Small Amplitude Limit Research Articles

Maximal canards in a slow–fast Rosenzweig–MacArthur model with intraspecific competition among predators

Using geometric singular perturbation theory, this paper investigates the canard phenomenon of a slow–fast Rosenzweig–MacArthur model. The model incorporates intraspecific competition among predators, assuming predator reproduction occurs much slower than prey. We demonstrate the occurrence of maximal canards between attracting and repelling slow manifolds as a bifurcation parameter varies. Additionally, we derive an analytic expression to approximate the bifurcation parameter value at which a maximal canard occurs. The method employed for this analysis relies on the blowup method. This involves finding a quasihomogeneous blowup map to desingularize the nonnormally hyperbolic point. Subsequently, charts are utilized to express the blowup in local coordinates, calculate local data, investigate the dynamics of the blown-up vector fields, and establish connections across charts. Furthermore, we provide numerical simulations to illustrate the canard explosion phenomenon in the model. Through parameter variation, we observe that the model transitions from a small amplitude limit cycle to small amplitude canard cycles (canards without a head), then to large amplitude canard cycles (canards with a head), and finally to a large amplitude relaxation cycle.

Obliquely Propagating Self-Gravitational Shock Waves in Non-Relativistic Degenerate Quantum Plasmas

A rigorous theoretical investigation has been carried out on the propagation of non-linear self-gravitational shock waves (SGSHWs) in a magnetized super dense degenerate quantum plasma system (DQPS) composed of inertia-less non-relativistic degenerate electrons and inertial non-degenerate extremely heavy nuclei/element. The nonlinear propagation of these SGSHWs in the plasma system under consideration is studied by the standard reductive perturbation technique, which is valid for a small finite amplitude limit. The nonlinear dynamics of the SGSHWs are found to be governed by the Burgers equation. The Burgers equation is derived analytically and solved numerically. It has been found that the considered plasma model supports positive potential shock waves only. The fundamental properties (amplitude, steepness, etc.) of these SGSHWs are significantly modified by the variation of kinematic viscosity, obliqueness and number density of the plasma species. The results of our present investigation can be applied to astrophysical compact objects like neutron stars. Journal of Engineering Science 15(1), 2024, 21-29

A Polynomial System of Degree Four with an Invariant Square Containing At Least Five Limit Cycles

We consider a class of polynomial systems of degree four with four real invariant straight lines that form a square, called this an invariant square, and also that contains in its interior at least five small amplitude limit cycles for a certain choice of the parameters. Moreover, we will obtain the necessary and sufficient conditions for the critical point inside the square to be a center.

Limit cycle bifurcation from a zero-Hopf equilibrium for a class of 3-dimensional Kolmogorov systems

A zero-Hopf equilibrium point p of a 3-dimensional autonomous differential system in R3 is an equilibrium point such that the eigenvalues of the linear part of the system at p are 0 and ±ωi with ω≠0. A zero-Hopf bifurcation takes place when from a zero-Hopf equilibrium bifurcate some small-amplitude limit cycles moving the parameters of the system.Polynomial Kolmogorov systems are the natural extension of the polynomial Lotka–Volterra systems of degree 2 to higher degree. The Kolmogorov systems have been studied intensively due to their applications for modeling many natural phenomena.In this paper we characterize, up to first order in the averaging theory, the eight distinct zero-Hopf bifurcations which can exhibit the class of 3-dimensional Kolmogorov systems of degree 3, providing an explicit approximation of the bifurcated small-amplitude limit cycles, together with information about their kind of stability.From each one of this eight different zero-Hopf bifurcations emerges one or two limit cycle using the averaging theory of first order. Moreover we provide an explicit example of each one of these eight zero-Hopf bifurcations.

Modelling surface waves on shear current with quadratic depth-dependence

The currents in the ocean have a serious impact on ocean dynamics, since they affect the transport of mass and thus the distribution of salinity, nutrients and pollutants. In many physically important situations the current depends quadratic-ally on the depth. We consider a single layer of fluid and study the propagation of the surface waves in the presence of depth-dependent current with quadratic profile. We select the scale of parameters and quantities, which are typical for the Boussinesq propagation regime (long wave and small amplitude limit) and we also derive the well known KdV model for the surface waves interacting with current.

Hybrid finite-amplitude periodic modes for two uniformly magnetized spheres.

