Showing papers in "International Journal of Bifurcation and Chaos in 2010"
TL;DR: A practical implementation of a memristor based chaotic circuit that employs the four fundamental circuit elements — the resistor, capacitor, inductor and the Memristor using off-the-shelf components is provided.
Abstract: This paper provides a practical implementation of a memristor based chaotic circuit. We realize a memristor using off-the-shelf components and then construct the memristor along with the associated chaotic circuit on a breadboard. The goal is to construct a physical chaotic circuit that employs the four fundamental circuit elements — the resistor, capacitor, inductor and the memristor. The central concept behind the memristor circuit is to use an analog integrator to obtain the electric flux across the memristor and then use the flux to obtain the memristor's characterstic function.
522 citations
TL;DR: A chaotic attractor has been observed with an autonomous circuit that uses only two energy-storage elements: a linear passive inductor and alinear passive capacitor and a nonlinear active memristor.
Abstract: A chaotic attractor has been observed with an autonomous circuit that uses only two energy-storage elements: a linear passive inductor and a linear passive capacitor. The other element is a nonlinear active memristor. Hence, the circuit has only three circuit elements in series. We discuss this circuit topology, show several attractors and illustrate local activity via the memristor's DC vM - iM characteristic.
440 citations
TL;DR: An attack is proposed that reveals the secret permutation that is used to shuffle the pixels of a round input that makes Fridrich's chaotic image encryption algorithm vulnerable against chosen-ciphertext attacks.
Abstract: We cryptanalyze Fridrich’s chaotic image encryption algorithm. We show that the algebraic weaknesses of the algorithm make it vulnerable against chosen-ciphertext attacks. We propose an attack that reveals the secret permutation that is used to shuffle the pixels of a round input. We demonstrate the effectiveness of our attack with examples and simulation results. We also show that our proposed attack can be generalized to other well-known chaotic image encryption algorithms.
204 citations
TL;DR: An expression of the first order Melnikov function is derived, which can be used to study the number of limit cycles bifurcated from the periodic orbits of piecewise Hamiltonian systems on the plane.
Abstract: In this paper, the general perturbation of piecewise Hamiltonian systems on the plane is considered. When the unperturbed system has a family of periodic orbits, similar to the perturbations of smooth system, an expression of the first order Melnikov function is derived, which can be used to study the number of limit cycles bifurcated from the periodic orbits. As applications, the number of bifurcated limit cycles of several concrete piecewise systems are presented.
138 citations
TL;DR: A new dynamical model of cancer growth is developed, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to cancer growth.
Abstract: In this study, we develop a new dynamical model of cancer growth, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to c...
137 citations
TL;DR: An unusual three-dimensional autonomous quadratic Lorenz-like chaotic system which, surprisingly, has two stable node-type of foci as its only equilibria is reported, which contains the diffusionless Lorenz system and the Burke–Shaw system.
Abstract: This paper reports the finding of an unusual three-dimensional autonomous quadratic Lorenz-like chaotic system which, surprisingly, has two stable node-type of foci as its only equilibria. The new system contains the diffusionless Lorenz system and the Burke–Shaw system, and some others, as special cases. The algebraic form of the new chaotic system is similar to the other Lorenz-type systems, but they are topologically nonequivalent. To further analyze the new system, some dynamical behaviors such as Hopf bifurcation and singularly degenerate heteroclinic and homoclinic orbits, are rigorously proved with simulation verification. Moreover, it is proved that the new system with some specified parameter values has Silnikov-type homoclinic and heteroclinic chaos.
129 citations
TL;DR: Using semi-tensor product of matrices, a matrix expression for multivalued logic is proposed, where a logical variable is expressed as a vector, and a logical function is express as a multilinear function.
Abstract: Using semi-tensor product of matrices, a matrix expression for multivalued logic is proposed, where a logical variable is expressed as a vector, and a logical function is expressed as a multilinear...
124 citations
TL;DR: Control laws are derived analytically to achieve synchronization of the chaotic fractional-order Chen, Rossler and modified Chua systems through chaos synchronization via a suitable linear controller applied to the response system.
