Hamilton Energy Research Articles

Modelling and dynamic analysis of a novel seven-dimensional Hamilton conservative hyperchaotic systems with wide range of parameter

The study of chaotic attractors has been a hot issue in complex science research in recent years. However, most of the current research has focused on low-dimensional dissipative systems. High-dimensional conservative systems have both conservative and hyperchaotic properties, the phase space is integer dimensional and does not have attractors, and the trajectories expand in multiple directions, thus having higher complexity and spatial ergodicity. In addition, the high dimensional conservative system with wide parameter range not only has better dynamic characteristics, but also has a good application prospect in the field of information security. In this paper, a novel seven-dimensional Hamiltonian conservative hyperchaotic system (7D-HCHCS) is constructed. The dynamical properties of this system are described by analyzing the rate of change of phase space volume, phase trajectory diagram, Poincaré map, Lyapunov exponential spectrum (LEs), bifurcation diagram, equilibrium point, and system complexity. A new pseudo-random number generator (PRNG) is designed on this basis, and the key stream generated by this PRNG passes the NIST test. Besides, the phase diagrams and Poincaré map under a wide range of parameters are compared. The results show that the proposed system satisfies the Hamilton energy conservation and can generate hyperchaotic flow. It also has good pseudorandom characteristics, ergodicity under a large range of control parameters, which also has good prospects in the field of information security.

Global dynamical analysis of the integer and fractional 4D hyperchaotic Rabinovich system

In this paper, the dynamical behavior of integer and fractional 4D hyperchaotic Rabinovich system is studied. We use the Lagrange coefficient method to solve an optimization problem analytically so that find an accurate ultimate bound set (UBS) for the 4D hyperchaotic Rabinovich system. Furthermore, bifurcation diagrams, Lyapunov exponents, global attractive sets (GASs), and positive invariant sets (PISs) of the fractional-order system are studied. Finally, using the Mittag-Leffler function and Lyapunov function method, the Mittag-Leffler GAS and Mittag-Leffler PIS of the proposed system are estimated. Our investigations indicate that there is a close relationship between changing the parameters and the dynamical behavior of the system, Hamilton energy consumption and changing the bound of the variables. The corresponding boundedness is numerically verified to show the effectiveness of the theoretical analysis.

Phase synchronization between neurons under nonlinear coupling via hybrid synapse

Biological neurons and nonlinear circuits contain certain inner field energy, and an equivalent Hamilton energy function can be obtained to discern the dependence of firing modes on the energy flow. In this paper, a voltage-controlled resistor is used to connect two neural circuits, and hybrid synapse is activated to control the synchronization stability and mode transition in neurons under phase lock. Realistic stimuli can be combined signals within certain frequency band rather than simple periodic stimulus, and the firing patterns will show multiple modes by applying mixed signals. Similar coherence resonance is detected in an isolated neuron driven by the filtered signals. Therefore, chaotic signals are encoded to represent filtered waves with finite frequency band, and then the mixed signals are used to excite two neural circuits via nonlinear coupling. Hamilton energy for the coupled neurons via hybrid synapse is estimated to detect the controllability of synchronization between neurons. The nonlinear resistor along the coupling channel can reproduce the biophysical property of hybrid synapse accompanying with field coupling, which mainly contributes the energy exchange between neurons. It is confirmed that this hybrid synapse is effective to control the synchronization stability and phase lock even the neurons presenting with different firing patterns/modes. The results confirmed that nonlinear coupling via specific components is helpful to design hybrid synapse, and it is effective to block the bursting synchronization between neurons driven by filtered chaotic signals. Therefore, activation of this hybrid synapse can prevent the occurrence of seizure in nervous system.

Energy encoding in a biophysical neuron and adaptive energy balance under field coupling

Some nonlinear circuits can present similar dynamical properties in the outputs as biological neurons by taming the external stimulus and bifurcation parameter, and they can be used for further investigation on neural activities. In this work, a photoelectric converter such as phototube is incorporated into the Fitzhugh-Nagumo neural circuit and a light-sensitive neuron is obtained to describe the nonlinear response in visual neuron. Furthermore, an additive induction coil is also connected to the phototube in this branch circuit of this neural circuit for detecting the disturbance from external magnetic field, and this functional neuron can perceive external illumination and changes of external magnetic field synchronously. Scale transformation is applied for the circuit equation and equivalent photoelectric neuron model is obtained. The intrinsic field energy of the neural circuit is described by equivalent Hamilton energy and the dependence firing modes on energy is clarified. When neurons are clustered closely, energy diversity can trigger field coupling and the synaptic connection will be regulated in the coupling intensity adaptively. An artificial hybrid is developed by combining capacitor and inductor in the coupling channel, and synchronization approach and energy balance is controlled completely. These results provide potential explanation for knowing the growth mechanism for synapse connection and external energy injection can be effective to control the collective behaviors of neurons.

