The double scroll

In this paper the experimental validation of a novel, modified double scroll chaotic attractor circuit, employed as a true random binary generator (TRBG) is presented. The double scroll attractor is modeled on Chua's circuit constituted as independent chaotic oscillator using passive only nonlinear device for nonlinear operation leading to chaotic behavior. The output from the chaotic circuit which is a correlated binary sequence is scrambled with a pseudo random binary sequence generator (PRBSG) topology to yield a true random binary source for key stream generation. The modified chaotic circuit has been first modeled in PSpice software. The randomness attributes of the modified generator, obtained from both the PSpice model and hardware implementation, using the PRBSG de-correlator were successfully tested by the well known NIST Test Suite and Diehard Test Set for statistical validation. Output binary streams from the proposed modified generator were examined for randomness using both Test Suites with all tests successfully passed for both PSpice model and experimental chaotic circuit generator. A physical TRBG has been constructed on the basis of the proposed PRBSG modification with all statistical tests successfully passed confirming theoretical expectations. (6 pages)

We study the statistics of the amplitude of the synchronization error in chaotic electronic circuits coupled through linear feedback. Depending on the coupling strength, our system exhibits three qualitatively different regimes of synchronization: weak coupling yields independent oscillations; moderate to strong coupling produces a regime of intermittent synchronization known as attractor bubbling; and stronger coupling produces complete synchronization. In the regime of moderate coupling, the probability distribution for the sizes of desynchronization events follows a power law, with an exponent that can be adjusted by changing the coupling strength. Such power-law distributions are interesting, as they appear in many complex systems. However, most of the systems with such a behavior have a fixed value for the exponent of the power law, while here we present an example of a system where the exponent of the power law is easily tuned in real time.

We present an algorithm for finding low-order periodic points of chaotic maps from possibly very short time series. No information about the map other than the time series is used. The method finds all the periodic points of a piecewise linear approximation of the map. We present examples showing the effectiveness of the method for the Henon and Ikeda maps and a chaotic electronic circuit, including a ``cycle expansion'' calculation of the Hausdorff dimension for the Henon map.

We have seen briefly in the previous chapter how tori and so-called "strange attractors" in phase space may provide the underlying mathematical representations of the processes involved in the creation of turbulence. Stationary and temporally periodic solutions, which we have covered in depth in Chapters 2 to 14, provide the simplest examples of attractors--the phase space limits of model trajectories--and their properties are relatively easy to describe analytically. However, with the need to consider more complicated temporal flows when modeling the transition to turbulence, it becomes essential to define quantitative measures for characterizing and categorizing the various types of attractors. In this chapter we introduce the Lyapunov exponents (Oseledec, 1968), the Lyapunov dimension (Kaplan and Yorke, 1979), and the correlation dimension (Grassberger and Procaccia, 1983a,b). These quantities give information about the attractor structure, solution transitions, and predictability characteristics of nonlinear models. We investigate how these quantitative measures of attractor dynamics might be helpful in distinguishing different types of chaotic attractors, and thereby possibly distinguishing some of the different routes to chaos. Moreover, because the seventh modeling principle suggests that a model must have sufficient complexity to represent a phenomenon in adequate detail, these quantitative measures may help us evaluate the adequacy of a particular model. Specifically, we illustrate the utility of the Lyapunov and correlation dimensions in characterizing attractors in several low-order spectral models, including the Lorenz (1963) model, the seven-coefficient generalized Lorenz system of Chang and Shirer (1984), and a related eleven-coefficient model.

