Small Amplitude Limit Research Articles

Hopf bifurcation in 3-dimensional polynomial vector fields

In this work we study the local cyclicity of some polynomial vector fields in R3. In particular, we give a quadratic system with 11 limit cycles, a cubic system with 31 limit cycles, a quartic system with 54 limit cycles, and a quintic system with 92 limit cycles. All limit cycles are small amplitude limit cycles and bifurcate from a Hopf type equilibrium. We introduce how to find Lyapunov constants in R3 for considering the usual degenerate Hopf bifurcation with a parallelization approach, which enables to prove our results for 4th and 5th degrees.

A Note on Small Amplitude Limit Cycles of Liénard Equations Theory

In this paper, we demonstrate using a counterexample for a theorem of the small amplitude limit cycles in some Liénard systems and show that that there will be no solutions unless we add an extra condition. A new condition is derived for some specific Liénard systems where a violation of the small amplitude limit cycles theorem takes place.

Center condition and bifurcation of limit cycles for quadratic switching systems with a nilpotent equilibrium point

In this work, a new perturbation approach is developed based on Bogdanov-Takens bifurcation theory, which enables the Poincaré-Lyapunov method for switching systems with linear type centers to be applied for studying the center conditions of planar switching polynomial systems associated with a nilpotent equilibrium point. The new method is then applied to consider a class of quadratic switching nilpotent systems, and a complete classification is given on the conditions of the nilpotent equilibrium point to be a center. Moreover, based on one of the center conditions, an example is constructed to show the existence of seven small-amplitude limit cycles around the nilpotent equilibrium point, which is a new lower bound on the number of limit cycles in such systems.

Maximum Number of Small Limit Cycles in Some Rational Liénard Systems with Cubic Restoring Terms

In this paper, we obtain an upper bound for the number of small-amplitude limit cycles produced by Hopf bifurcation in one particular type of rational Liénard systems in the form of [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomials in [Formula: see text] with degrees [Formula: see text] and [Formula: see text], respectively. Furthermore, we show that the upper bound presented here is sharp in the case of [Formula: see text].

Global bifurcation of solitary waves for the Whitham equation

The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, traveling-wave solution, and his conjecture was recently verified in the periodic case by Ehrnström and Wahlén. In the present paper we prove it for solitary waves. Like in the periodic case, the proof is based on global bifurcation theory but with several new challenges. In particular, the small-amplitude limit is singular and cannot be handled using regular bifurcation theory. Instead we use an approach based on a nonlocal version of the center manifold theorem. In the large-amplitude theory a new challenge is a possible loss of compactness, which we rule out using qualitative properties of the equation. The highest wave is found as a limit point of the global bifurcation curve.

Localized patterns in a generalized Swift–Hohenberg equation with a quartic marginal stability curve

Abstract In some pattern-forming systems, for some parameter values, patterns form with two wavelengths, while for other parameter values, there is only one wavelength. The transition between these can be organized by a codimension-three point at which the marginal stability curve has a quartic minimum. We develop a model equation to explore this situation, based on the Swift–Hohenberg equation; the model contains, amongst other things, snaking branches of patterns of one wavelength localized in a background of patterns of another wavelength. In the small-amplitude limit, the amplitude equation for the model is a generalized Ginzburg–Landau equation with fourth-order spatial derivatives, which can take the form of a complex Swift–Hohenberg equation with real coefficients. Localized solutions in this amplitude equation help interpret the localized patterns in the model. This work extends recent efforts to investigate snaking behaviour in pattern-forming systems where two different stable non-trivial patterns exist at the same parameter values.

Small amplitude ion-acoustic solitons with regularized κ-distributed electrons

A theoretical investigation of ion-acoustic solitons in an unmagnetized plasma consisting of cold ions and regularized κ-distributed electrons has been carried out. The properties of stationary solitary structures are briefly studied by the reductive perturbation method in the small amplitude limit. It is found that in the small κ region, the ion-acoustic solitary waves propagate more slowly in the case of regularized κ distribution than in the case of standard κ distribution. The regularized κ-distributed electrons affect the width and amplitude of the solitary waves. As the cutoff parameter α increases, the amplitudes of both the compressive and rarefactive solitary waves decrease, and their widths also decrease. In addition, for a given value of κ, positive potential solitons will appear at large α, while negative potential solitons will appear at small α. The results of this paper may be useful for understanding nonlinear electrostatic phenomena in space plasmas.

