Helmholtz Theorem Research Articles

The electromagnetic field of a circular parallel plate capacitor excited by an AC current and AC voltage supply

Abstract This study investigates the electromagnetic field (EMF) distribution of an ideal circular parallel plate capacitor excited by a time-harmonic power source. Considering the lead wire and capacitor as a charged whole, we formulate the boundary value problems of the Helmholtz equation for the EMF in the lead wire space and capacitor space, respectively. First, we solve for the EMF generated by the total current in the lead wire space of an AC current. Following this, we solve for the EMF boundary value problem of a capacitor filled with linear, uniform, isotropic, and non-magnetic lossy dielectric under an AC current excitation, using the continuity of the total current as a basis. Second, the EMF distributions in the capacitor space and the lead wire space under an AC voltage excitation are provided, following the principles of the generalized Helmholtz theorem. Third, the EMF distribution is discussed when the electromagnetic ``standing-wave phenomenon" occurs in the ideal dielectric capacitor, and identify the ``resonance phenomenon" and the ``current-stopping phenomenon" of the capacitor's EMF when excited by an AC current and AC voltage, respectively. We also present the corresponding ``resonance frequency" and ``current-stopping frequency". Finally, we explore the quasi-stable solution of the capacitor's EMF under low-frequency condition and the static solutions of the electric and magnetic fields under DC excitation and static voltage excitation. Our findings suggest that the existing formula for the capacitor's EMF approximates our analytical solution under quasi-stable condition.

Simulating the motion of a mechanical arm driven by neural circuit

A simple electromechanical model is constructed in this work to investigate the dynamical behavior of a mechanical arm driven by a light-sensitive neural circuit, simulating the operation of a micromechanical device implanted in the brain under neuronal operation. The physical equation describing the relationship between neural firing and mechanical motion is provided, the dimensionless model is obtained using the scale transformation, and the Hamilton energy of the electromechanical system is calculated based on Helmholtz theorem. It is found that the variation of photocurrent, magnetic field and damping coefficient causes the neuron and mechanical arm to interact with each other to generate a variety of oscillations when the phototube is used as a voltage/current source. Furthermore, it is demonstrated that the conversion of field energy and mechanical energy is another effective method for estimating and controlling the oscillation modes of the electromechanical model. The results explain how a micromassager connected to a neuron processes information from both environment and itself, and provide an insight for the design of implantable neurostimulators to restore brain or muscle function.

A memristive map neuron under noisy electric field

From a neural circuit to a biophysical neuron model, standard scale transformation requires the combination of physical variables including capacitance and resistance associated with a capacitor C and a resistor R, respectively. The voltage across the capacitor is used to mimic the membrane potential for neurons. Therefore, most of the neuron models contain one or two capacitive variables for the membrane potential. In fact, the reference time scale (RC) can be replaced by combinating parameters for inductor and resistor during approach of scale transformation for physical variables in the neural circuit. In this study, a simple nonlinear circuit is built by applying two inductors, a charge-controlled memristor (CCM), a nonlinear resistor and a resistor. The circuit equations are derived to develop a memristive oscillator with exact description of energy function. Furthermore, an equiavlent memristive map is obtained by applying linear transformation on the sampled variables from the memristive oscillator. The Hamilton energy function for memristive oscillator is calculated by using the Helmholtz's theorem, and then the memristive map is given in energy function in discrete form. The memristive map has rich dynamic characteristics and can identify coherence resonance under applying noisy electric field. This scheme will provide a theoretical guidance for building nonlinear circuits and maps without using capacitors.

