Small Amplitude Limit Research Articles

A study of fixed points and Hopf bifurcation of Fitzhugh-Nagumo model

In this article, a class of FitzHugh-Nagumo model is studied. First we are finding all necessary conditions for the parameters of system such that we have just one stable fixed point which presents the resting state for this famous model. After that, by using the Hopf’s theorem, we proof analytically the existence of a Hopf bifurcation that is a critical point where a system’s stability switches and a periodic solution arises. More precisely, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues cross the complex plane imaginary axis. With reasonably generic assumptions for the dynamical system, a small-amplitude limit cycle branches from the fixed point.

Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model

Applying the Lyapunov–Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift–Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction–diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg–Landau amplitude equations, rigorously validating the standard weakly unstable approximation and the Eckhaus criterion.

Nonlinear interaction between a two-level atom and a weak polychromatic field

The interaction between two-level media and a polychromatic field is considered in the limit of small field amplitudes. The obtained third-order corrections allow us to describe a nonlinear response to the field and intermodal interaction. The obtained polarization spectra are compared to numerical solutions to the density matrix equation in the rotating wave approximation. The results can be used in nonlinear comb spectroscopy.

Bifurcations of Multi-Vortex Configurations in Rotating Bose–Einstein Condensates

We analyze global bifurcations along the family of radially symmetric vortices in the Gross–Pitaevskii equation with a symmetric harmonic potential and a chemical potential µ under the steady rotation with frequency $${\Omega}$$ . The families are constructed in the small-amplitude limit when the chemical potential µ is close to an eigenvalue of the Schro dinger operator for a quantum harmonic oscillator. We show that for $${\Omega}$$ near 0, the Hessian operator at the radially symmetric vortex of charge $${m_0 \in \mathbb{N}}$$ has m 0(m 0+1)/2 pairs of negative eigenvalues. When the parameter $${\Omega}$$ is increased, 1+m 0(m 0-1)/2 global bifurcations happen. Each bifurcation results in the disappearance of a pair of negative eigenvalues in the Hessian operator at the radially symmetric vortex. The distributions of vortices in the bifurcating families are analyzed by using symmetries of the Gross–Pitaevskii equation and the zeros of Hermite–Gauss eigenfunctions. The vortex configurations that can be found in the bifurcating families are the asymmetric vortex (m 0 = 1), the asymmetric vortex pair (m 0 = 2), and the vortex polygons $${(m_0 \geq 2)}$$ .

Solitary structures in an inhomogeneous plasma with pseudo-potential approach

The set of nonlinear partial differential equations for the coupled ion acoustic and drift waves is reduced to the KdV equation, which is finally transformed into the form of energy integral equation of a pseudo particle in small amplitude limit. It is pointed out that this approach is convenient for choosing appropriate plasma parameters and numerically obtaining drift solitary wave profiles as compared to the solution of the KdV equation, particularly, in non-uniform plasmas. Electrons are assumed to follow the Kappa distribution function. It is found that the solitons amplitude decreases corresponding to steeper density and temperature gradients because of the restriction on the validity of local approximation. Deviation of electrons from thermal equilibrium distribution is supportive for the formation of electrostatic solitary structures by the coupled nonlinear ion acoustic and drift waves. The estimates of the width of the solitons formed by these coupled nonlinear electrostatic waves in the F-region ionosphere are found to be a few meters in agreement with the satellite observations.

Nucleus-acoustic envelope solitons and their modulational instability in a degenerate quantum plasma system

The existence and the basic features of nucleus-acoustic (NA) envelope bright and dark solitons in a degenerate quantum plasma system (DQPSs) (containing non-relativistically degenerate nuclei and inertialess ultra-relativistically degenerate electrons and positrons), have been theoretically investigated by deriving the nonlinear Schrödinger (NLS) equation. The reductive perturbation method, which is valid for a small but finite amplitude limit, is employed. The plasma parametric regimes for both modulationally stable and unstable NA waves (NAWs) are observed. It is found that the growth rate of the modulationally unstable NAWs is significantly modified by the number density of nucleus species. It is also observed that the modulationally stable (unstable) NAWs give rise to stable dark (bright) envelope solitons. The implications of our results obtained from our present investigation in astrophysical and laboratory DQPSs are briefly discussed.