Abstract: The chaotic dynamics of fractional-order systems has attracted much attention recently. Chaotic synchronization of fractional-order systems is further studied in this paper. We investigate the chaos synchronization of two identical systems via a suitable linear controller applied to the response system. Based on the stability results of linear fractional-order systems, sufficient conditions for chaos synchronization of these systems are given. Control laws are derived analytically to achieve synchronization of the chaotic fractional-order Chen, Rossler and modified Chua systems. Numerical simulations are provided to verify the theoretical analysis.
114 citations
TL;DR: Three well-known theorems related to the simplest codimension-one bifurcations occurring in discrete time dynamical systems, such as the fold, flip and Neimark–Sacker bifURcations are recalled and analyzed.
Abstract: We recall three well-known theorems related to the simplest codimension-one bifurcations occurring in discrete time dynamical systems, such as the fold, flip and Neimark–Sacker bifurcations, and analyze these bifurcations in presence of certain degeneracy conditions, when the above mentioned theorems are not applied. The occurrence of such degenerate bifurcations is particularly important in piecewise smooth maps, for which it is not possible to specify in general the result of the bifurcation, as it strongly depends on the global properties of the map. In fact, the degenerate bifurcations mainly occur in piecewise smooth maps defined in some subspace of the phase space by a linear or linear-fractional function, although not necessarily only by such functions. We also discuss the relation between degenerate bifurcations and border-collision bifurcations.
107 citations
TL;DR: Numerical investigations about the relative abundance and structure of chaotic phases in autonomous dissipative flows, i.e. in continuous-time dynamical systems described by sets of ordinary differential equations, are reported.
Abstract: This manuscript reports numerical investigations about the relative abundance and structure of chaotic phases in autonomous dissipative flows, i.e. in continuous-time dynamical systems described by sets of ordinary differential equations. In the first half, we consider flows containing "periodicity hubs", which are remarkable points responsible for organizing the dynamics regularly over wide parameter regions around them. We describe isolated hubs found in two forms of Rossler's equations and in Chua's circuit, as well as surprising infinite hub cascades that we found in a polynomial chemical flow with a cubic nonlinearity. Hub cascades converge orderly to accumulation points lying on specific parameter paths. In sharp contrast with familiar phenomena associated with unstable orbits, hubs and infinite hub cascades always involve stable periodic and chaotic orbits which are, therefore, directly measurable in experiments. In the last part, we consider flows having no hubs but unusual phase diagrams: a cubic...
106 citations
TL;DR: This paper numerically study the bifurcations and chaotic behaviors in the fractional-order simplified Lorenz system using the time-domain scheme and presents complex dynamics with interesting characteristics.
Abstract: The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we numerically study the bifurcations and chaotic behaviors in the fractional-order simplified Lorenz system using the time-domain scheme. Chaos does exist in this system for a wide range of fractional orders, both less than and greater than three. Complex dynamics with interesting characteristics are presented by means of phase portraits, bifurcation diagrams and the largest Lyapunov exponent. Both the system parameter and the fractional order can be taken as bifurcation parameters, and the range of existing chaos is different for different parameters. The lowest order we found for this system to yield chaos is 2.62.
TL;DR: In this paper, the authors consider the ten most populated urban areas in United Kingdom and study what would be an optimal layout of transport links between these urban areas from the "plasmodium's point of view".
Abstract: Plasmodium of Physarum polycephalum is a single cell visible by unaided eye. During its foraging behavior the cell spans spatially distributed sources of nutrients with a protoplasmic network. Geometrical structure of the protoplasmic networks allows the plasmodium to optimize transfer of nutrients between remote parts of its body, to distributively sense its environment, and make a decentralized decision about further routes of migration. We consider the ten most populated urban areas in United Kingdom and study what would be an optimal layout of transport links between these urban areas from the "plasmodium's point of view". We represent geographical locations of urban areas by oat flakes, inoculate the plasmodium in Greater London area and analyze the plasmodium's foraging behavior. We simulate the behavior of the plasmodium using a particle collective which responds to the environmental conditions to construct and minimize transport networks. Results of our scoping experiments show that during its col...
TL;DR: This paper aims to an present account of some problems considered in the past years in Dynamical Systems, new research directions and also provide some open problems.
Abstract: This paper aims to an present account of some problems considered in the past years in Dynamical Systems, new research directions and also provide some open problems.
TL;DR: The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor, named as a contraction for memory resistor.
Abstract: The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical...