Dynamical Analysis of A New Memristive Map with Offset Boosting and Extreme Multistability

Abstract In this paper, a chaotic circuit is designed based on a charge-controlled memristor, and the Hamilton energy function is obtained from the Helmholtz theorem. The system equation was discretized by redefining the variables to obtain a three-dimensional memristive map and the corresponding energy function. Then, the effects of parameters and initial values on the memristive map are analyzed using conventional dynamical analysis. There are multiple types of quasi-periodic and chaotic states of the memristive map under different parameters, and the energy evolution of different states is shown. In addition, offset boosting and homogeneous extreme multistability are found in the map, which can be controlled to offset the attractor by changing specific parameters and initial values. The complexity of the chaotic sequence is also analyzed as the parameters and initial values are varied over the chosen range. Finally, the map is implemented on PSIM circuit simulation and DSP digital circuit respectively. This research will provide a reference for secure communication.

On the direct quantization of Maxwell field

In this paper, we apply the generalized integration constants (GCI) method (Belhadi 2023 https://arxiv.org/abs/2303.08236), in field theory to quantize Maxwell and the Klein–Gordon free fields. The study is performed in both position and momentum spaces, to obtain the equal-time Dirac brackets among the fields and their conjugate momenta. The idea is to compute the brackets near the initial instant using the Taylor polynomial expansion, and then deduce directly their expressions at any later time. In the case of the Maxwell field, the interdependence of the field components (constraints) requires the use of the Helmholtz theorem to separate the transversal and longitudinal parts. Our work finishes with the study of the O(3) nonlinear sigma model using the GCI approach.

Boundary analysis and energy feedback control of fractional-order extended Malkus–Robbins dynamo system

In this paper, the fractional-order extended Malkus-Robbins dynamo system (FOEMRD) is introduced. Additionally, the stability, complexity diagrams, Lyapunov dimension, Lyapunov exponents, and bifurcation diagrams of the system are studied. Also, we estimate the global Mittag-Leffler positive invariant sets and global Mittag-Leffler attractive sets of the proposed system. In addition, the Hamilton energy function of the FOEMRD system is computed by the Helmholtz theorem. Next, we design an energy feedback controller to control the chaos. Due to the relationship of Hamilton energy with the occurrence of chaos, the energy control method is very efficient and powerful. The applicability of the presented theories was proven with numerical simulations and related diagrams.

Interaction of stream-wise vortices generated by swirler grid

A system of stream-wise vortices has been created using a grid of swirling elements with alternating orientations (like a chessboard). The particle image velocimetry method has been used to map the velocity field in several planes perpendicular to the stream. The mesh-based Reynolds number is 1.35×104 and 2.71×104, respectively. The stream-wise development of turbulent kinetic energy (TKE) shows first an increase in a distance of x≈10M, followed by power-law decay. Individual vortices are detected in each snapshot. The radial profile of TKE transformed to a vortex coordinate system is almost constant, either with maximum as in static frame or zero as observed by previous research. The properties of detected vortices are studied statistically: the meandering amplitude expressed as the standard deviation of vortex positions grows roughly as ∼ex, i.e., faster than expected random-walk growth ∼x. Vortex circulation decays exponentially as predicted by classical Helmholtz theorem. The interaction between neighboring vortices is expressed via correlation of selected quantities. Correlation of energy develops downstream from anticorrelation to a positive correlation. The strongest correlation is observed between the first vortex circulation and the second vortex position perpendicular to their connection line. Other correlations are weak.

Multiple firing patterns, energy conversion and hardware implementation within Hindmarsh-Rose-improved neuron model

The transmission of information between neurons is accomplished in living organisms through synapses. The memristor is an electronic component that simulates the tunability of the strength of biological synaptic connections in artificial neural networks. This article constructs a novel type of locally active memristor and verifies by nonlinear theoretical analysis, locally active analysis and circuit simulation. The designed memristor is simulated as a biological autapse of Hindmarsh-Rose(HR) neuron to obtain the improved HR neuron model of memristive autapse, and the Hamilton energy is obtained according to Helmholtz theorem. By varying the external forcing current and the memristive autapse strength, this article analyses the changes of the Hamilton energy and explores its self-excited and hidden firing behavior. The analog circuit simulation and digital circuit implementation of the HR model confirm the consistency between the mathematical model and the actual behavior, which can advance the field of neuroscience and artificial intelligence.

