A new oscillator with mega-stability and its Hamilton energy: Infinite coexisting hidden and self-excited attractors

Wave propagation in a light-temperature neural network under adaptive local energy balance

External electric and mechanical stimuli can induce shape deformation in excitable media because of its intrinsic flexible property. When the signals propagation in the media is described by a neural network, creation of heterogeneity or defect is considered as the effect of shape deformation due to accumulation or release of energy in the media. In this paper, a temperature-light sensitive neuron model is developed from a nonlinear circuit composed of a phototube and a thermistor, and the physical energy is kept in capacitive and inductive terms. Furthermore, the Hamilton energy for this function neuron is obtained in theoretical way. A regular neural network is built on a square array by activating electric synapse between adjacent neurons, and a few of neurons in local area is excited by noisy disturbance, which induces local energy diversity, and continuous coupling enables energy propagation and diffusion. Initially, the Hamilton energy function for a temperature-light sensitive neuron can be obtained. Then, the finite neurons are applied noise to obtain energy diversity to explore the energy spread between neurons in the network. For keeping local energy balance, one intrinsic parameter is regulated adaptively until energy diversity in this local area is decreased greatly. Regular pattern formation indicates that local energy balance creates heterogeneity or defects and a few of neurons show continuous parameter shift for keeping energy balance in a local area, which supports gradient energy distribution for propagating waves in the network.

Investigation of the Simplest Megastable Chaotic Oscillator with Spatially Triangular Wave Damping

The simplest megastable chaotic system is built by employing a piecewise-linear damping function which is periodic over the spatial domain. The unforced oscillator generates an infinite number of nested limit cycles with constant distances whose strength of attraction decreases gradually as moving to outer ones. The attractors and the basins of attraction of the proposed system are almost compatible with those of the system with sinusoidal damping. However, the nonzero Lyapunov Exponent of the latter is consistently below that of the former. A comparative bifurcation analysis is carried out for periodically forced systems, showing the chaotic behavior of coexisting attractors in specific values of parameters. Changing the bifurcation parameter results in expansion, contraction, merging, and separation of the coexisting attractors, make it challenging to find the corresponding basins. Three symmetric pairs of attractors are observed; each one consists of two symmetric attractors (with respect to the origin) with almost the same values of the corresponding Lyapunov Exponent.

A Novel Compound-Coupled Hyperchaotic Map for Image Encryption

Considering a nonlinear dynamic oscillator, a high Lyapunov exponent indicates a high degree of randomness useful in many applications, including cryptography. Most existing oscillators yield very low Lyapunov exponents. The proposed work presents a general strategy to derive an n-D hyperchaotic map with a high Lyapunov exponent. A 2D case study was analyzed using some well-known nonlinear dynamic metrics including phase portraits, bifurcation diagrams, finite time Lyapunov exponents, and dimension. These metrics indicated that the state of the novel map was more scattered in the phase plane than in the case of some traditional maps. Consequently, the novel map could produce output sequences with a high degree of randomness. Another important observation was that the first and second Lyapunov exponents of the proposed 2D map were both positive for the whole parameter space. Consequently, the attractors of the map could be classified as hyperchaotic attractors. Finally, these hyperchaotic sequences were exploited for image encryption/decryption. Various validation metrics were exploited to illustrate the security of the presented methodology against cryptanalysts. Comparative analysis indicated the superiority of the proposed encryption/decryption protocol over some recent state-of-the-art methods.

New Type Modelling of the Circumscribed Self-Excited Spherical Attractor

The fractal–fractional derivative with the Mittag–Leffler kernel is employed to design the fractional-order model of the new circumscribed self-excited spherical attractor, which is not investigated yet by fractional operators. Moreover, the theorems of Schauder’s fixed point and Banach fixed existence theory are used to guarantee that there are solutions to the model. Approximate solutions to the problem are presented by an effective method. To prove the efficiency of the given technique, different values of fractal and fractional orders as well as initial conditions are selected. Figures of the approximate solutions are provided for each case in different dimensions.

