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This review paper describes different lumped circuitry realizations of the chaotic dynamical systems having equilibrium degeneration into a plane object with topological dimension of the equilibrium structure equals one. This property has limited amount (but still increasing, especially recently) of third-order autonomous deterministic dynamical systems. Mathematical models are generalized into classes to design analog networks as universal as possible, capable of modeling the rich scale of associated dynamics including the so-called chaos. Reference state trajectories for the chaotic attractors are generated via numerical analysis. Since used active devices can be precisely approximated by using third-level frequency dependent model, it is believed that computer simulations are close enough to capture real behavior. These simulations are included to demonstrate the existence of chaotic motion.

In multi-carrier differential chaos shift keying (MC-DCSK) system, channel noises pollute both the reference and data signals, resulting in deteriorated performance. To reduce noises in received signals in MC-DCSK, a novel noise reduction MC-DCSK (NR-MC-DCSK) system is proposed in this paper. The proposed system utilizes duplicated chaotic samples, rather than different ones, as the reference. At the receiver side, identical samples can be averaged before correlation detection, which helps decrease the noise interferences and thus brings performance improvement. Theoretical bit error rate (BER) expressions are derived and verified by simulation results for additive white Gaussian noise and multipath Rayleigh fading channels. Finally, comparisons to MC-DCSK and other DCSK-based systems are given to confirm the superiority of the proposed system in BER performance.

Chaotic communication requires the knowledge of corresponding phase relationship between the primary phase of Duffing oscillator’s internal driving force and the primary phase of the undetected signal. Currently, there is no method of noncoherent demodulation for DPSK (Differential Phase Shift Keying) signal and mobile communication signal by Duffing oscillator. To solve this problem, this study presents a noncoherent demodulation method based on the Duffing oscillators array and Duffing oscillator optimization. We first present the model of Duffing oscillator and its sensitivity to undetected signal primary phase. Then the zone partition is proposed to identify the Duffing oscillator’s phase trajectory, and subsequently, the mathematical model and implementation method of the Duffing oscillators array are outlined. Thirdly, the Duffing oscillator optimization and its adaptive strobe technique are proposed, also their application to DPSK signal noncoherent demodulation are discussed. Finally, the design of new concept DPSK chaotic digital receiver based on the Duffing oscillators array is presented, together with its simulation results obtained by using SystemView simulation platform. The simulation results suggest that the new concept receiver based on the Duffing oscillator optimization of Duffing oscillators array owns better SNR (signal-to-noise ratio) threshold property than typical existing receivers (chaotic or nonchaotic) in the AWGN (additive white Gaussian noise) channel and multipath Rayleigh fading channel. In addition, the new concept receiver may detect mobile communication signal.

High-order Lorenz systems with five, six, eight, nine, and eleven equations are derived by choosing different numbers of Fourier modes upon truncation. For the original Lorenz system and for the five high-order Lorenz systems, solutions are numerically computed, and periodicity diagrams are plotted on $(\sigma ,r)$ parameter planes with $\sigma ,r\in [0,1000]$, and $b=8/3$. Dramatic shifts of patterns are observed among periodicity diagrams of different systems, including the appearance of expansive areas of period 2 in the fifth-, eighth-, ninth-, and 11th-order systems and the disappearance of the onion-like structure beyond order 5. Bifurcation diagrams along with phase portraits offer a closer look at the two phenomena.

This paper presents the Chen system as a controlled weather model. Mathematically, the Chen system is dual to the Lorenz system via time reversal. Physically, the Chen system can be viewed as a controlled weather model from the anti-control perspective. This paper illustrates the physical principle of this controlled weather model, and develops an engineering design of the model for real indoor climate (temperature-humidity) regulation, with a perspective on outdoor weather control application.

We analyze an extended version of the dynamical mean-field Ising model. Instead of classical physical representation of spins and external magnetic field, the model describes traders' opinion dynamics. The external field is endogenized to represent a smoothed moving average of the past state variable. This model captures in a simple set-up the interplay between instantaneous social imitation and past trends in social coordinations. We show the existence of a rich set of bifurcations as a function of the two parameters quantifying the relative importance of instantaneous versus past social opinions on the formation of the next value of the state variable. Moreover, we present a thorough analysis of chaotic behavior, which is exhibited in certain parameter regimes. Finally, we examine several transitions through bifurcation curves and study how they could be understood as specific market scenarios. We find that the amplitude of the corrections needed to recover from a crisis and to push the system back to “normal” is often significantly larger than the strength of the causes that led to the crisis itself.