Amplitude-dependent boundary modes in topological mechanical lattices

Boundary modes localized at the edge or on the interface of topological mechanical lattices are analogous to their electronic counterparts in topological insulators and are robust and immune to structural imperfections. Most studies on the boundary modes in mechanical lattices are based on band theories, focusing on amplitude-independent behaviors at different frequency ranges. However, highly distorted topological mechanical lattices exhibit amplitude-dependent nonlinear effects that are difficult to characterize. In this study, a topological mechanical lattice exhibiting amplitude-dependent boundary modes is explored to uncover the evolution and connection between boundary modes in the linear and nonlinear regimes. Both discrete and continuum models are developed. An iterative method for the discrete model is proposed to numerically capture the boundary mode evolution while, based upon the continuum model, an analytical solution of the decay length scale of the boundary modes is obtained and can reduce to the linear band theory prediction at the small amplitude limit. A design strategy is presented for programmable topological polarization in the lattice that can be used to encode mechanical information. The obtained results provide insight into the modulation of topological protected mechanical response and stimulation of spatially ordered modes.

Effect of non adiabatic dust charge fluctuation on nonplanar dust acoustic waves in superthermal polarized plasma

Some properties of cylindrical/spherical dust acoustic solitary waves have been studied in collisionless dusty plasma with variable charges under the influence of polarization force by reductive perturbation method. It is shown that the collisionless damping due to dust charge fluctuation causes the dust acoustic wave propagation to be described by the nonplanar damped Korteweg de Vries equation. By use of weighted residual method, a progressive wave solution to nonplanar damped KdV equation is presented for the first time. It is shown that dust charge fluctuation produce damping which is responsible for wave attenuation. The solution shows that as one goes towards the axis of the cylinder or centre of the sphere, the wave attenuates faster and the effects are shown in some 2D figures. Within a small amplitude limit, the influence of dust charge fluctuation, polarisation power, superthermality and ion temperature on the time evolution of dust acoustic solitons is also demonstrated. In laboratory experiments and space regions, findings obtained here may shed light on the salient characteristics of DA waves.

Collisionless damping of nonplanar dust acoustic waves due to dust charge fluctuation in nonextensive polarized plasma

A rigorous theoretical investigation has been made of the cylindrical/spherical dust acoustic waves in collisionless dusty plasma with variable charges under the influence of polarization force by reductive perturbation method. It is shown that the collisionless damping due to dust charge fluctuation causes the dust acoustic wave propagation to be described by the nonplanar damped Korteweg–de Vries equation. By use of weighted residual method, a progressive wave solution to nonplanar damped KdV equation is presented for the first time. It is shown that dust charge fluctuation produce damping which is responsible for wave attenuation. The solution shows that as one goes towards the axis of the cylinder or centre of the sphere, the wave attenuates faster and the effects are shown in some 2D figures. Within a small amplitude limit, the influence of dust charge fluctuation, polarisation power, nonextensivity and ion temperature on the time evolution of dust acoustic solitons is also demonstrated. In laboratory experiments and space regions, findings obtained here may shed light on the salient characteristics of DA waves.

