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No isolated system may ever support complexity. Emergent phenomena may however appear in an open system, if, as established by the Edge of Chaos theory, some of its constitutive elements feature the capability to amplify infinitesimal fluctuations in energy, provided an external source supplies them with a sufficient amount of DC power, which is known to be a signature for locally-active behaviour. In particular, complex behaviours, including static and dynamic pattern formation, may emerge in arrays of identical diffusively-coupled cells, if and only if the basic unit is poised on a particular sub-domain of the Local Activity regime, referred to as Edge of Chaos, within which a quiet state hides in fact a high degree of excitability. Here we show, for the first time, that these counterintuitive phenomena may emerge in a basic memristor cellular neural network, consisting of two identical diffusively-coupled second-order cells. The proposed bio-inspired array represents the simplest ever-reported open system, which reproduces the shocking phenomenon, reported by Smale in 1974, when, while studying a model from cellular biology, he observed two identical reaction cells, “mathematically dead” on their own, pulsating together upon diffusive coupling. Impressively, the bio-inspired two-cell reaction-diffusion network contains only nine circuit elements, specifically two DC voltage sources, three linear resistors, two linear capacitors, and two functional niobium oxide (NbO) memristors from NaMLab. Applying the theory of Local Activity to an accurate model of the memristor oscillator, a comprehensive picture for its local and global dynamics may be drawn, providing a systematic method to tune the design parameters of the two-cell array to enable diffusion-driven instabilities therein. [ABSTRACT FROM AUTHOR]

In this technical note, a novel approach to trajectory tracking control of a complex network is presented, based on the idea and technique of inverse optimal pinning control. A control law that stabilizes the tracking error dynamics is developed. This controller allows different coupling strengths, and simultaneously minimizing an associated cost functional. Chaotic systems are considered to illustrate the applicability of the new methodology and the proposed controller effectiveness via simulations. [ABSTRACT FROM AUTHOR]

The key intent of the work is to analyze complex dynamics and synchronization phenomena in a 1-D array of N diffusively coupled memristor-based oscillatory/chaotic circuits, i.e., each uncoupled oscillator is a 3rd–order memristor-based Chua’s circuit obtained by replacing the nonlinear resistor with an ideal flux-controlled memristor. It is shown that the state space \mathbb {R}^{4N} in the voltage–current domain of the array can be decomposed in \infty ^{N}~3N$ -dimensional manifolds which are positively invariant for the nonlinear dynamics. Moreover, on each manifold the array obeys a different reduced-order dynamics in the flux-charge domain. These basic properties imply that two main types of bifurcations can occur, i.e., standard bifurcations on a fixed invariant manifold induced by changing the circuit parameters and bifurcations due to the variation of initial conditions and invariant manifold, but for fixed circuit parameters. The latter bifurcation phenomena are referred to as bifurcations without parameters. The reduced dynamics on invariant manifolds, and their analytic expressions, are the key tools for a comprehensive analysis of synchronization phenomena in the array of memristor-based Chua’s circuits. The main results are proved via a recently introduced technique for studying memristor-based circuits in the flux-charge domain. [ABSTRACT FROM AUTHOR]

Motivated by the concept of circuit design in digital circuit, this paper proposes a one-dimensional (1D) nonlinear model (1D-NLM) for producing 1D discrete-time chaotic maps. Our previous works have designed four nonlinear operations of generating new chaotic maps. However, they focus only on discussing individual nonlinear operations and their properties, but fail to consider their relationship among these operations. The proposed 1D-NLM includes these existing nonlinear operations, develops two new nonlinear operations, discusses their relationship among different nonlinear operations, and investigates the properties of different combinations of these operations. To show the effectiveness of 1D-NLM in generating new chaotic maps, as examples, we provide four new chaotic maps and study their dynamics properties from following three aspects: equilibrium point, stability, and bifurcation diagram. Performance evaluations are provided using the Lyapunov exponent, Shannon entropy, correlation dimension, and initial state sensitivity. The evaluation results show that these new chaotic maps have more complex chaotic behaviors than existing ones. To demonstrate the performance of 1D-NLM in practical applications, we use a pseudo-random number generator (PRNG) to compare new and existing chaotic maps. The randomness test results indicate that new chaotic map generated by 1D-NLM shows better performance than existing ones in designing PRNG. [ABSTRACT FROM PUBLISHER]

Simulation of large-scale nonlinear dynamical systems on hardware with a high resemblance to their mathematical equivalents has been always a challenge in engineering. This paper presents a novel current-input current-output circuit supporting a systematic synthesis procedure of log-domain circuits capable of computing bilateral dynamical systems with considerably low power consumption and acceptable precision. Here, the application of the method is demonstrated by synthesizing four different case studies: 1) a relatively complex 2-D nonlinear neuron model; 2) a chaotic 3-D nonlinear dynamical system Lorenz attractor having arbitrary solutions for certain parameters; 3) a 2-D nonlinear Hopf oscillator, including bistability phenomenon sensitive to initial values; and 4) three small neurosynaptic networks comprising three FHN neuron models variously coupled with excitatory and inhibitory synapses. The validity of our approach is verified by nominal and Monte Carlo simulated results with realistic process parameters from the commercially available AMS 0.35- \mu \textm technology. The resulting continuous-time, continuous-value, and low-power circuits exhibit various bifurcation phenomena, nominal time-domain responses in good agreement with their mathematical counterparts and fairly acceptable process variation results (less than 5% STD). [ABSTRACT FROM PUBLISHER]