Firing behaviors of memristor-based Rulkov neuron map using energy method

Neural firing behavior is essentially the energy transmission between neurons, which can represent the information process of nervous system. Consequently, it is crucial to delve deeply into the intricate firing activity and energy characteristics of neurons. The Helmholtz theorem offers an accurate energy description for continuous neurons. However, the energy of discrete neurons remains under-explored due to lacking an effective theoretical framework. In this paper, we introduce a novel Rulkov neuron map that incorporates a charge-controlled memristor for imitating the effect of electromagnetic radiation. The Hamilton energy method is utilized to investigate the firing behaviors of this neuron map. A pivotal step in our analysis is meticulously scaling the parameters and variables to transform the neuron map into a continuous oscillator, which is essential for obtaining its energy function. In the exploration of the Rulkov neuron map, a range of intricate firing behaviors, including mixed-mode firing, bursting firing and coexistence firing, are observed. Notably, the occurrence of coherence resonance is detected under noise excitation. These behaviors are characterized by the time evolution of membrane potential and Hamilton energy spectrum. Furthermore, FPGA-based experiments are conducted to confirm the numerical results.

Hamiltonian energy in a modified Hindmarsh-Rose model.

This paper investigates the Hamiltonian energy of a modified Hindmarsh-Rose (HR) model to observe its effect on short-term memory. A Hamiltonian energy function and its variable function are given in the reduced system with a single node according to Helmholtz's theorem. We consider the role of the coupling strength and the links between neurons in the pattern formation to show that the coupling and cooperative neurons are necessary for generating the fire or a clear short-term memory when all the neurons are in sync. Then, we consider the effect of the degree and external stimulus from other neurons on the emergence and disappearance of short-term memory, which illustrates that generating short-term memory requires much energy, and the coupling strength could further reduce energy consumption. Finally, the dynamical mechanisms of the generation of short-term memory are concluded.

An adaptive energy regulation in a memristive map linearized from a circuit with two memristive channels

Nonlinear circuits can show multistability when a magnetic flux-dependent memristor (MFDM) or a charge-sensitive memristor (CSM) is incorporated into a one branch circuit, which helps estimate magnetic or electric field effects. In this paper, two different kinds of memristors are incorporated into two branch circuits composed of a capacitor and a nonlinear resistor, thus a memristive circuit with double memristive channels is designed. The circuit equations are presented, and the dynamics in this oscillator with two memristive terms are discussed. Then, the memristive oscillator is converted into a memristive map by applying linear transformation on the sampled time series for the memristive oscillator. The Hamilton energy function for the memristive oscillator is obtained by using the Helmholtz theorem, and it can be mapped from the field energy of the memristive circuit. An energy function for the dual memristive map is suggested by imposing suitable weights on the discrete energy function. The dynamical behaviors of the new memristive map are investigated, and an adaptive law is proposed to regulate the firing mode in the memristive map. This work will provide a theoretical basis and experimental guidance for oscillator-to-map transformation and discrete map energy calculation.

Dynamics and synchronization of neural models with memristive membranes under energy coupling

Dynamical modeling of neural systems plays an important role in explaining and predicting some features of biophysical mechanisms. The electrophysiological environment inside and outside of the nerve cell is different. Due to the continuous and periodical properties of electromagnetic fields in the cell during its operation, electronic components involving two capacitors and a memristor are effective in mimicking these physical features. In this paper, a neural circuit is reconstructed by two capacitors connected by a memristor with periodical mem-conductance. It is found that the memristive neural circuit can present abundant firing patterns without stimulus. The Hamilton energy function is deduced using the Helmholtz theorem. Further, a neuronal network consisting of memristive neurons is proposed by introducing energy coupling. The controllability and flexibility of parameters give the model the ability to describe the dynamics and synchronization behavior of the system.