We analyze a system of two uniformly magnetized spheres, one fixed and the other free to slide in frictionless contact with the surface of the first. The centers of the two magnets, and their magnetic moments, are restricted to a plane. We search for sets of initial conditions that yield finite-amplitude oscillatory periodic solutions. We extend two small-amplitude base modes, one with orbital and spin motions that are in phase and the other out of phase, to finite amplitudes and show that the motion for arbitrary oscillatory solutions can be considered to be a nonlinear superposition of these base modes. Some solutions are pure periodic finite-amplitude extensions of one base mode, while others are hybrid finite-amplitude superpositions of the two modes. Hybrid modes with rational frequency ratios are periodic and come in families defined by their frequency ratios. We further characterize hybrid periodic modes by identifying two symmetry classes that describe their relative phases. We see continuous transitions between one finite-amplitude base mode and the other, with one mode gradually transforming into the other. We also calculate frequency spectra of nonperiodic modes, show that the two base modes have well-defined frequencies even for nonperiodic states, and show that periodic solutions can give clues about the behavior of nearby nonperiodic solutions. In the limit of small amplitudes, we confirm that the computed frequencies of these modes agree with small-amplitude analytical results. We also generate a Lyapunov exponent heatmap that reflects periodic and nonperiodic regions of state space.

Wronskian solutions, bilinear Bäcklund transformation, quasi-periodic waves and asymptotic behaviors for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

In this paper, we investigate the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which is widely used in fluid mechanics and theoretical physics. The N-soliton solution is obtained via the Hirota’s bilinear method. The Wronskian solution is derived by using the Wronskian technique for the bilinear form. Through the exchange formula, we deduce the bilinear Bäcklund transformation consisting of four equations and six parameters. In order to consider the quasi-periodic wave having complex structure, one-, two- and three-periodic waves are investigated systemically by combining the Hirota’s bilinear method with Riemann theta function. Furthermore, the corresponding graphs of periodic wave are presented by considering the geometric properties between the characteristic lines. The propagation characteristics of periodic waves are investigated by virtue of the characteristic lines. Finally, the asymptotic relationships between quasi-periodic wave solutions and soliton solutions are established theoretically under a condition of the small amplitude limit. The analytical method used in this paper can be applied in other integrable systems.

An upper bound for the number of small-amplitude limit cycles in non-smooth Liénard system

The non-smooth Liénard system, a well-known nonlinear model, appears in a natural way in physics, chemistry, biology, and so on, in which periodic phenomena play a relevant role. In this paper, we investigate the small-amplitude limit cycles generated by the Hopf bifurcation of the non-smooth Liénard system. By utilizing the Picard–Fuchs equation, we gain the upper bounds of the number of small-amplitude limit cycles for a generic non-smooth Liénard system. Finally, an example is given to illustrate the efficiency of the theoretical results.

Bifurcation of Limit Cycles from a Focus-Parabolic-Type Critical Point in Piecewise Smooth Cubic Systems

In this paper, we investigate the maximum number of small-amplitude limit cycles bifurcated from a planar piecewise smooth focus-parabolic type cubic system that has one switching line given by the x-axis. By applying the generalized polar coordinates to the parabolic subsystem and computing the Lyapunov constants, we obtain 11 weak center conditions and 9 weak focus conditions at (0,0). Under these conditions, we prove that a planar piecewise smooth cubic system with a focus-parabolic-type critical point can bifurcate at least nine limit cycles. So far, our result is a new lower bound of the cyclicity of the piecewise smooth focus-parabolic type cubic system.

A polynomial system of degree four with an invariant triangle containing at least four small amplitude limit cycles

In this work, the existence of a polynomial system of degree four with an invariant triangle containing at least four small-amplitude limit cycles is proved. This result improves the result obtained in [2].

Perturbation of a piecewise regular-singular Liénard system

In this paper we discuss the perturbation of a generalized Liénard system, defined in two regions. For x≤0 the system is regular, while for x > 0 the system is singularly perturbed. An essential difference with other papers is that a degenerate canard point appears which requires a blow-up analysis.We prove that under generic conditions the perturbed system has at most one small-amplitude limit cycle and at most N canard limit cycles, where N is the number of local extrema of the function F(x) in the singular part of the Liénard system. For critical values of the control parameters the mechanism of a canard explosion is discussed.As an application we prove that the upper bound for the number of limit cycles in a Gause predator-prey system with a general monotonic functional response function with cut-off is two times the number of maxima of the natural prey growth function after the cut-off.