Analysis of electron acoustic waves interaction in the presence of homogeneous unmagnetized collision-free plasma

In this research, the transmission and interaction of nonlinear electron acoustic waves (EAWs) in such an unmagnetized, homogeneous, collision-free plasma composed of hot and cold electrons together with stationary ions throughout in the background have been analyzed. For the small-amplitude limit, the Korteweg–de Vries (KdV) equation for (EAWs) have been extracted. For electron acoustic solitary waves (EASWs), using the new extended direct algebraic approach, soliton solutions have also documented. The parametric analysis demonstrated that the hot to cold electron ratio and hot electron superthermal play a key role in changing the (EASWs) amplitude. The family of semi-bright solitons, dark singular solitons, Type 1 as well as 2 single solitons, trigonometric, intermingled hyperbolic and rational solitons was constructed and tested with the assistance of the innovative package software of numerical computations. The results show that the method is clear and efficient, produces analytical results in a generalized form, and these findings can also help resolve the difficulties and predicaments in the relevant disciplines of plasma physics and may be useful for studying the relationship between two (EASWs) in astrophysical and laboratory plasma. The solutions presented in this prototype are the latest in a literature review. For physical interpretation, some randomly selected solutions are shown graphically. Conclusions are held at the end.

Computational fluid-structure interaction of a restrained ogive-cylindrical body with a blunt elliptical base at a high incidence

The fluid-structure interaction of an elastically restrained inclined tangent ogive-cylindrical body with a blunt elliptical base is investigated numerically. The flow is three-dimensional, compressible, and laminar, and the slender body is allowed to yaw at high incidence. The resulting response exhibits an intricate bifurcation structure that includes bistable periodic (finite amplitude) and nonstationary (small amplitude) limit cycles for moderate angles of attack, and nonstationary (finite amplitude) oscillations for high angles of attack.

Modulated equations of Hamiltonian PDEs and dispersive shocks

Motivated by the ongoing study of dispersive shock waves in non integrable systems, we propose and analyze a set of wave parameters for periodic waves of a large class of Hamiltonian partial differential systems—including the generalised Korteweg–de Vries equations and the Euler–Korteweg systems—that are well-behaved in both the small amplitude and large wavelength limits. We use this parametrisation to determine fine asymptotic properties of the associated modulation systems, including detailed descriptions of eigenmodes. As a consequence, in the solitary wave limit we prove that modulational instability is decided by the sign of the second derivative—with respect to speed, fixing the endstate—of the Boussinesq moment of instability; and, in the harmonic limit, we identify an explicit modulational instability index of Benjamin–Feir type.

Delay dynamics of a levitating motor with two-limit control strategy

In a recent work (Shayak, 2019), I have proposed a new comparator-based control algorithm for a magnetically levitated motor. The rotor dynamics are governed by a sixth order nonlinear differential equation, whose stability analysis is treated as given. Here we consider this device from a dynamical systems viewpoint. We first present a simplified model which is a second order nonlinear delay differential equation. We then see that in this equation, a fixed point which is unstable in the absence of control gets converted to a small-amplitude stable limit cycle in its presence. Extrapolating from the simplified model, we find that insufficient damping can create an instability but it can be countered by increasing the inverter voltage, decreasing the inverter switching period, and relaxing the displacement tolerance value. Extensive simulation results show that all predictions made on the basis of the simplified model are applicable to the original system.

Eighteen limit cycles around two symmetric foci in a cubic planar switching polynomial system

In this paper, we present a cubic planar switching polynomial system with Z2-symmetry, and prove that such a system can exhibit at least 9 small-amplitude limit cycles around each of two symmetric foci, giving a total 18 limit cycles. This is a new lower bound for the number of limit cycles bifurcating in cubic switching polynomial systems around foci, simultaneously obtained around the same time when more limit cycles are achieved by perturbing a cubic switching integral system.

Effects of the ionic masses and positron density on the damped behavior in nonthermal collisional plasmas

Wave characteristics of solitons with damped structures in a four-component plasma fluid having positive and negative ions, nonthermal distributed positrons and electrons have been studied. The damped Kadomtsev–Petviashvili (DKP) equation has been obtained in a small amplitude limit. The nonlinear criticality of DKP is examined for related Earth’s ionosphere plasma parameters. The effects of the positron density ratio, the ionic mass ratio, the electron density ratio, the index of non-thermality and frequency parameters of collisions on the formation of both damped structures of compressive and rarefactive types are studied. It is a value mention that the results performed in this work may be used in the plasma of (D–F) Earth’s ionosphere regions.