Energy function and complex dynamics from a jerk system

Jerk, as a mathematical concept, is used in mechanics to describe the rate of change of acceleration and plays a crucial role in the design of control systems for machines and vehicles. Therefore, it is important to master the various states and the energy released during the change of acceleration. This is why a new simple jerk function introduced afterward, energy released, is derived from a Hamilton function using the Helmholtz theorem. The condition of having a stable or unstable rate of change of acceleration is established using Hopf bifurcation theory. Some two-parameter stability charts are then computed for a suitable selection region of the study. Using some nonlinear analysis metrics, in the unstable region of the study, the occurrence of phenomena is found, such as reverse period doubling bifurcation, antimonotonicity, and hysteresis involving the coexistence of the states in the considered jerk system. An electronic circuit is built and used to implement the mathematical expression of the jerk equation and validate the result of the theoretical investigation.

Energy and synchronization between two neurons with nonlinear coupling.

Consensus and synchronous firing in neural activities are relative to the physical properties of synaptic connections. For coupled neural circuits, the physical properties of coupling channels control the synchronization stability, and transient period for keeping energy diversity. Linear variable coupling results from voltage coupling via linear resistor by consuming certain Joule heat, and electric synapse coupling between neurons derives from gap junction connection under special electrophysiological condition. In this work, a voltage-controlled electric component with quadratic relation in the i-v (current-voltage) is used to connect two neural circuits composed of two variables. The energy function obtained by using Helmholtz theorem is consistent with the Hamilton energy function converted from the field energy in the neural circuit. Chaotic signals are encoded to approach a mixed signal within certain frequency band, and then its amplitude is adjusted to excite the neuron for detecting possible occurrence of nonlinear resonance. External stimuli are changed to trigger different firing modes, and nonlinear coupling activates changeable coupling intensity. It is confirmed that nonlinear coupling behaves functional regulation as hybrid synapse, and the synchronization transition between neurons can be controlled for reaching possible energy balance. The nonlinear coupling is helpful to keep energy diversity and prevent synchronous bursting because positive and negative feedback is switched with time. As a result, complete synchronization is suppressed and phase lock is controlled between neurons with energy diversity.

On the Physical Significance of the Helmholtz Theorem in a Static Electromagnetic Field

On the Physical Significance of the Helmholtz Theorem in a Static Electromagnetic Field

A neuron model with nonlinear membranes.

One-layer membrane separates the gradient field in and out of the cell, while some two-layer membranes filled with excitable media/material are important to regulate the energy flow when ions are propagated and diffused. The intracellular and extracellular media can be effectively separated by the membrane. It is important to clarify and describe the biophysical function and then the capacitive property can be reproduced in equivalent neural circuit. Here, we suggest the cell membrane has certain thickness and becomes flexible under external stimuli, therefore, it is considered as a kind of nonlinear media. To mimic the physical property of the two-layer cell membrane, a nonlinear resistor is used to connect two linear circuits, which is used to describe the electrical characteristic of two sides of the cell membrane, respectively. The combination of two linear circuits via a nonlinear resistor can describe the energy characteristic and firing mode in the flexible membrane of biophysical neurons. Circuit equations are defined and converted into equivalent nonlinear oscillator like a neuron. The voltage difference for the two capacitors can be consistent with the membrane potential for the neuron. The Hamilton energy function for this neuron can be mapped from the field energy in the electronic components, and it is also derived by using Helmholtz's theorem. The neuron can show similar spiking and bursting firing patterns, and uncertain diversity in membrane potentials is effective to support continuous firing patterns and mode transition under external stimulus. Furthermore, noisy disturbance is applied to induce coherence resonance. The results indicate that the lower coefficient variability and higher average energy level supports periodic firing in the neuron under coherence resonance. Therefore, this neuron model with nonlinear membranes (or two-layer form) is more suitable for identifying the biophysical property of biological neuron.

