The discovery of simple chaotic systems with complex dynamics has always been an interesting research work. This brief aims to construct an extremely simple chaotic system with infinitely many ...coexisting chaotic attractors. The system consists of five terms with two nonlinearities, and has an infinite number of unstable equilibria owing to its sinusoidal nonlinearity. The most prominent feature of the system is that it coexists infinitely many chaotic attractors for different initial values and fixed system parameters. To our best knowledge, there is no 3-D system with such a simple mathematical model can produce infinitely many coexisting chaotic attractors. The phenomenon of coexisting attractors of the new system is numerically investigated. The circuit and microcontroller-based implementation of the system are presented as well.
A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm ...of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10 different input distributions or histograms.
This brief presents a novel control scheme to achieve fast fixed-time system stabilization. Based on fixed-time stability theory, a novel fixed-time stable system is presented. Using the proposed ...fixed-time stable system, a fast fixed-time nonsingular terminal sliding mode control method is derived. Our control scheme achieves system stabilization within bounded time independent of the initial condition and has an advantage in convergence rate over the existing result of the fixed-time stable control method. The proposed control strategy is applied to suppress chaotic oscillation in power systems, and its effectiveness as well as superiority is verified through numerical simulation. The proposed control strategy can be applied to address the control and synchronization problem for other complex systems.
Chaotic systems with memristor are favored by academia because of diversity of dynamics. This brief reports a novel two-memristor-based 4D chaotic system. Numerical simulation shows that the system ...can yield infinite coexisting attractors. The generation mode of infinite coexisting attractors is investigated as well. Finally, the circuit and microcontroller-based implementation of two-memristor-based chaotic system are also given, corresponding experimental results fit well with numerical simulations.
This paper investigates spatiotemporal nonlinear dynamics and chaos in a dissipative mechanical Duffing-type system subjected to external stimulus. A nonlinear wave equation with cubic nonlinearity ...governs the system dynamics. A perturbation description is employed to build mathematical tools that represent different aspects of system dynamics, from local to global behaviors. Lyapunov exponents are defined from the different perturbations allowing the evaluation of local, convective and mean exponents. Different dynamical regimes are investigated considering homogeneous and heterogeneous spatial stimuli. Distinct dynamical responses are observed including periodic, quasi-periodic and chaotic behaviors. A novel concept of chaotic wave is employed to explain the spatial transport of chaos through the media considering heterogeneous conditions. Chaotic wave velocity is measured by the convective Lyapunov exponents.
•Spatiotemporal nonlinear dynamics of a dissipative Duffing-type system considering external stimulus.•Perturbaton tools are employed to define local, mean and convective Lyapunov exponents.•Regular and irregular dynamics are investigated.•A novel concept of chaotic wave is employed to explain the spatial transport of chaos through the media.
In view of the problem that dissipative chaos has attractors and is easy to be attacked by reconstruction, which leads to the security defects of encryption algorithm based on dissipative chaos, we ...design a new general form of n-dimensional conservative chaos according to the generalized Hamiltonian system. Taking four-dimensional (4D) as an example, numerical verification and performance analysis show that the conservative chaos has excellent chaotic characteristics such as wide ergodicity, no attractors, no chaotic degradation, and it can resist reconstruction and other attacks. Based on this 4D conservative chaos, we propose a new image encryption algorithm, which includes the plaintext related dynamic scrambling method and the dynamic diffusion mechanism of quadrilateral rule (MQR). Moreover, the initial values of the system are controlled by the external key stream and the internal key stream, so that the generation of ciphertext information are closely related with that of plaintext information, part of the ciphertext information, pseudo-random sequence and the key stream, which can increase the ability of the algorithm to resist plaintext and other attacks. Experimental simulation and performance analysis show that the encryption algorithm has better security and real time communication.
The significance of neuronal discharge symmetry lies in its reflection of the stability and orderliness within the neuron's activity. This symmetry is correlated with various issues, including the ...biophysical properties within neurons, network structures, and synaptic transmission. Memristors are highly analogous to neuronal synapses due to their unique memory properties, allowing them to emulate biological neural synapses. In this work, a memristor is employed into the Hindmarsh‐Rose (HR) neuron as a synapse for the construction of a memristive HR neuron. Accompanied with the introducing of the absolute value and signum function, the derived neurons exhibit complex coexisting symmetric firing patterns. The phenomenon of symmetric firing can effectively simulate the depolarization and hyperpolarization processes of neurons, providing a new approach to study the diversity of the brain.
In this work, a memristor is employed into the HR neuron as a synapse for the construction of a memristive HR neuron. Accompanied with the introducing of the absolute value and signum function, the derived neurons exhibit complex coexisting symmetric firing patterns.
There has recently been an increase in the number of new chaotic system designs and chaos-based engineering applications. In this study, since homoclinic and heteroclinic orbits did not exist and ...analyses like Shilnikov method could not be used, a 3D chaotic system without equilibrium points was included and thus different engineering applications especially for encryption studies were realized. The 3D chaotic system without equilibrium points represents a new different phenomenon and an almost unexplored field of research. First of all, chaotic system without equilibrium points was examined as the basis and electronic circuit application of the chaotic system was realized and oscilloscope outputs of phase portraits were obtained. Later, chaotic system without equilibrium points was modelled on Labview Field Programmable Gate Array (FPGA) and then FPGA chip statistics, phase portraits and oscilloscope outputs were derived. With another study, VHDL and RK-4 algorithm were used and a new FPGA-based chaotic oscillators design was achieved. Results of Labview-based design on FPGA- and VHDL-based design were compared. Results of chaotic oscillator units designed here were gained via Xilinx ISE Simulator. Finally, a new chaos-based RNG design was achieved and internationally accepted FIPS-140-1 and NIST-800-22 randomness tests were run. Furthermore, video encryption application and security analyses were carried out with the RNG designed here.
The exponential growth of the out-of-time-ordered correlator (OTOC) has been proposed as a quantum signature of classical chaos. The growth rate is expected to coincide with the classical Lyapunov ...exponent. This quantum-classical correspondence has been corroborated for the kicked rotor and the stadium billiard, which are one-body chaotic systems. The conjecture has not yet been validated for realistic systems with interactions. We make progress in this direction by studying the OTOC in the Dicke model, where two-level atoms cooperatively interact with a quantized radiation field. For parameters where the model is chaotic in the classical limit, the OTOC increases exponentially in time with a rate that closely follows the classical Lyapunov exponent.
•One method of making a simple and effective chaotic system by using different outputs sequences of two same existing 1D chaotic maps was introduced.•A novel encryption system of ...linear-nonlinear-linear structure based on total shuffling was proposed•The experimental results show the proposed encryption algorithm has excellent performance in noise and attacks
This paper introduces a method of making a simple and effective chaotic system by using a difference of the output sequences of two same existing one-dimension (1D) chaotic maps. Simulations and performance evaluations show that the proposed system is able to produce a one-dimension (1D) chaotic system with better chaotic performances and larger chaotic ranges compared with the previous chaotic maps. To investigate its applications in image encryption, a novel encryption system of linear-nonlinear-linear structure based on total shuffling is proposed. The experiment demonstrated the accuracy of the encryption algorithm. Experiments and security analysis prove that the algorithm has an excellent performance in image encryption and various attacks.
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