Estimate physical reliability in Hindmarsh-Rose neuron

Reliable neural circuit can address the field energy in exact form and equivalent Hamilton energy is obtained for theoretical models. Furthermore, the energy dependence on firing activities can be discerned by changing the excitability and forcing current applied on the neuron. In this paper, the distinct energy function for the original Hindmarsh-Rose (HR) neuron is obtained by using Helmholtz theorem and then an equivalent two-variable HR model is proposed. By applying additive noise, stochastic resonance is induced. The intrinsic energy function is approached and energy shunting from each electric component is confirmed. The energy proportion for magnetic field and electric field is changed when mode transition is controlled by the external stimulus. Therefore, the physical property for each electric component in the equivalent neural circuit is predicted, and then the reliability of our neural circuit is confirmed in experiment.

A memristive neuron and its adaptability to external electric field.

Connecting memristors into any neural circuit can enhance its potential controllability under external physical stimuli. Memristive current along a magnetic flux-controlled memristor can estimate the effect of electromagnetic induction on neural circuits and neurons. Here, a charge-controlled memristor is incorporated into one branch circuit of a simple neural circuit to estimate the effect of an external electric field. The field energy kept in each electric component is respectively calculated, and equivalent dimensionless energy function H is obtained to discern the firing mode dependence on the energy from capacitive, inductive, and memristive channels. The electric field energy HM in a memristive channel occupies the highest proportion of Hamilton energy H, and neurons can present chaotic/periodic firing modes because of large energy injection from an external electric field, while bursting and spiking behaviors emerge when magnetic field energy HL holds maximal proportion of Hamilton energy H. The memristive current is modified to control the firing modes in this memristive neuron accompanying with a parameter shift and shape deformation resulting from energy accommodation in the memristive channel. In the presence of noisy disturbance from an external electric field, stochastic resonance is induced in the memristive neuron. Exposed to stronger electromagnetic field, the memristive component can absorb more energy and behave as a signal source for energy shunting, and negative Hamilton energy is obtained for this neuron. The new memristive neuron model can address the main physical properties of biophysical neurons, and it can further be used to explore the collective behaviors and self-organization in networks under energy flow and noisy disturbance.

Dynamics analysis and Hamilton energy control of a class of Filippov neuron model

Electromagnetic induction plays a key role in regulating the electrical activity, excitability, and bistable structure of neurons. In this paper, a discontinuous control strategy with membrane potential as the threshold is introduced to the HR neuron model under the influence of electromagnetic field (EMFN model), and then we establish a Filipov EMFN neuron model to realize the regulation effect of electromagnetic field on the neuron system. Specific work of this paper reads as follows: first, the existence and stability of equilibrium points of two subsystems are analyzed by using MatCont software; second, the bistable region and its internal mechanism are discussed in detail by two-parameter bifurcation analysis; third, the mechanism of bistability and a series of complex sliding mode dynamics including sliding segment and sliding bifurcations are further revealed with the help of the fast–slow variable dissection method; finally, based on Hamilton energy, the regulation of threshold on chaotic discharge in the Filippov EMFN neuron model is explored. The numerical simulation results show that the EMFN neuron model can produce the corresponding sliding limit cycle and sliding bursting behaviors under the influence of the threshold control strategy, meanwhile, the chaotic behavior of the new system can be controlled effectively within a certain range. The results provide ideas for controlling the effects of electromagnetic fields on the biological neuronal system and exploring the regulation mechanism of neurological diseases.