Initial condition-offset regulating synchronous dynamics and energy diversity in a memristor-coupled network of memristive HR neurons

Memristors can be thought of as special connection synapses with internal states that regulate synchronous behavior or energy propagation between neurons in a neural network. By connecting two memristive Hindmarsh-Rose (mHR) neurons with one memristor coupling, this paper synthesizes a memristor-coupled mHR neuron (MC-mHRN) network without equilibrium point. For this network, the synchronization condition regulated by the initial condition-offset (ICO) of the memristive channel are derived theoretically, and the ICO-regulating synchronous dynamics is evaluated numerically. Furthermore, the Hamiltonian energy of mHR neuron is studied by Helmholtz's theorem, and the ICO-regulating energy diversity is analyzed numerically. It is demonstrated that the coupling strength and initial condition of the memristive channel can regulate the synchronous dynamics and energy diversity in the MC-mHRN network, thus achieving complete synchronization as well as energy balance. Consequently, the two mHR neurons in hidden chaotic firing modes deliver field energy via memristive channel until the fully synchronized energy balance is achieved. In addition, a digital platform of the MC-mHRN network is fabricated, and the experimental results validate the numerical ones of the synchronous dynamics.

The global dynamics of a new fractional-order chaotic system

This paper investigates the global dynamics of a new 3-dimensional fractional-order (FO) system that presents just cross-product nonlinearities. Firstly, the FO forced Lorenz-84 system is introduced and the stability of its equilibrium points, as well as the chaos control for their stabilization, are addressed. Secondly, dynamical behavior is further analyzed and bifurcation diagrams, phase portraits, and largest Lyapunov exponent (LE) are discussed. Then, the global Mittag-Leffler attractive sets (MLASs) and Mittag-Leffler positive invariant sets (MLPISs) of the FO forced Lorenz-84 system are presented. Finally, the Hamilton energy function (HEF) of the Lorenz-84 system is calculated by using the Helmholtz theorem. The calculation of the Hamilton energy has an essential role on the estimation of chaos in dynamical systems, the guidance of orbits, and stability. In fact, any control action on the dynamical system completely changes the HEF. Numerical simulations are presented for illustrating the theoretical findings.

Consistency between modulational instability and energy localization in time-delay-memristive neural network

In this letter, we exclusively study the consistency between modulational instability and energy localization in a Hindmarsh-Rose–type memristive neural network under time-delay effect. Analytically, we find the suitable values of time-delay and memristor strength that support and control modulational instability. Also the Hamiltonian energy of the system is computed using the Helmholtz theorem. Numerical simulations are done to verify the analytical predictions. As a result, emission of action potential arises only in the domain where energy is highly localized, thus attesting that analytical predictions are in perfect agreement with numerical findings. These findings give more insight on relationship between modulational instability and energy localization in an excitable media along with the underlying biological phenomena that are involved.

Complex dynamics, released energy, and multistability in a single-layered graphene sheet under periodic loads

Graphene was found in 2004. It’s a carbon allotrope built around a single layer of atoms organized in a two-dimensional (2D) honeycomb lattice nanostructure. In the contribution, the complex behavior of a graphene sheet under periodic loads is investigated using the usual nonlinear analysis tools. From the dynamic states and the material’s symmetry investigations, it is found that the graphene model considered in this work can exhibit the simultaneous existence of up to six different patterns by varying initial conditions for fixed parameters. The Helmholtz theorem is also used to calculate the Hamilton energy of excited graphene. As a result, when the damping coefficient and the nonlocal parameter were close to zero, the system’s energy was null. This supports the fact that when graphene is in its resting state, vibrations are null inside the material. In contrast, when the damping coefficient and the nonlocal parameter were simultaneously increased, the energy of the graphene sheet increased until it reached a peak value, justifying the oscillations observed in the material. The dependency of the material on the initial condition is also used to support the coexistence of the energy found.