This paper provides a rigorous mathematical proof that the double scroll is indeed chaotic. Our approach is to derive a linearly equivalent class of piecewise-linear differential equations which includes the double scroll as a special case. A necessary and sufficient condition for two such piecewise-linear vector fields to be linearly equivalent is that their respective eigenvalues be a scaled version of each other. In the special case where they are identical, we have exact equivalence in the sense of linear conjugacy. An explicit normalform equation in the context of global bifurcation is derived and parametrized by their eigenvalues. Analytical expressions for various Poincare maps are then derived and used to characterize the birth and the death of the double scroll, as well as to derive an approximate one-dimensional map in analytic form which is useful for further bifurcation analysis. In particular, the analytical expressions characterizing various half-return maps associated with the Poincare map are used in a crucial way to prove the existence of a Shilnikov-type homoclinic orbit, thereby establishing rigorously the chaotic nature of the double scroll. These analytical expressions are also fundamental in our in-depth analysis of the birth (onset of the double scroll) and death (extinction of chaos) of the double scroll. The unifying theme throughout this paper is to analyze the double scroll system as an unfolding of a large family of piecewise-linear vector fields in R^3 . Using this approach, we were able to prove that the chaotic dynamics of the double scroll is quite common, and is robust because the associated horseshoes predicted from Shilnikov's theorem are structurally stable. In fact, it is exhibited by a large family (in fact, infinitely many linearlyequivalent circuits) of vector fields whose associated piecewise-linear differential equations bear no resemblance to each other. It is therefore remarkable that the normalized eigenvalues, which is a local concept, completely determine the system's global qualitative behavior.

In some previous papers, we have already demonstrated how the problem of strange attractors (SA) was not confined only to the nature of their dynamics and how the aim of control did not depend only on the fact, that such a dynamics was considered as favourable or wrong for the functioning of the system modelled by these SA. The topology in the phase space of a SA is of the same importance because ergodic theory allows us to take into account the mean values of the various trajectories in this SA. As far as biomathematical models are concerned, these mean values may correspond either to physiological values or to pathological ones. In this regard rather than trying to eliminate (or re-establish) a chaotic dynamics (CD), control should sometimes try to displace the SA in order to let the mean values return to some physiological value. This paper reports a theoretical study of such a type of SA-control by using a so-called "model for the regulation of agonistic antagonistic (AA) couples" or "AA networks", including equations governing the values of certain control variables and has initially been connected to periodic dynamics; we recall these principles of control in case of an elementary AA-model (imbalanced limit-cycles). In case of AA-networks showing a CD, it is also possible to re-establish the balance between the variables with or without eliminating the CD of the attractor; in the second case, the control variables disclose a CD by themselves.

A model is proposed for the inputoutput nonlinearity and the gain-frequency characteristic of a typical bipolar operational amplifier circuit. Using this model a closed-form expression is derived for the output signal of the operational amplifier circuit excited by a multisinusoidal input signal. Using this expression, the RFI demodulation in the operational amplifier circuit is predicted.

Synchronization of chaotic models involving multiple drives and responses has numerous practical applications in cryptography and information processing. Existing research on synchronizing multiple chaotic systems is currently limited to twelve components. This study introduces a novel higher-order synchronization method, hexa compound combination, that synchronizes an assembly of nineteen n-dimensional chaotic models. Well-known synchronization methods, such as double compound, triple compound, and quad compound, serve as particular instances of this new strategy. Thus, our research significantly advances the understanding of multi-leveled chaos synchronization. In addition, we also present a non-uniformly conservative system classified into a rare category B, analyze its dynamic properties, and utilize it for achieving the proposed synchronization. Numerical results are provided through graphical representations to illustrate the efficacy of the new synchronization approach by comparing it with other techniques. Furthermore, we emulate the corresponding virtual schematic circuit of the newly designed system to evaluate its real-world applicability and utility.

This paper focuses on the modeling and symmetry breaking analysis of the large class of chaotic electronic circuits utilizing an antiparallel diodes pair as nonlinear device necessary for chaotic oscillations. A new and relatively simple autonomous jerk circuit is used as a paradigm. Unlike current approaches assuming identical diodes (and thus a perfect symmetric circuit), we consider the more realistic situation where both antiparallel diodes present different electrical properties in spite of unavoidable scattering of parameters. Hence, the nonlinear component synthesized by the diodes pair exhibits an asymmetric current–voltage characteristic which engenders the explicit symmetry break of the whole electronic circuit. The mathematical model of the new circuit consists of a continuous time 3D autonomous system with exponential nonlinear terms. We investigate the chaos mechanism with respect to model parameters and initial conditions as well, both in the symmetric and asymmetric modes of operation by using bifurcation diagrams and phase space trajectories plots as main indicators. We report period doubling route to chaos, spontaneous symmetry breaking, merging crisis and the coexistence of two, four, or six mutually symmetric attractors in the symmetric regime of operation. More intriguing and complex nonlinear behaviors are revealed in the asymmetric system such as coexisting asymmetric bubbles of bifurcation (a new kind of phenomenon discovered in this work), hysteretic dynamics, critical phenomena, and coexisting multiple (i.e. two, three, four, or five) asymmetric attractors for some appropriately selected values of parameters. Laboratory experimental tests are conducted to support the theoretical analysis. The results obtained in this work clearly indicate that chaotic circuits with back to back diodes can demonstrate much more complex dynamics than what is reported in the relevant literature and thus should be reconsidered accordingly following the method described in this paper.