Dynamics of Chua’s circuit with a smooth nonlinearity is studied by means of interval arithmetic methods. We analyze behaviors of the system for several parameter values in the periodic and chaotic regions. For parameter values in the chaotic region, we find lower bounds for the topological entropy of a corresponding return map and prove that the system is chaotic in the topological sense. We find low-period orbits embedded in chaotic attractors and estimate the true value of the topological entropy. We construct a trapping region enclosing the spiral attractor and discuss how to prove the existence of a trapping region for the case of the double-scroll attractor. [ABSTRACT FROM PUBLISHER]

A technique is introduced for identifying uncertain and/or unknown parameters of chaotic and hyper-chaotic systems via using a synchronization-based parameter observer. The proposed technique is based on designing a state feedback controller and then solving for the unknown parameters using some simple parameter adaptive laws that require access to some or all of the states depending on the dynamical model of the systems. Two detailed cases utilizing chaotic and hyper-chaotic systems are used to exemplify the proposed observer when partial identification of the unknown parameters is considered, where the results demonstrated the effectiveness of the proposed method via comparing it with other methods reported in the literature, Furthermore, the feasibility of the designed observer on unknown-structure model and complex-variable model are verified through the theoretical analyses and simulation results, and the advantages and limitations of the synchronization-based parameter observer approach are discussed in detail. [ABSTRACT FROM AUTHOR]

Traditionally, chaotic systems are built on the domain of infinite precision in mathematics. However, the quantization is inevitable for any digital devices, which causes dynamical degradation. To cope with this problem, many methods were proposed, such as perturbing chaotic states and cascading multiple chaotic systems. This paper aims at developing a novel methodology to design the higher-dimensional digital chaotic systems (HDDCS) in the domain of finite precision. The proposed system is based on the chaos generation strategy controlled by random sequences. It is proven to satisfy the Devaney's definition of chaos. Also, we calculate the Lyapunov exponents for HDDCS. The application of HDDCS in image encryption is demonstrated via FPGA platform. As each operation of HDDCS is executed in the same fixed precision, no quantization loss occurs. Therefore, it provides a perfect solution to the dynamical degradation of digital chaos. [ABSTRACT FROM PUBLISHER]

A novel non-coherent multi-level differential chaos shift keying (DCSK) modulation scheme is proposed in this paper. This new scheme is based on both the transmitted-reference technique and M-ary orthogonal modulation, where each data-bearing signal is chosen from a set of orthogonal chaotic wavelets constructed by a reference signal. Thanks to this signaling design, the new scheme can achieve a higher attainable data rate, lower energy loss in reference transmission, increased bandwidth efficiency, better data security and better bit error rate (BER) performance as compared to the conventional DCSK. Unlike other DCSK-based systems that separate the reference and data-bearing signals using the TDMA scheme, this new system employs I/Q channels to send these two signals in a parallel and simultaneous manner, making the system easily extendable to multi-carriers. This transmission mechanism not only can further increase data rate but also can remove all radio frequency delay lines from detectors. Analytical BER expressions of the proposed system are derived for both additive white Gaussian noise (AWGN) and multipath Rayleigh fading channels. Relevant simulation results are given and compared to non-coherent binary/M-ary DCSK systems. In addition, the impacts of various system parameters on noise performance are discussed. Both theoretical analysis and simulation results confirm the promising benefits of the new design. [ABSTRACT FROM PUBLISHER]

A configuration of two memristors connected with opposite polarity is proposed to realize an adaptive law for consensus and synchronization. This configuration allows one to couple two systems so that the memristor memory variables regulate the coupling strength and dynamically evolve as a function of the mismatches between the units. A physical-based model capturing the main feature of TiO2 memristors is used for our study. Adaptive consensus of two integrators and adaptive synchronization of two chaotic circuits are investigated, and, then, extended to the case of more than two units coupled into a network configuration. The approach provides an effective strategy for the analog implementation of adaptive laws at the circuit level, as the proposed coupling configuration consists of only two components. [ABSTRACT FROM AUTHOR]

In this note, it is shown that some of the results reported in the above titled paper on the behavior analysis of a fractional order system are not consistent with reality. To support this claim, theoretical justifications and numerical results are presented. Also, it is analytically explained that why such inconsistencies have been occurred in the aforementioned paper. [ABSTRACT FROM AUTHOR]

This paper introduces a discrete wheel-switching chaotic system (DWSCS). Changing the controlling sequence of the wheel switch allows DWSCS to generate a large number of new chaotic sequences from a set of existing chaotic maps (seed maps). Simulations and analysis demonstrate the DWSCS's characteristics and chaotic behaviors. An FPGA design of DWSCS shows its effectiveness in hardware implementation. We also propose a pseudo-random number generator using DWSCS whose excellent performance has been shown in experiments and comparisons. [ABSTRACT FROM AUTHOR]