TL;DR: The authors examined the network characteristics based on the phonological similarities in the lexicons of several languages and found that the properties of these networks suggest explanations for various aspects of linguistic processing and hint at deeper organization within the human language.
Abstract: The network characteristics based on the phonological similarities in the lexicons of several languages were examined. These languages differed widely in their history and linguistic structure, but commonalities in the network characteristics were observed. These networks were also found to be different from other networks studied in the literature. The properties of these networks suggest explanations for various aspects of linguistic processing and hint at deeper organization within the human language.
TL;DR: It is found that spatial variance increases as the lake approaches the point of transition to a eutrophic state, and some of these indicators are not early enough to avert the undesired impending shift.
Abstract: The task of providing leading indicators of catastrophic regime shifts in ecosystems is fundamental in order to design management protocols for those systems. Here we address the problem of lake eutrophication (that is, nutrient enrichment leading to algal blooms) using a simple spatial lake model. We discuss and compare different spatial and temporal early warning signals announcing the catastrophic transition of an oligotrophic lake to eutrophic conditions. In particular, we consider the spatial variance and its associated patchiness of eutrophic water regions. We found that spatial variance increases as the lake approaches the point of transition to a eutrophic state. We also analyze the spatial and temporal early warnings in terms of the amount of information required by each and their respective forewarning times. From the consideration of different remedial procedures that can be followed after these early signals we conclude that some of these indicators are not early enough to avert the undesired impending shift.
TL;DR: In this article, the authors give conditions on T that guarantee the existence of an invariant curve emanating from when both eigenvalues of the Jacobian of T at are nonzero and at least one of them has absolute value less than one, and establish that an increasing curve that separates into invariant regions.
Abstract: Let T be a competitive map on a rectangular region , and assume T is C1 in a neighborhood of a fixed point . The main results of this paper give conditions on T that guarantee the existence of an invariant curve emanating from when both eigenvalues of the Jacobian of T at are nonzero and at least one of them has absolute value less than one, and establish that is an increasing curve that separates into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. These results, known in hyperbolic case, have been used to determine the basins of attraction of hyperbolic equilibrium points and to establish certain global bifurcation results when switching from competitive coexistence to competitive exclusion. The emphasis in applications in this paper is on planar systems of difference equations with nonhyperbolic equilibria, where we establish a precise description of the basins of attraction of finite or infinite number of equilibrium points.
TL;DR: This work recalls Leonov's approach and explains why it works, and slightly improves the approach by avoiding an unnecessary coordinate transformation, and demonstrates that the approach can be used not only for the calculation of border-collision bifurcation curves.
Abstract: 50 years ago (1959) in a series of publications by Leonov, a detailed analytical study of the nested period adding bifurcation structure occurring in piecewise-linear discontinuous 1D maps was presented. The results obtained by Leonov are barely known, although they allow the analytical calculation of border-collision bifurcation subspaces in an elegant and much more efficient way than it is usually done. In this work we recall Leonov's approach and explain why it works. Furthermore, we slightly improve the approach by avoiding an unnecessary coordinate transformation, and also demonstrate that the approach can be used not only for the calculation of border-collision bifurcation curves.
TL;DR: This paper presents an efficient method for finding horseshoe in dynamical systems by using several simple results on topological horseshoes using a series of points from an attractor.
Abstract: This paper presents an efficient method for finding horseshoes in dynamical systems by using several simple results on topological horseshoes. In this method, a series of points from an attractor o...
TL;DR: It is shown that sensible improvements in the final results can be expected if the underlying probability distribution is "extracted" via appropriate consideration regarding causal effects in the system's dynamics.
Abstract: A generalized Statistical Complexity Measure (SCM) is a functional that characterizes the probability distribution P associated to the time series generated by a given dynamical system. It quantifies not only randomness but also the presence of correlational structures. We review here several fundamental issues in such a respect, namely, (a) the selection of the information measure ; (b) the choice of the probability metric space and associated distance ; (c) the question of defining the so-called generalized disequilibrium ; (d) the adequate way of picking up the probability distribution P associated to a dynamical system or time series under study, which is indeed a fundamental problem. In this communication we show (point d) that sensible improvements in the final results can be expected if the underlying probability distribution is "extracted" via appropriate consideration regarding causal effects in the system's dynamics.