There are numerous phenomena in the space, biological and ecological sciences whose discrete evolution can be effectively modeled by exponentially decaying discrete dynamical systems, which can be represented as follows: Let F : Rn x P → Rn be a smooth map, where Rn is Euclidean n-space representing the state variables x and P is a subset of Rm representing the parameter space comprised of m-vectors denoted by µ . The discrete dynamics of the system is given by the iterates of the map fµ : Rn → Rn , where fµ (x) := F(x,µ), with the exponentially decaying behavior embodied in the property that there exists a positive constant M such that |f µ(x)| ≤ Me -|x| for all (x,µ ) ∈ Rn x P. It is not difficult to show that these exponentially decaying dynamical systems have attracting sets that can have remarkable properties. For example, the attractor may be a strange attractor – a set with non-integer fractal dimension on which the dynamics is chaotic - and variations in the parameter can cause bifurcations that change the fundamental nature of the chaotic regimes. Known results on these strange attractors will be discussed, several new developments will be presented and some challenging open problems will be outlined. AMS Subject Classification: 37C05; 37G10; 37N25; 92D25. Keywords: Bifurcation, chaotic dynamics, discrete dynamical system simulation, fractal dimension, Lyapunov exponents, strange attractors, orbital trajectories, Dynamical Systems, exponentially decaying

In this paper, some complex nonlinear behaviors in a four-dimensional multiscroll autonomous memristor based chaotic system are investigated. This system is derived from the three-dimensional autonomous charge-controlled Muthuswamy–Chua simplest chaotic circuit. The system can generate four different coexisting attractors for a fixed set of parameters and different initial conditions. This phenomenon is relatively rare given that we have four different attractors namely: an equilibrium point, a stable limit cycle, a 16-peak limit cycle and a strange attractor that coexist in the system within a wide range of parameters. The nonlinear phenomenon of transient chaos is studied and revealed numerically in Matlab and Pspice environments. The complex transient dynamics of this memristive system under different initial states shows that the transient time depends strongly on the initial conditions. Moreover, this model displays spiking and bursting oscillations. The bursting behavior is classified according to the dynamics of separated slow and fast subsystems. It is shown to be of the fold-Hopf type. These complex dynamical behaviors of this system are investigated by means of numerical simulations and via Pspice circuit simulations. The use of bifurcation diagrams, Lyapunov exponents diagrams, power spectrums, phase portraits, time series, isospike diagram, basin of attraction, clearly shows these complex phenomena.

A single fractal number cannot fully describe the complex structure of a strange attractor embedded in phase space of dynamic systems. Hence it is essential to seek a procedure through which fractal dimension provides more information about the underlying dynamic system, in order to serve such a purpose correlation integrals are calculated at every row or off-diagonal from the distance matrix obtained from a set of points on a strange attractor. The slope of each integral is plotted versus the number of row or off-diagonal considered in the calculation. Hence, an asymptotic convergence to correlation dimension is obtained in this manner. More information is revealed for the dynamic system including its lacunarity.

The structure of a strange attractor

This paper discusses the explanatory significance of the equilibrium concept in the context of an example of extremely complicated dynamical behavior. In particular, numerical evidence is presented for the existence of chaotic dynamics on a "strange attractor" in the evolutionary game dynamics introduced by Taylor and Jonker [also known as the "replicator dynamics"]. This phenomenon is present already in four strategy evolutionary games where the dynamics takes place in a simplex in three dimensional space-the lowest number of dimensions in which such a strange attractor is possible. From a dynamical point of view, it is the attractor-rather than the equilibrium-that is of prime interest.

Implementation of chaotic secure communication systems based on OPA circuits