Creation of heterogeneity or defects in a memristive neural network under energy flow

The field energy in neuron can be changed under shape deformation because of external energy injection into the media. The electromagnetic field superposition enables energy pumping when neurons in the same region are stimulated, and synaptic connection can be created adaptively for keeping local energy balance. All neurons will keep energy balance with the adjacent neurons in the network by continuous energy propagation and exchange, and identical neurons will develop a homogeneous state while non-identical neurons will support formation of gradient spatial patterns. In this paper, the simple Fitzhugh–Nagumo (FHN) neural circuit is improved by incorporating a thermistor and a phototube in different branch circuits synchronously, and this neuron becomes sensitive to light and temperature. The Hamilton energy for this functional neuron is obtained to discern the dependence of firing modes on the energy level. These functional neurons are clustered to build a regular network via memristive synapses, and the creation and growth of memristive synapse can be controlled by energy diversity between adjacent neurons. It is found that heterogeneity in the network can be formed when more energy is accumulated locally, while the occurrence of local defects results from energy release and lower energy in a local area. As a result, shape deformation occurs and some parameters for the neurons covered with heterogeneity or defects show certain changes with time. The collective behaviors of the network are investigated by calculating the membrane potentials for neurons, Hamilton energy, coupling intensity and synchronization factor. The results indicate that energy diversity between neurons controls the occurrence of heterogeneity and defects in the network and local energy injection can control the firing patterns of network because the wave propagation can be regulated by the heterogeneity and defects.

Formation of local heterogeneity under energy collection in neural networks

Static distribution of intracellular and extracellular ions can induce the spatial distribution of electric field in the cell, while stochastic diffusion and propagation of ions can generate channel current accompanied by the magnetic field. In fact, the field energy (including magnetic and electric fields) in each cell or neuron can be changed under external stimuli or the deformation of shape. Furthermore, energy pumping occurs, and the synaptic connection is created adaptively to keep energy balance when more neurons are clustered in the same region because of electromagnetic field superposition. For a synchronous and homogeneous network, all neurons keep energy balance with the adjacent neurons, and the network becomes uniform and identical. For cardiac tissue and biological media, local heterogeneity such as sinoatrial node can emit continuous wave front, and target waves are propagated to regulate the neural activities and heartbeat rhythm. In this paper, the Hamilton energy in a simple neuron model is calculated according to the Helmholtz theorem, and more neurons are clustered to build a regular network (chain network and square array). The creation and growth of synapse connection are controlled to pump energy, and local heterogeneity is formed by taming one intrinsic parameter when more energy is accumulated in a few neurons in the network. These results indicate that asymmetrical energy transport will induce the occurrence of heterogeneity in the network, and adaptive synaptic regulation on neurons is activated to keep local energy balance before reaching perfect synchronization. When external stimuli with diversity in intensity are applied, neurons are excited with energy diversity, and possible shape deformation is induced to keep local energy balance. As a result, heterogeneity is developed in a local area.

Dynamical analysis and boundedness for a generalized chaotic Lorenz model

<abstract><p>The dynamical behavior of a 5-dimensional Lorenz model (5DLM) is investigated. Bifurcation diagrams address the chaotic and periodic behaviors associated with the bifurcation parameter. The Hamilton energy and its dependence on the stability of the dynamical system are presented. The global exponential attractive set (GEAS) is estimated in different 3-dimensional projection planes. A more conservative bound for the system is determined, that can be applied in synchronization and chaos control of dynamical systems. Finally, the finite time synchronization of the 5DLM, indicating the role of the ultimate bound for each variable, is studied. Simulations illustrate the effectiveness of the achieved theoretical results.</p></abstract>

Hidden Dynamics, Multistability and Synchronization of a Memristive Hindmarsh–Rose Model

Reliable neuron models play an essential role in identifying the electrical activities, global bifurcation patterns, and dynamic mechanisms of neurons in electromagnetic environments. Considering that memristive autapse can characterize the self-induced effect of neurons, a five-dimensional Hindmarsh–Rose (HR) neuron model involving electric and magnetic fields is established. The detailed existence and stability analyses for equilibrium points are performed, and the complex time-varying stability, saddle-node bifurcation, and Hopf bifurcation behaviors are demonstrated. Interestingly, the bistable structures consisting of quiescent state and periodic bursting modes near the subcritical Hopf bifurcation and counterintuitive dynamic phenomena can be induced via appropriately adjusting the memristive current. Accordingly, the mechanism of positive feedback autaptic current decreases its firing frequency, while negative feedback autaptic current promotes its excitability and is revealed by the fast–slow dynamic analysis. Generally, the system possesses period-adding bifurcation patterns and comb-shaped chaotic structures as demonstrated by the numerical results. Importantly, it can be confirmed that the electrical activities and multistability of the system can be accurately predicted by analyzing the global dynamic behaviors of the Hamilton energy. Furthermore, it is verified that the unidirectional coupling controller involving energy is far more efficient and consumes lower energy than electrical synaptic coupling in achieving complete synchronization with mismatched parameters. These results provide potential guidance and help for further research in computational neuroscience and the design and control of intelligent sensors.