Large-amplitude dust acoustic shocklets in non-Maxwellian dusty plasmas

The formation and propagation of fully nonlinear dust-acoustic (DA) waves and shocks are studied in a non-Maxwellian thermal dusty plasma which is composed of Maxwellian electrons and nonthermal energetic ions with a neutralizing background of negatively charged dust grains. For this purpose, we have solved dust dynamical equations along with quasineutrality equation by using a diagonalization matrix technique. A set of two characteristic wave equations is obtained, which admits both analytical and numerical solutions. Taylor expansion in the small-amplitude limit (Φ≪1) leads to nonlinear effective phase and shock speeds accounting for nonthermal energetic ions. It is numerically shown that DA pulses can be developed into DA shocklets involving the negative electrostatic potential, dust fluid velocity, and dust number density. These structures are significantly influenced by the ion-nonthermality, dust thermal correction, and temporal variations. However, the amplitudes of solitary and shock waves are found smaller in case of Cairns-distributed ions as compared to Kappa-distributed ions due to smaller linear and nonlinear effective phase speeds that cause smaller nonlinearity effects. The present results should be useful for understanding the nonlinear characteristics of large-amplitude DA excitations and nonstationary shocklets in a laboratory non-Maxwellian dusty plasma, where nonthermal energetic ions are present in addition to Maxwellian electrons.

On the uniqueness of the limit cycle for the Liénard equation with f(x) not sign-definite

The problem of uniqueness of limit cycles for the Liénard equation x ̈ + f ( x ) x ̇ + g ( x ) = 0 is investigated. The classical assumption of sign-definiteness of f ( x ) is relaxed. The effectiveness of our result as a perturbation technique is illustrated by some constructive examples of small amplitude limit cycles, coming from bifurcation theory.

An Oceanic Ultra-Violet Catastrophe, Wave-Particle Duality and a Strongly Nonlinear Concept for Geophysical Turbulence

There is no theoretical underpinning that successfully explains how turbulent mixing is fed by wave breaking associated with nonlinear wave-wave interactions in the background oceanic internal wavefield. We address this conundrum using one-dimensional ray tracing simulations to investigate interactions between high frequency internal waves and inertial oscillations in the extreme scale separated limit known as “Induced Diffusion”. Here, estimates of phase locking are used to define a resonant process (a resonant well) and a non-resonant process that results in stochastic jumps. The small amplitude limit consists of jumps that are small compared to the scale of the resonant well. The ray tracing simulations are used to estimate the first and second moments of a wave packet’s vertical wavenumber as it evolves from an initial condition. These moments are compared with predictions obtained from the diffusive approximation to a self-consistent kinetic equation derived in the ‘Direct Interaction Approximation’. Results indicate that the first and second moments of the two systems evolve in a nearly identical manner when the inertial field has amplitudes an order of magnitude smaller than oceanic values. At realistic (oceanic) amplitudes, though, the second moment estimated from the ray tracing simulations is inhibited. The transition is explained by the stochastic jumps obtaining the characteristic size of the resonant well. We interpret this transition as an adiabatic ‘saturation’ process which changes the nominal background wavefield from supporting no mixing to the point where that background wavefield defines the normalization for oceanic mixing models.

Liénard Equation and Its Generalizations

In this paper, we first present a survey of the known results on limit cycles and center conditions for Liénard differential systems. Next we propose a generalization of such systems and we study their center conditions and the number of small-amplitude limit cycles that can bifurcate from the origin. Computing the focal values and using Gröbner bases we find the center conditions for such systems up to a certain degree. We also establish a conjecture about the center conditions for such systems when they have arbitrary degree.

Soliton and quasi-periodic wave solutions for b-type Kadomtsev–Petviashvili equation

In this paper, truncated Laurent expansion is used to obtain the bilinear equation of a nonlinear evolution equation. As an application of Hirota’s method, multisoliton solutions are constructed from the bilinear equation. Extending the application of Hirota’s method and employing multidimensional Riemann theta function, one and two-periodic wave solutions are also obtained in a straightforward manner. The asymptotic behavior of one and two-periodic wave solutions under small amplitude limits is presented, and their relations with soliton solutions are also demonstrated.