A new autonomous memristive megastable oscillator and its Hamiltonian-energy-dependent megastability

Multistability is a special issue in nonlinear dynamics. In this paper, a three-dimensional autonomous memristive chaotic system is presented, with interesting multiple coexisting attractors in a nested structure observed, which indicates the megastability. Furthermore, the extreme event is investigated by local riddled basins. Based on Helmholtz's theorem, the average Hamiltonian energy with respect to initial-dependent dynamics is calculated and the energy transition explains the occurrence mechanisms of the megastability and the extreme event. Finally, by configuring initial conditions, multiple coexisting megastable attractors are captured in PSIM simulations and FPGA circuits, which validate the numerical results.

Periodic offset boosting for attractor self-reproducing

The special regime of multistability of attractor self-reproducing is deeply decoded based on the conception of offset boosting in this letter. Attractor self-reproducing is essentially originated from periodic initial condition-triggered offset boosting. Typically, a trigonometric function is applied for attractor self-reproducing. The position, size, and clone frequency determine the selected periodic function. Specifically, in-depth investigation on three elements of sinusoidal quantity is taken into account and then a universal law of attractor self-reproducing is built: the original position of an attractor determines the initial phase and the size of attractor sets the amplitude, while the reproducing interval between two attractors determines the frequency of the trigonometric function. It is found that the product of amplitude and frequency is a constant determined by the reproducing periodic function. The positive and negative switching of the slope in sinusoidal function also leads to the waste of phase space since in general there is no attractor reproduced at the region with negative slope except that new polarity balance is reconstructed paying back the attractor with conditional symmetry. Three-element-oriented offset boosting makes attractor self-reproducing more designable, achievable, and adjustable, which brings great convenience to engineering applications.

Multistability analysis and nonlinear vibration for generator set in series hybrid electric vehicle through electromechanical coupling

The non-linear analysis of undesired vibrations observed on hybrid electric vehicle (HEV) powertrains is hardly developed in the literature. In this paper, a mathematical modeling of the vibrations observed at the level of the electromechanical coupling between the internal combustion engine and the generator in the series architecture of HEVs, named (SHEVs), is established using the Lagrangian theory. The stability and instability motions of this SHEV are perfectly detailed using amplitude-frequency response curves. An analysis of the electromagnetic torque amplitude of the new SHEV demonstrates the presence of multistability with the coexistence of two or three different types of attractors. In addition, this new SHEV model has other dynamic regimes of chaotic and periodic oscillations. Coexisting bifurcations with parallel branches, hysteresis, and period-doubling are also discovered. A unique contribution of this work is the abundance and complicated dynamical behaviors found in such types of systems compared with some rare cases previously reported on HEV powertrain models. The simulation results obtained using non-linear analysis tools sufficiently demonstrate that the objectives of this paper are achieved.

Window of multistability and its control in a simple 3D Hopfield neural network: application to biomedical image encryption

In this contribution, the problem of multistability control in a simple model of 3D HNNs as well as its application to biomedical image encryption is addressed. The space magnetization is justified by the coexistence of up to six disconnected attractors including both chaotic and periodic. The linear augmentation method is successfully applied to control the multistable HNNs into a monostable network. The control of the coexisting four attractors including a pair of chaotic attractors and a pair of periodic attractors is made through three crises that enable the chaotic attractors to be metamorphosed in a monostable periodic attractor. Also, the control of six coexisting attractors (with two pairs of chaotic attractors and a pair of periodic one) is made through five crises enabling all the chaotic attractors to be metamorphosed in a monostable periodic attractor. Note that this controlled HNN is obtained for higher values of the coupling strength. These interesting results are obtained using nonlinear analysis tools such as the phase portraits, bifurcations diagrams, graph of maximum Lyapunov exponent, and basins of attraction. The obtained results have been perfectly supported using the PSPICE simulation environment. Finally, a simple encryption scheme is designed jointly using the sequences of the proposed HNNs and the sequences of real/imaginary values of the Julia fractals set. The obtained cryptosystem is validated using some well-known metrics. The proposed method achieved entropy of 7.9992, NPCR of 99.6299, and encryption time of 0.21 for the 256*256 sample 1 image.