Energy computation and multistability in a class of second-order chaotic systems with simple nonlinearities: numerical, experimental and analytical results

The present work is aimed at the design, implementation and analysis of generic new second-order non-autonomous chaotic systems with simple nonlinear functions that have algebraically simple representations in mathematical form. These systems involve simple second-order non-autonomous ordinary differential equations with different simple nonlinearities like . The present study deals with numerical simulation, experimental analog circuit simulation results to demonstrate the ordered and chaotic phenomena being exhibited by these systems. Of most interest the striking phenomenon of multistability is uncovered for certain nonlinearities. Coexisting attractors are harnessed through the offset boosting approach. Also, the Hamilton energy is established through the Helmholtz theorem. It is found that both methods can be exploited as a non-bifurcation approach in characterizing multistability in the model. Further, analytical solutions developed for systems with certain nonlinearities are presented and the chaotic behavior of the systems studied using the solutions validating the numerical and experimental results are reported.

Analytical studies on complete, lag and anticipation synchronization in cascaded circuits with numerical and experimental confirmation

We report in this paper, the explicit analytical solutions developed for studying the different types of synchronization behaviors such as complete, lag and anticipation synchronization observed in cascaded circuits without any delay factors in the system. The Pecora–Carroll method of synchronization of unidirectionally coupled systems, which has not been earlier studied through explicit analytical solutions is reported in this paper. The emergence of different synchronization phenomena and the nature of evolution of the synchronization behavior has been discussed. Further, it has also been confirmed that the evolution of anticipation from lag through complete synchronization along with the attenuation, amplification of the subsystem signal occurs as a generalized phenomenon irrespective of the nature of the chaotic attractor state of the drive and the system parameters. The Hamilton energy of the cascaded system for different types of synchronization is reported. The numerical simulation and electronic circuit experimental results are presented to confirm the analytical results.

Hamilton energy balance and synchronization behaviors of two functional neurons.

The nervous system is composed of various functional neurons, some of which perceive sound or light, and these physical signals can be converted into bioelectrical signals. From the biophysical point of view, piezoelectric ceramic embedded in neuronal circuits can detect the external auditory waves, while phototube can capture light signals, so as to obtain two functional neurons with auditory recognition and light-dependent recognition. Considering the two identical or different functional neurons are connected by an induction coil to stimulate magnetic field coupling, and there will be energy diversity when they are driven by different initial conditions or external stimulation. Thus, synaptic connections can be activated and awakened in an adaptive manner when field energy is exchanged, and the coupling channel remains open until the energy diversity between neurons is controlled at a limited threshold. For this purpose, a criterion of the coupling strength increases exponentially is proposed to discuss the enhancement of neuronal synaptic connections. It is found that two neurons can be coupled adaptively to achieve complete synchronization, quasi-synchronization or intermittent quasi-synchronization. These results could help in designing functional assistive devices for patients with hearing or vision impairment.

The bounded sets, Hamilton energy, and competitive modes for the chaotic plasma system

The bounded sets, Hamilton energy, and competitive modes for the chaotic plasma system

Dynamic analysis of porosity dependent functionally graded sigmoid piezoelectric (FGSP) plate

This paper analyzes the vibration behaviour of the functionally graded sigmoid piezoelectric (FGSP) plate subjected to electrical and mechanical loading of a porous and non-porous plate. The sigmoid law is used for the variations of material properties along with the direction of thickness with different porosity types, i.e. even and uneven types. The plate incorporates the geometrical nonlinearity with von Kármán strains and First-order shear deformation theory (FSDT) based displacement field. For getting the nonlinear governing equation from Hamilton's principle or energy principle and higher-order numerical formulation with nine nodded isoperimetric elements and the six degrees of freedom per node. The perfect FGSP plate shows multi-periodic excitation with two strong attractors for SSSS boundary conditions, while quasi-periodic responses and 3-T periodic responses are observed for CCCC and CFFF boundary conditions. For evenly distributed porous SSSS FGSP square plate, the center displacement increases with an increase of the porosity from 0.2 to 0.4, while for un-evenly distributed porous center displacement reduces with a rise of the porous exponent from 0.2 to 0.4. The current analysis outcomes can be used for perfect and porous functionally graded piezoelectric smart structure applications under different electromechanical loading environments.