Bäcklund transformations and Riemann–Bäcklund method to a (3 + 1)-dimensional generalized breaking soliton equation

In this paper, the bilinear method is employed to investigate the N-soliton solutions of a (3 + 1)-dimensional generalized breaking soliton equation. Three sets of bilinear Backlund transformations are obtained by means of gauge transformation. The Riemann–Backlund method is further extended to the (3 + 1)-dimensional nonlinear integrable systems. The quasiperiodic wave solutions of the (3 + 1)-dimensional generalized breaking soliton equation are systematically analyzed. The asymptotic properties of the quasiperiodic solutions are discussed by using a limiting procedure. The one-periodic and two-periodic waves tend to the 1-soliton and 2-soliton under a small amplitude limit, respectively. The dynamical characteristics of the one- and two-periodic waves are summarized by selecting different parameters. Furthermore, we obtain some new types of the quasiperiodic wave solutions of the variable coefficient (3 + 1)-dimensional generalized breaking soliton equation. These solutions present the dynamical behaviors of C-type, anti-C-type and Z-type periodic waves moving on the background of the periodic waves of bell type.

Ion‐acoustic solitary waves in e‐p‐i plasmas with (r, q)‐distributed electrons and kappa‐distributed positrons

AbstractIn this paper, we have studied the propagation of non‐linear ion‐acoustic waves in a plasma comprising of (r, q)‐distributed electrons and kappa‐distributed positrons. We have investigated the effect of complete electron distribution profile on the propagation of small, as well as arbitrary, amplitude solitons (via pseudopotential technique) by using generalized (r, q) distribution, which exhibits a spiky and flat top nature at low energies and a super‐thermal tail at high energies. Interestingly, for negative values of r, solitons are formed with both polarities, positive (compressive) and negative (rarefactive), separately within a small amplitude limit and exist simultaneously in an arbitrary amplitude limit. We also found that the propagation of solitons has been affected by the change in parameters r, q, positron concentration, and electron to positron temperature ratio. The results presented in this study add to the fundamental understanding of the complete profile of the electron distribution function, high‐ and low‐energy parts, and in the formation of compressive and rarefactive small and finite amplitude solitons in both space and astrophysical plasmas.

Self-Gravitational Shock Structures in Self-Gravitating, Super-Dense, Degenerate Quantum Plasmas

The nonlinear propagation of self-gravitational shock structures (SGSSs) in a self-gravitating, super-dense, degenerate quantum plasma system (containing non-degenerate extremely heavy nuclei and ultra-relativistic degenerate electrons) has been investigated. The well-known reductive perturbation technique, which is valid in the small but finite amplitude limit, has been used to examine the nonlinear propagation of these SGSSs in such degenerate quantum plasma systems. The nonlinear dynamics of these SGSSs has been found to be governed by the Burgers equation, which is derived analytically and solved numerically in planar coordinates. These SGSSs in such plasma systems are shown to be formed due to the presence of a viscous force (which is the source of dissipation) acting on inertial extremely heavy nuclei species of the plasma system. The fundamental properties (viz., amplitude, width, speed, etc.) of these SGSSs are influenced in the ultra-relativistic limit. Our considered plasma model and the numerical analysis of the Burgers equation can be applied to astrophysical compact objects like neutron stars.

Small Amplitude Limit Cycles and Local Bifurcation of Critical Periods for a Quartic Kolmogorov System

In this paper small amplitude limit cycles and the local bifurcation of critical periods for a quartic Kolmogrov system at the single positive equilibrium point (1, 1) are investigated. Firstly, through the computation of the singular point values, the conditions of the critical point (1, 1) to be a center and to be the highest degree fine singular point are derived respectively. Then, we prove that the maximum number of small amplitude limit cycles bifurcating from the equilibrium point (1, 1) is 7. Furthermore, through the computation of the period constants, the conditions of the critical point (1, 1) to be a weak center of finite order are obtained. Finally, we give respectively that the number of local critical periods bifurcating from the equilibrium point (1, 1) under the center conditions. It is the first example of a quartic Kolmogorov system with seven limit cycles and a quartic Kolmogorov system with four local critical periods created from a single positive equilibrium point.