Solitary kinetic Alfvén waves in nonextensive electron-positron-ion plasma

In low but finite β (thermal to magnetic pressure ratio) collisionless nonextensive electron-positron-ion plasma, we have derived linear dispersion relation of kinetic Alfvén waves (KAWs) and have also investigated solitary KAWs through Sagdeev pseudopotential approach in small amplitude limit. Linear analysis of KAWs shows a decrease in frequency of the wave with increase in each of the parameters, viz. r (equilibrium positron-to-ion density ratio), α (electron-to-positron temperature ratio) and q (nonextensive parameter). Nonlinear analysis of KAWs reveals that the nonextensive plasma model considered here supports only compressive solitary waves at sub-Alfvénic speeds. The effects of q, r, α and a (soliton parallel velocity) on the characteristic properties of solitary KAWs are depicted in graphical plots. The results of our present investigation may have relevance to the understanding of low-frequency electromagnetic noises/localized fluctuations in strongly magnetized electron-positron-ion plasmas having nonextensive electrons and positrons.

Self-gravitational perturbation in super dense degenerate quantum plasmas

Linear and nonlinear propagation of self-gravitational perturbation mode in a super dense degenerate quantum plasma (containing heavy nuclei/element and degenerate electrons) has been investigated. The linear dispersion relation for this mode (associated with self-gravitational potential) has been derived and analyzed. On the other hand, the nonlinear propagation of this mode in such a degenerate quantum plasma system is examined by the reductive perturbation method, which is valid for a small but finite amplitude limit. It has been found that the nonlinear dynamics of small but finite amplitude self-gravitational perturbation mode in spherical geometry is governed by the modified Korteweg-de Vries (mKdV) equation with negative dispersion coefficient, and the numerical solutions of this mK-dV equation have been analyzed to identify the basic features of spherical self-gravitational potential structures that may form in such a super dense degenerate quantum plasma system. The implications of our results in astrophysical compact objects like neutron stars are briefly discussed.

Bifurcation of limit cycles in a cubic-order planar system around a nilpotent critical point

In this paper, bifurcation of limit cycles is considered for planar cubic-order systems with an isolated nilpotent critical point. Normal form theory is applied to compute the generalized Lyapunov constants and to prove the existence of at least 9 small-amplitude limit cycles in the neighborhood of the nilpotent critical point. In addition, the method of double bifurcation of nilpotent focus is used to show that such systems can have 10 small-amplitude limit cycles near the nilpotent critical point. These are new lower bounds on the number of limit cycles in planar cubic-order systems near an isolated nilpotent critical point. Moreover, a set of center conditions is obtained for such cubic systems.

Instability Analysis of Positron-Acoustic Waves in a Magnetized Multi-Species Plasma

The nonlinear propagation of electrostatic excitations and their multi-dimensional instability in a magnetized, degenerate electron-positron-ion (EPI) plasma system (containing inertial cold positrons, relativistic degenerate electrons and hot positrons, and negatively charged immobile heavy ions) are theoretically investigated. The reductive perturbation method is employed to derive the Zakharov–Kuznetsov equation which admits a localized solitary wave solution for small but finite amplitude limit, and the multi-dimensional instability of the positron acoustic solitary waves (PASWs) is studied by the small-k perturbation expansion method. It is found that the basic characteristics (viz. phase speed, amplitude, width) of the PASWs are significantly affected by the degree of obliqueness, relativistic degeneracy, and plasma particle number densities. The instability criterion and its growth rate, which are depending on the magnetic field and the propagation directions of both the PASWs, and their perturbation modes are discussed. The present analysis can be helpful in understanding the nonlinear phenomenon in dense astrophysical as well as space plasma systems, especially in pulsar environments.