TL;DR: It is shown that the corresponding solutions of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/2 generate a random dynamical system for which the existence and uniqueness of a random attractor is proved.
Abstract: In this paper, the asymptotic behavior of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/2 is studied. In particular, it is shown that the corresponding solutions generate a random dynamical system for which the existence and uniqueness of a random attractor is proved.
TL;DR: This paper initiates a novel approach for generating multi-wing butterfly chaotic attractors from the generalized first and second kinds of Lorenz-type systems, and it is especially pointed out that this is the first time in the literature that a maximal 10-wing Butterfly chaotic attractor is experimentally verified by analog circuits.
Abstract: Lorenz system, as the first classical chaotic system, has been intensively investigated over the last four decades. Based on the sawtooth wave function, this paper initiates a novel approach for generating multi-wing butterfly chaotic attractors from the generalized first and second kinds of Lorenz-type systems. Compared with the traditional ring-shaped multi-scroll Lorenz chaotic attractors, the proposed multi-wing butterfly chaotic attractors are much easier to be designed and implemented by analog circuits. The dynamical behaviors of these multi-wing butterfly chaotic systems are further studied. Theoretical analysis shows that every index-2 saddle-focus equilibrium corresponds to a unique wing in the butterfly attractors. Finally, a module-based unified circuit diagram is constructed for realizing various multi-wing butterfly attractors. It should be especially pointed out that this is the first time in the literature that a maximal 10-wing butterfly chaotic attractor is experimentally verified by analog circuits.
TL;DR: A four-step teaching/learning method in which two groups of students manually built Chua's circuit and used it as a source for extraordinary images and music shows that manipulation of the circuit and related artistic activities can provide a new, creative and enjoyable approach to science learning.
Abstract: Chua's circuit is a nonlinear dynamic circuit that has assumed a paradigmatic role in mathematical, physical and experimental demonstrations of chaos. Yet even today, complexity and chaos are seen as challenging topics reserved for specialists. Is it possible for young people in junior and senior high school to acquire a difficult concept such as chaos? In order to study this issue, we developed a four-step teaching/learning method in which two groups of students (one from junior and one from senior high school) manually built Chua's circuit and used it as a source for extraordinary images and music. All this allowed students to interact with strange attractors, discovering their fascination and beauty. Both groups of students created 3-D models of attractors, modified the control parameters, explored the sound and music of chaos and, during the process, discovered the connection between science and art. Briefly, they learned through engaging activities, which allowed them to acquire physical and mathematical knowledge of chaos through various modalities. The results show that manipulation of the circuit and related artistic activities can provide a new, creative and enjoyable approach to science learning.
TL;DR: It was found that the evolution of the attractor is governed by a complex interplay between smooth and nonsmooth bifurcations, and the interactions between a number of coexisting orbits.
Abstract: In this work the strange behavior of an impact oscillator with a one-sided elastic constraint discovered experimentally is compared with the predictions obtained using its mathematical model. Extensive experimental investigations undertaken on the rig developed at the Aberdeen University reveal different bifurcation scenarios under varying excitation frequency near grazing which were recorded for a number of values of the excitation amplitude. In the paper, particular attention is paid to the chaotic oscillations recorded near grazing frequency when a nonimpacting orbit becomes an impacting one under increasing excitation frequency. It was found that the evolution of the attractor is governed by a complex interplay between smooth and nonsmooth bifurcations, and the interactions between a number of coexisting orbits. The occurrence of coexisting attractors is manifested in the experimental results through discontinuous transitions from one orbit to another via boundary crisis. In some cases, the basins of attraction have a fractal structure. Detailed numerical exploration also revealed coexisting unstable periodic orbits. These stable and unstable coexisting orbits are often born far from the parameter values at which they influence the system dynamics. The very rich dynamics of the bilinear oscillator close to grazing is demonstrated and typical mechanisms of the attractors' appearance and disappearance are explained using stability analysis.
TL;DR: An extended linear state observer (here addressed as Generalized Proportional Integral (GPI) observer) is proposed for the accurate estimation of the phase variables and the perturbation input of the nonlinear output dynamics.