Energy harvesting from a cantilever beam with a spring-loaded oscillating mass system and base excitation

The present study deals with the mathematical modeling of a nonlinear energy harvester (EH) to analyze it's frequency bandwidth for optimal use. The exhibited unique piezoelectric energy harvester (PEH) contains a cantilever beam as resonator with a moving mass. To fulfill the mechanism, the mass needs to be connected to the fixed end of the beam with a spring that acts as a mechanical amplifier. A novel analytical approach with 1:2 internal resonance (IR) including kinematic nonlinearity is addressed for energy harvesting. The occurrence of strong nonlinear coupling is the result of constant interconnection between the beam and moving mass. A discrete electro-mechanical coupled equation is derived using Galerkin's method followed by Hamilton's energy method. To analyze the effect of input parameters on the frequency bandwidth, MATLAB code is improved by implementing the “Method of multiple scales” (MMS) for furthering the energy output of the system. Due to IR, the solution tilted to two branches in the designed EH, witnessing broader frequency bandwidth. The influence of different system parameters such as spring stiffness, mass, and velocity of moving mass changed the position of central frequency resulting in the change in symmetry of frequency response.

Free vibration analysis of functionally graded porous circular and annular plates using differential quadrature method

The research aims to find the fundamental frequencies of functionally graded (FG) porous circular and annular plates while accounting for three alternative porosity distributions in the thickness direction: uniform, O-shape, and X-shape. In the thickness direction, the material characteristics of FG plates are considered to vary constantly. The material characteristics are determined using Voigt's micromechanical model, which uses the power-law distribution approach with an arbitrary power index. The first-order shear deformation theory (FSDT) is used to derive the mathematical formulation of functionally graded plates. The equilibrium equation is determined using Hamilton's energy principle, and the problem is solved using the differential quadrature method (DQM). The established solution approach for non-dimensional frequencies of FG circular/annular plates is validated using convergence studies concerning the number of nodes. The FG circular/annular plate's non-dimensional frequency is determined and compared to current literature results. The natural frequency of FG porous circular/annular plates is studied in-depth in terms of thickness to radius ratio, material properties, porosity distribution, and boundary conditions.

Response mechanism in a functional neuron under multiple stimuli

Based on an auditory neuron model developed from the piezoelectric neural circuit, two different stimuli (F1, F2) are respectively used and then combined (F1+F2) synchronously to trigger different firing modes (S1, S2, S3). The Hamilton energy for each firing mode is calculated to discern the effect from the external stimulus, and the final firing mode is mainly controlled by one of the stimuli which can induce a higher Hamilton energy in the neuron. The external forcing inducing spiking has priority than the external stimulus inducing bursting to be responded by the neuron. The involvement of periodic firing modes is helpful to induce nonlinear resonance and thus new periodic firing modes can be generated even it presents lower Hamilton energy. The changes in the profile of periodic attractors confirm that external stimuli for inducing spiking and bursting can be perceived sensitively. In case of chaotic firing modes resulting from chaotic driving, the contribution from spiking, bursting and even periodic forcing can be detected because the profile of chaotic attractors is also changed greatly. A correlation function is defined to discern the effect of external stimuli on the selection of firing modes in the neuron driven by multi-channel stimuli. The results confirmed that the neuron is more sensitive to the external stimulus which can release higher Hamilton energy and/or induce high regularity in the neural activities. This study discovers the potential response mechanism in the auditory neuron under external stimuli.

Hierarchical Fuzzy Optimal Controller for Hamilton Energy of a Generalized Chaotic Lorenz System with Hidden Attractors

Hierarchical Fuzzy Optimal Controller for Hamilton Energy of a Generalized Chaotic Lorenz System with Hidden Attractors