Structure of the streaming flow generated by a sphere in a fluid undergoing rectilinear oscillation

A solid body in a viscous fluid undergoing oscillatory motion naturally produces a steady secondary flow due to convective inertia. This phenomenon is embodied in the streaming flow generated by a sphere in an unbounded fluid executing rectilinear oscillations. We review the considerable literature on this canonical problem and summarise exact and asymptotic formulas in the small-amplitude limit. These analytical formulas are used to explore the characteristic flow structure of this problem and clarify previously unreported features. A single, toroidal-shaped vortex exists in each hemisphere regardless of the oscillation frequency, which can drive a counter-flow away from the sphere. The vortex centre moves monotonically away from the sphere with decreasing oscillation frequency, and engulfs the entire flow domain for $\beta \equiv \omega R^2/\nu < 16.317$ , where $\omega$ is the angular oscillation frequency, $R$ the sphere radius, and $\nu$ the fluid kinematic viscosity. This seemingly abrupt change in flow structure at the critical frequency $\beta _{critical} =16.317$ , and its quantification, appear to have not been reported previously. We perform a direct numerical simulation of the Navier–Stokes equations, to (1) confirm existence of this critical frequency at finite amplitude, and (2) examine its variation with amplitude. This reveals a universal relationship between the critical frequency and oscillation amplitude, clarifying previous reports on the structure of this streaming flow. The critical frequency is shown to be identical for the streaming flow and the cycle-averaged particle paths, establishing that the critical frequency is accessible directly using standard measurements.

Quasi-periodic waves and asymptotic behavior for a coupled nonlinear Klein-Gordon equation

With the Hirota’s bilinear method, N N -periodic wave solutions to the coupled nonlinear Klein-Gordon equation are constructed in terms of theta function. The asymptotic behaviors under small amplitude limits are also deduced strictly for the one and two-periodic waves. Moreover, a numerical scheme is presented for the case of N ≥ 3 N\geq 3 . Some three-periodic wave solutions are calculated numerically as examples.

Nonlinear coherent structures of nucleus-acoustic wave excitations in multi-nuclei quantum plasmas with ultra-relativistically degenerate electrons and positrons

The propagation dynamics of the nucleus-acoustic waves (NAW) in a quantum plasma composed of nondegenerate inertial light nuclei, stationary heavy nuclei, and ultra-relativistically degenerate electrons and positrons has been theoretically investigated within the framework of the Boussinesq equation, which is valid for a bi-directional propagation of a small but finite amplitude limit. The N-soliton solution of the Boussinesq equation is derived using Hirota's method. It is found that positive potential structures exist in the sonic and supersonic regimes, whereas negative potential structures are found to be present in the subsonic regime. Pertinent plasma properties are analyzed for one-, two-, and three-soliton solutions in terms of different parameters. In addition to the typical solitary wave solutions, our findings indicate that the nonlinear NAW has breather structures. The three- and four-soliton solutions are used to construct the elastic interaction solutions of the breather–soliton and breather–breather, respectively. The findings are discussed in the context of ultra-relativistic astrophysical plasmas.

Bifurcation of Limit Cycles and Center in 3D Cubic Systems with Z3-Equivariant Symmetry

This paper focuses on investigating the bifurcation of limit cycles and centers within a specific class of three-dimensional cubic systems possessing Z3-equivariant symmetry. By calculating the singular point values of the systems, we obtain a necessary condition for a singular point to be a center. Subsequently, the Darboux integral method is employed to demonstrate that this condition is also sufficient. Additionally, we demonstrate that the system can bifurcate 15 small amplitude limit cycles with a distribution pattern of 5−5−5 originating from the singular points after proper perturbation. This finding represents a novel contribution to the understanding of the number of limit cycles present in three-dimensional cubic systems with Z3-equivariant symmetry.

Bifurcations driven by generalist and specialist predation: mathematical interpretation of Fennoscandia phenomenon.

In this paper, we revisit a predator-prey model with specialist and generalist predators proposed by Hanski et al. (J Anim Ecol 60:353-367, 1991) , where the density of generalist predators is assumed to be a constant. It is shown that the model admits a nilpotent cusp of codimension 4 or a nilpotent focus of codimension 3 for different parameter values. As the parameters vary, the model can undergo cusp type (or focus type) degenerate Bogdanov-Takens bifurcations of codimension 4 (or 3). Our results indicate that generalist predation can induce more complex dynamical behaviors and bifurcation phenomena, such as three small-amplitude limit cycles enclosing one equilibrium, one or two large-amplitude limit cycles enclosing one or three equilibria, three limit cycles appearing in a Hopf bifurcation of codimension 3 and dying in a homoclinic bifurcation of codimension 3. In addition, we show that generalist predation stabilizes the limit cycle driven by specialist predators to a stable equilibrium, which clearly explains the famous Fennoscandia phenomenon.

Magnetosonic waves propagation in a magnetorotating quantum plasma.

Employing the quantum magnetohydrodynamic (QMHD) model, the basic properties of magnetosonic waves were investigated in a magnetorotating quantum plasma. The contemplated system considered a combined effects of quantum tunneling and degeneracy forces, dissipation influence, spin magnetization, in addition to the Coriolis force. Fast and slow magnetosonic modes were obtained and examined in the linear regime. Their frequencies are significantly modified due to the rotating parameters (frequency and angle) in addition to quantum correction effects. The nonlinear Korteweg-de Vries-Burger equationwas derived using the reductive perturbation approach in a small amplitude limit. The aspects of magnetosonic shock profiles were explored analytically by applying the Bernoulli equationapproach and numerically using the Runge-Kutta method. The regarded plasma parameters due to the investigated effects were found to play major roles in specifying the nature of monotonic and oscillatory shock waves' structures and their features. Our results may be applicable in magnetorotating quantum plasma in astrophysical environments such as neutron stars and white dwarfs.