Abstract: The problem of synchronization of chaotic oscillators, using state observers, is handled through the use of a simplified perturbed linear integration model of the chaotic system output dynamics. The linear simplified model of the chaotic system does not entitle approximate linearizations, nor state coordinate transformations, but, simply, a pure linear integration model with additive unknown but bounded perturbation inputs lumping all the output dynamics nonlinearities. An extended linear state observer (here addressed as Generalized Proportional Integral (GPI) observer) is proposed for the accurate estimation of the phase variables and the perturbation input of the nonlinear output dynamics. The effectiveness of the approach is tested in the synchronization of two study cases: The Genesio–Tesi chaotic system and the Rossler oscillator. As an application of the estimation process, a coding–decoding process involving encrypted messages, in transmitted phase variables, is implemented using a Rossler chaotic system.
TL;DR: A general method of generating continuous fractal interpolation surfaces by iterated function systems on an arbitrary data set over rectangular grids and estimate their Box-counting dimension is presented.
Abstract: We present a general method of generating continuous fractal interpolation surfaces by iterated function systems on an arbitrary data set over rectangular grids and estimate their Box-counting dimension.
TL;DR: This work has demonstrated that the application of Leonov's technique is not resticted to that particular bifurcation structure, and the presented map replacement approach, which is an extension of Leonova's technique, allows the analytical calculation of border-collision b ifurcation curves for periodic orbits with high periods and complex symbolic sequences using appropriate composite maps.
Abstract: The complex bifurcation structure in the parameter space of the general piecewise-linear scalar map with a single discontinuity — nowadays known as nested period adding structure — was completely studied analytically by N. N. Leonov already 50 years ago. He used an elegant and very efficient recursive technique, which allows the analytical calculation of the border-collision bifurcation curves, causing the nested period adding structure to occur. In this work, we have demonstrated that the application of Leonov's technique is not resticted to that particular bifurcation structure. On the contrary, the presented map replacement approach, which is an extension of Leonov's technique, allows the analytical calculation of border-collision bifurcation curves for periodic orbits with high periods and complex symbolic sequences using appropriate composite maps and the bifurcation curves for periodic orbits with much lower periods.
TL;DR: In this paper, it was shown that the necessary condition for the existence of a stable regime of maintained oscillations in a device of radio engineering completely analogous to the triode is the presence in the phase plane of stable limit cycle.
Abstract: At the beginning of the twentieth century while Henri Poincare (1854–1912) was already deeply involved in the developments of wireless telegraphy, he was invited, in 1908, to give a series of lectures at the Ecole Superieure des Postes et Telegraphes (today Sup'Telecom). In the last part of his presentation, he established that the necessary condition for the existence of a stable regime of maintained oscillations in a device of radio engineering completely analogous to the triode is the presence in the phase plane of stable limit cycle. The aim of this work is to prove that the correspondence highlighted by Andronov between the periodic solution of a nonlinear second-order differential equation and Poincare's concept of limit cycle has been carried out by Poincare himself, 20 years ago in some forgotten conferences of 1908.
TL;DR: A distance for marked point process data is defined and it is shown that foreign exchange tick data have serial dependence using recurrence plots and the random shuffle surrogate method.
Abstract: Recurrence plots are effective in analyzing nonstationary time series. Further, it is desirable to make the recurrence plot-based analysis applicable to marked point process data such as foreign exchange tick data. In this paper, we define a distance for marked point process data and establish the basis for further analyses. We also show that foreign exchange tick data have serial dependence using recurrence plots and the random shuffle surrogate method.
TL;DR: This paper describes the global dynamics of the Lorenz system and provides the global phase portrait of the system in the Poincare ball (i.e. in the compactification of ℝ3 with the sphere 𝕊2 of the infinity).
Abstract: In this paper by using the Poincare compactification of ℝ3 we describe the global dynamics of the Lorenz system \begin{eqnarray*} \dot{x}= s(-x+y),\quad \dot{y} = rx-y-xz, \quad \dot{z} =-bz+xy, \end{eqnarray*} having some invariant algebraic surfaces. Of course (x, y, z) ∈ ℝ3 are the state variables and (s, r, b) ∈ ℝ3 are the parameters. For six sets of the parameter values, the Lorenz system has invariant algebraic surfaces. For these six sets, we provide the global phase portrait of the system in the Poincare ball (i.e. in the compactification of ℝ3 with the sphere 𝕊2 of the infinity).