dBV.
It is shown that anomalous diffusion (i.e., nonlinear growth of mean square displacements) can be caused by a specifically chaotic mechanism. It depends on deterministic diffusion and intermittency and gives rise to mean square displacements which grow asymptotically like tnu with 0
An abstract is not available.
We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions Dq, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities Dq. We prove that lim q→0Dq = fractal dimension (D), limq→1Dq = information dimension (σ) and Dq=2 = correlation exponent (v). Dq with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > Dq for any q′ > q. For homogeneous fractals Dq = Dq. A particularly interesting dimension is Dq=∞. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D∞ is a non-trivial number. All the other generalized dimensions are bounded between the fractal dimension and D∞.
We propose a fast chaotic cryptographic scheme based on iterating a logistic map. In particular, no random numbers need to be generated and the look-up table used in the cryptographic process is updated dynamically. Simulation results show that the proposed method leads to a substantial reduction in the encryption and decryption time. As a result, chaotic cryptography becomes more practical in the secure transmission of large multi-media files over public data communication network.
The occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated. It is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor. We call such collisions crises. Phenomena associated with crises include sudden changes in the size of chaotic attractors, sudden appearances of chaotic attractors (a possible route to chaos), and sudden destructions of chaotic attractors and their basins. This paper presents examples illustrating that crisis events are prevalent in many circumstances and systems, and that, just past a crisis, certain characteristic statistical behavior (whose type depends on the type of crisis) occurs. In particular the phenomenon of chaotic transients is investigated. The examples discussed illustrate crises in progressively higher dimension and include the one-dimensional quadratic map, the (two-dimensional) Hénon map, systems of ordinary differential equations in three dimensions and a three-dimensional map. In the case of our study of the three-dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four, respectively. Based on the examples presented the following conjecture is proposed: almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises.
This paper derives a super-chaotification (or hyper-chaotification) scheme by making all Lyapunov exponents of the controlled dynamical system positive via the controller of some simple triangular function.
It is possible to encrypt a message (a text composed by some alphabet) using the ergodic property of the simple low-dimensional and chaotic logistic equation. The basic idea is to encrypt each character of the message as the integer number of iterations performed in the logistic equation, in order to transfer the trajectory from an initial condition towards an ϵ-interval inside the logistic chaotic attractor.
The generalized Tsallis statistics produces a distribution function appropriate to describe the interior solar plasma, thought as a stellar polytrope, showing a tail depleted with respect to the Maxwell-Boltzmann distribution and reduces to zero at energies greater than about 20kBT. The Tsallis statistics can theoretically support the distribution suggested in the past by Clayton and collaborators, which shows also a depleted tail, to explain the solar neutrino counting rate.
We propose a modified version of the chaotic cryptographic method based on iterating a logistic map. Simulation results show that the distribution of the ciphertext is flatter and the encryption time is shorter. Moreover, the trade-off between the spread of the distribution of ciphertext and the encryption time can be controlled by a single parameter.
In a recent study Jaganathan and Sinha [Jaganathan R, Sinha S. A q-deformed nonlinear map. Phys Lett A 2005;338:277-87] have introduced a scheme for the q-deformation of nonlinear maps using the logistic map as an example and shown that the q-logistic map exhibits a wide spectrum of dynamical behaviours including the co-existence of attractors (which is a rare phenomenon in one-dimensional maps). In this paper, we aim to analyze another famous one-dimensional map - the Gaussian map (a known one-dimensional map exhibiting co-existing attractors) subject to the same q-deformation scheme. We compare the dynamical behaviour of the Gaussian map and q-deformed Gaussian map with a special attention on the regions of the parameter space, where these maps exhibit co-existing attractors. An important conclusion of the present study is that the appearance of co-existing attractors for a particular choice of system parameters can be understood as a consequence of the presence of multiple fixed points in one-dimensional nonlinear maps; however the converse is not always true.
In the present paper, our aim is to determine both upper and lower bounds for all the Lyapunov exponents of a given finite-dimensional discrete map. To show the efficiency of the proposed estimation method, two examples are given, including the well-known Henon map and a coupled map lattice.
We present a rigorous analysis and numerical evidence indicating that a re- cently developed methodology for detecting unstable periodic orbits is capable of yielding all orbits up to periods limited only by the computer precision. In particular, we argue that an efficient convergence to every periodic orbit can be achieved and the basin of attraction can be made finite and accessible for typical or particularly chosen initial conditions.
We point out a connection between anomalous quantum transport in an optical lattice and Tsallis' generalized thermostatistics. Specifically, we show that the momentum equation for the semiclassical Wigner function that describes atomic motion in the optical potential, belongs to a class of transport equations recently studied by Borland [PLA 245, 67 (1998)]. The important property of these ordinary linear Fokker--Planck equations is that their stationary solutions are exactly given by Tsallis distributions. Dissipative optical lattices are therefore new systems in which Tsallis statistics can be experimentally studied. Comment: 4 pages, 1 figure
A generalization of the functions Γ(n) and xn
F. H. Jackson, "A generalization of the functions (n) and x n," Proc. R.
Soc. London 74, 64-72 (1904).
- J P Boon
- C Tsallis
J. P. Boon and C. Tsallis, "Special issue overview Nonextensive statistical
mechanics: new trends, new perspectives," Europhys. News 36, 185-186
(2005).