Oscillatory and transient dynamics of a slow–fast predator–prey system with fear and its carry-over effect

In almost every ecological system, growth of various interacting species evolve in different time scales and the implementation of this time scale difference in the corresponding mathematical model exhibits some rich and complex oscillatory dynamics. In this article, we consider a predator–prey model with Beddington–DeAngelis functional response in which the prey reproduction is affected by the predation induced fear and its carry-over effect. Considering the growth of prey species occurs on a faster time scale than that of predator, the proposed system reduces to a ‘slow–fast predator–prey’ system. Using the geometric singular perturbation theory and asymptotic expansion technique, we investigate the system both analytically and numerically, and observe a wide range of rich and complex dynamics such as canard cycles (with or without head) near the singular Hopf-bifurcation threshold and relaxation oscillation cycles. The system experiences a canard explosion through which a rapid transition from small amplitude limit cycle to large amplitude limit cycle occurs in a tiny parametric interval. These types of complex oscillatory dynamics are absent in non slow–fast systems. Furthermore, it is shown that the interplay between fear and its carry-over effect, and the variation of time scale parameter may lead to a regime shift of the oscillatory dynamics. We also study the impact of fear and its carry-over effect on the properties of long transient dynamics. Thus our study provides some valuable biological insights of a slow–fast predator–prey system which will aid in understanding the interplay between fear and its carry-over effect.

Integrability, breather, lump and quasi-periodic waves of non-autonomous Kadomtsev–Petviashvili equation based on Bell-polynomial approach

This article describes the infinite conservation law, quasi-periodic wave, breather, lump, and characteristic of integrability via bilinear Bäcklund and lax pairs of the non-autonomous Kadomtsev–Petviashvili (NKP) equation (in the presence of an external force and damping). Bell-polynomial is deduced from Bäcklund transformation from which infinite conservation law for NKP equation is derived. Obtaining well-known quasi-periodic solutions is an important aspect of this investigation. A limiting procedure is used for analyzing the asymptotic behavior of multi-periodic waves. This method proves rigorously that periodic waves tend toward soliton solutions under a small amplitude limit, which results in a relationship between periodic waves and soliton solutions. For the first time, the effect of external force in a quasi-periodic wave is demonstrated explicitly from the numerical understanding. Further, some complicated solution structures of the NKP equation such as lump and breather-type solitons are explored from the bilinear form of the said equation with the appropriate choice of polynomial functions. Important characteristics of lump and breather waves in different forcing backgrounds are graphically described. Finally, the lump wave of the NKP equation is explored from the breather wave in the limiting stage. It is also confirmed from the analytical results of the relevant motions that the velocity, maximum altitude, and interacting natures of the wave quantities are all influenced by the damping and forcing terms.

Global stability, sensitivity and passive control of low-Reynolds-number flows around NACA 4412 swept wings

The stability and sensitivity of two- and three-dimensional global modes developing on steady spanwise-homogeneous laminar separated flows around NACA 4412 swept wings are numerically investigated for different Reynolds numbers ${\textit {Re}}$ and angles of attack $\alpha$ . The wake dynamics is driven by the two-dimensional von Kármán mode whose emergence threshold in the $\alpha \unicode{x2013}{\textit {Re}}$ plane is computed with that of the three-dimensional centrifugal mode. At the critical Reynolds number, the Strouhal number, the streamwise wavenumber of the von Kármán mode and the spanwise wavenumber of the leading three-dimensional centrifugal mode scale as a power law of $\alpha$ . The introduction of a sweep angle attenuates the growth of all unstable modes and entails a Doppler effect in the leading modes’ dynamics and a shift towards non-zero frequencies of the three-dimensional centrifugal modes. These are found to be non-dispersive as opposed to the von Kármán modes. The sensitivity of the leading global modes is investigated in the vicinity of the critical conditions through adjoint-based methods. The growth-rate sensitivity map displays a region on the suction side of the wing, wherein a streamwise-oriented force has a net stabilising effect, comparable to what could have been obtained inside the recirculation bubble. In agreement with the predictions of the sensitivity analysis, a spanwise-homogeneous force suppresses the Hopf bifurcation and stabilises the entire branch of von Kármán modes. In the limit of small amplitudes, passive control via spanwise-wavy forcing produces a stabilising effect similar to that of a spanwise-homogeneous control and is more effective than localised spherical forces.