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the $D$-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

In this study, we first present a generalized Lorenz model (LM) with $M$ modes, where $M$ is an odd number that is greater than three. The generalized LM (GLM) is derived based on a successive extension of the nonlinear feedback loop (NFL) with additional high wavenumber modes. By performing a linear stability analysis with $\sigma =10$ and $b=8/3$, we illustrate that: (1) within the 3D, 5D, and 7D LMs, the appearance of unstable nontrivial critical points requires a larger Rayleigh parameter in a higher-dimensional LM and (2) within the 9DLM, nontrivial critical points are stable. By comparing the GLM with various numbers of modes, we discuss the aggregated negative feedback enabled by the extended NFL and its role in stabilizing solutions in high-dimensional LMs. Our analysis indicates that the 9DLM is the lowest order generalized LM with stable nontrivial critical points for all Rayleigh parameters greater than one. As shown by calculations of the ensemble Lyapunov exponent, the 9DLM still produces chaotic solutions. Within the 9DLM, a larger critical value for the Rayleigh parameter, $r_c=679.8$, is required for the onset of chaos as compared to a $r_c=24.74$ for the 3DLM, a $r_c=42.9$ for the 5DLM, and a $r_c=116.9$ for the 7DLM. In association with stable nontrivial critical points that may lead to steady-state solutions, the appearance of chaotic orbits indicates the important role of a saddle point at the origin in producing the sensitive dependence of solutions on initial conditions. The 9DLM displays the coexistence of chaotic and steady-state solutions at moderate Rayleigh parameters and the coexistence of limit cycle and steady-state solutions at large Rayleigh parameters. The first kind of coexistence appears within a smaller range of Rayleigh parameters in lower-dimensional LMs (i.e. $24.06<r<24.74$ within the 3DLM) but in a wider range of Rayleigh parameters within the 9DLM (i.e. $679.8<r<1058$). The second kind of coexistence has never been reported in high-dimensional Lorenz systems.

At present, the performance of the digital chaotic sequence generator is affected by the calculation precision of processor and quantization method, which leads to the quantized chaotic binary sequences emerging from some periodic phenomena in the local range of sequences. However, in view of the current methods for analyzing the periodicity and randomness of binary sequences, evaluating the periodic characteristics which exist in the local range of chaotic binary sequences is extremely difficult. Therefore, in this paper, we have proposed a novel algorithm called the GBSPD (General Binary Sequences Periodic Detection) algorithm, which can detect typical periodic phenomena and local periodic phenomena in the chaotic binary sequence. Moreover, it can accurately locate the position of periodic phenomena of binary sequences and perform a detailed statistical analysis. Furthermore, by using the GBSPD algorithm to detect the periodic characteristics of the quantified logistic chaotic binary sequences, the experimental results show that the algorithm can effectively locate multiple periodic phenomena in the binary sequences.

This paper is concerned with the frequency independent continuous sequences generated by the following discrete spatiotemporal system:

The nonlinear dynamics of a multigear pair with the time-varying gear mesh stiffness are investigated using an enhanced compliance-based methodology. In the proposed approach, Lagrangian theory and Runge–Kutta method are used to derive the equation of motion of the multigear pair and solve its dynamic response for various values of the gear mesh frequency, respectively. The simulation results obtained for the dynamic behavior of the multigear pair are compared with those obtained by using continuous (cosine, sine and offset sine function) and intermittent representations of the time-varying gear mesh stiffness. It is shown that periodic, quasi-periodic, aperiodic and chaos motions are induced at different values of the gear mesh frequency. In addition, the bifurcation diagram reveals the occurrence of both nonimpact motion and single-sided impact motion, and Lyapunov exponent can easily diagnose the chaos phenomenon of system.

In this paper, a new class of systems with nonclassical jump resonance behavior is presented. Although jump resonance has been widely studied in the literature, this contribution refers to systems presenting a multiple hysteresis jump resonance phenomenon, meaning that the frequency response of the system presents more hysteresis windows nested within the same range of frequency. The analytical conditions for observing this type of behavior are derived and a design strategy to obtain multiple hysteresis jump resonance in circuits and systems presented.

Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude.

AIMD and RED are two dominant algorithms for controlling Internet congestion. So this paper explores the periodic solutions and complex dynamical phenomena in the state-dependent round-trip delayed AIMD/RED network congestion model with heterogeneous flows, and its improved model. We first use the semi-analytical and semi-numerical method, known as the harmonic balance method with alternating frequency/time (HB-AFT) domain technique, to derive the analytical approximations of periodic oscillations of the system. The obtained results are compared with the numerical results by WinPP, and they show good consistency. At the same time, this suggests that the method used in this paper is correct and valid. Then for the sake of making the system more realistic, we improve the model by using the hyperbolic tangent function. We obtain the approximate solutions, and find some rich dynamical behaviors of this delayed heterogeneous system, including Period-1 to torus, Period-1 to Period-2 to Period-3 motions and two kinds of mechanisms of chaos, i.e. the windows of Period-2 and Period-3 orbits to chaos, where to the best knowledge of the authors, the former route has never been reported. The periodic oscillations may induce synchronization and further congestion, where chaotic oscillation usually means that the system is unstable and may even collapse. Hence, we need to avoid these abundant dynamics discovered in this paper because they are undesirable and harmful. The derived results can help researchers better understand the performance of the AIMD/RED system, and they can be a guide for choosing parameters in a suitable range in order to maintain the network stability and optimize system performance.