Relativistic quasi-solitons and embedded solitons with circular polarization in cold plasmas

The existence of localized electromagnetic structures is discussed in the framework of the 1-dimensional relativistic Maxwell-fluid model for a cold plasma with immobile ions. New partially localized solutions are found with a finite-difference algorithm designed to locate numerically exact solutions of the Maxwell-fluid system. These solutions are called quasi-solitons and consist of a localized electromagnetic wave trapped in a self-generated plasma density cavity with oscillations at its tails. They are organized in families characterized by the number of nodes p of the vector potential and exist in a continuous range of parameters in the plane, where V is the velocity of propagation and ω is the vector potential angular frequency. A parametric study shows that the familiar fully localized relativistic solitons are special members of the families of partially localized quasi-solitons. Soliton solution branches with p > 0 are therefore parametrically embedded in the continuum of quasi-solitons. On the other hand, geometric arguments and numerical simulations indicate that p = 0 solitons exist only in the limit of either small amplitude or vanishing velocity. Direct numerical simulations of the Maxwell-fluid model indicate that the p > 0 quasi-solitons (and embedded solitons) are unstable and lead to wake excitation, while p = 0 quasi-solitons appear stable. This helps explain the ubiquitous observation of structures that resemble p = 0 solitons in numerical simulations of laser-plasma interaction.

Zero-Hopf bifurcation in a Chua system

A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are ±ωi≠0 and 0. In general for a such equilibrium there is no theory for knowing when it bifurcates some small-amplitude limit cycles moving the parameters of the system. Here we study the zero-Hopf bifurcation using the averaging theory. We apply this theory to a Chua system depending on 6 parameters, but the way followed for studying the zero-Hopf bifurcation can be applied to any other differential system in dimension 3 or higher.In this paper first we show that there are three 4-parameter families of Chua systems exhibiting a zero-Hopf equilibrium. After, by using the averaging theory, we provide sufficient conditions for the bifurcation of limit cycles from these families of zero-Hopf equilibria. From one family we can prove that 1 limit cycle bifurcates, and from the other two families we can prove that 1, 2 or 3 limit cycles bifurcate simultaneously.

Symmetry breaking in an electrolytic cell under AC field and non-identical adsorbing electrodes

Broken symmetry of any kind in natural or synthetic systems is a burgeoning area of science. In this article, the temporal behavior of the bulk density, electric potential inside the sample, and the adsorbed ionic particles density in an electrolytic cell are shown to be non-symmetric and strongly affected by the surfaces dynamics. These parameters are analytically calculated by considering electrodes surfaces with non-identical adsorption-desorption phenomena. The system is submitted to an applied external voltage in the limit of small amplitude, where the time scales involved allow strong surface effects. The influence of the surfaces on the ionic distribution is performed in the framework of the Poisson-Nernst-Planck model by taking into account non-identical boundary conditions on the electrodes and on the electric potential, described by distinct kinetic equations. Such approach may be directly applied to explain strong surface dependence on the impedance spectroscopy, in hybrid aligned liquid crystal cells, fuel cells, biological and condensed matter systems in general.

Single electron transistor realised by suspended carbon nanotube

We have studied about single electron transistor realized by a suspended carbon nanotube. A detailed analysis of a single frequency resonance dip has shown that a broadening is obtained increasing the source drain voltage. Some of the observed effects in terms of a model in which the gate voltage acquires assigned time dependence. This phenomenological model, the back action of the nanotube motion on detected current was observed. Neglecting the dynamics of the resonator analysed the problem directly at mechanical resonance conditions only in the limit of small external antenna amplitudes in the linear response regime. We have observed that increasing the temperature the nonlinear effects in the current frequency response were washed out as a result of the increase of the intrinsic damping of the resonator and of the reduction of the intrinsic nonlinear terms of the effective self consistent force.

LIMIT CYCLE BIFURCATION FOR A NILPOTENT SYSTEM IN <i>Z</i><sub>3</sub>-EQUIVARIANT VECTOR FIELD

Our work is concerned with the problem on limit cycle bifurcation for a class of Z3-equivariant Lyapunov system of five degrees with three third-order nilpotent critical points which lie in a Z3-equivariant vector field. With the help of computer algebra system-MATHEMATICA, the first 5 quasiLyapunov constants are deduced. The fact of existing 12 small amplitude limit cycles created from the three third-order nilpotent critical points is also proved. Our proof is algebraic and symbolic, obtained result is new and interesting in terms of nilpotent critical points' Hilbert number in equivariant vector field.