Detection of chaos in experimental data is a crucial issue in nonlinear science. Historically, one of the first evidences of a chaotic behavior in experimental recordings came from laser physics. In a recent work, a Minimal Universal Model of chaos was developed by revisiting the model of laser with feedback, and a first electronic implementation was discussed. Here, we propose an upgraded electronic implementation of the Minimal Universal Model, which allows for a precise and reproducible analysis of the model’s parameters space. As a marker of a possible chaotic behavior the variability of the spiking activity that characterizes one of the system’s coordinates was used. Relying on a numerical characterization of the relationship between spiking activity and maximum Lyapunov exponent at different parameter combinations, several potentially chaotic settings were selected. The analysis via divergence exponent method of experimental time series acquired by using those settings confirmed a robust chaotic behavior and provided values of the maximum Lyapunov exponent that are in very good agreement with the theoretical predictions. The results of this work further uphold the reliability of the Minimal Universal Model. In addition, the upgraded electronic implementation provides an easily controllable setup that allows for further developments aiming at coupling multiple chaotic systems and investigating synchronization processes.

The presence of nonlinear feedback is an effective method for generating chaos in dynamical systems. Although physics provides a plethora of nonlinear relationships that can be exploited for this purpose, engineering chaotic electronic circuits with prescribed nonlinear terms remain a formidable challenge. In particular, the implementation of fractional exponentiation represents a particularly challenging task. In this paper, we present a solution to this problem in the form of a new circuit topology, which we have designated as the “444 circuit”, comprising only four transistors, four operational amplifiers, and four resistor pairs. In addition to multiplication and division, this circuit can perform complex algebraic operations, including the calculation of fractional powers. After introducing its operating principles and some conceptual aspects of its implementation, we demonstrate the performance of this unifying circuit structure by realizing a family of Lorenz-like systems.

This paper provides evidence of fractal, multifractal and chaotic behaviors in urban crime by computing key statistical attributes over a long data register of criminal activity. Fractal and multifractal analyses based on power spectrum, Hurst exponent computation, hierarchical power law detection and multifractal spectrum are considered ways to characterize and quantify the footprint of complexity of criminal activity. Moreover, observed chaos analysis is considered a second step to pinpoint the nature of the underlying crime dynamics. This approach is carried out on a long database of burglary activity reported by 10 police districts of San Francisco city. In general, interarrival time processes of criminal activity in San Francisco exhibit fractal and multifractal patterns. The behavior of some of these processes is close to $1/f$ noise. Therefore, a characterization as deterministic, high-dimensional, chaotic phenomena is viable. Thus, the nature of crime dynamics can be studied from geometric and chaotic perspectives. Our findings support that crime dynamics may be understood from complex systems theories like self-organized criticality or highly optimized tolerance.

The fractional derivative holds historical dependence or non-locality and it becomes a powerful tool in many real-world applications. But it also brings error accumulation of the numerical solutions as well as the theoretical analysis since many properties from the integer order case cannot hold. This paper defines the tempered fractional derivative on an isolated time scale and suggests a new method based on the time scale theory for numerical discretization. Some useful properties like composition law and equivalent fractional sum equations are derived for theoretical analysis. Finally, numerical formulas of fractional discrete systems are provided. As a special case for the step size $h=1$, a fractional logistic map with two-parameter memory effects is reported.

In recent years, some efforts have been devoted to nonlinear dynamics of fractional discrete-time systems. A number of papers have so far discussed results related to the presence of chaos in fractional maps. However, less results have been published to date regarding the presence of hyperchaos in fractional discrete-time systems. This paper aims to bridge the gap by introducing a new three-dimensional fractional map that shows, for the first time, complex hyperchaotic behaviors. A detailed analysis of the map dynamics is conducted via computation of Lyapunov exponents, bifurcation diagrams, phase portraits, approximated entropy and $C_0$ complexity. Simulation results confirm the effectiveness of the approach illustrated herein.

A new fractional two-dimensional quadric polynomial discrete chaotic map (2D-QPDM) with the discrete fractional difference is proposed. Afterwards, the new dynamical behaviors are observed, so that the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits of the proposed map are given, respectively. The new discrete fractional map is exploited into color image encryption algorithm and it is illustrated with several examples. The proposed image encryption algorithm is analyzed in six aspects which indicates that the proposed algorithm is superior to other known algorithms as a conclusion.