Enard System Research Articles

On the Number of Hyperelliptic Limit Cycles of Liénard Systems

In this paper, we study the maximum number, denoted by $H(m,n)$, of hyperelliptic limit cycles of the Li\'enard systems $$\dot x=y, \qquad \dot y=-f_m(x)y-g_n(x),$$ where, respectively, $f_m(x)$ and $g_n(x)$ are real polynomials of degree $m$ and $n$, $g_n(0)=0$. The main results of the paper are as follows: In term of $m$ and $n$ of the system, we obtain the upper bound and lower bound of $H(m,n)$ in all the possible cases. Furthermore, these upper bound can be reached in some cases.

Reduction of Kinetic Equations to Liénard–Levinson–Smith Form: Counting Limit Cycles

We have presented an unified scheme to express a class of system of equations in two variables into a Li\'enard-Levinson-Smith (LLS) oscillator form. We have derived the condition for limit cycle with special reference to Rayleigh and Li\'enard systems for arbitrary polynomial functions of damping and restoring force. Krylov-Boguliubov (K-B) method is implemented to determine the maximum number of limit cycles admissible for a LLS oscillator atleast in the weak damping limit. Scheme is illustrated by a number of model systems with single cycle as well as the multiple cycle cases.

Influence of time-delay feedback on extreme events in a forced Liénard system

A periodically forced Li\'enard system is capable of generating frequent large-amplitude chaotic bursts for a range of system and external forcing parameter values which are known as mixed mode oscillations. Particularly, if these large chaotic bursts occur infrequently and randomly, then they are characterized as extreme events. We present a numerical study of the effect of self-time-delay feedback on these extreme events in this system and interestingly find that extreme events can be completely eliminated from the system dynamics, even for smaller values of delay feedback strength. We show here that the number of extreme events is reduced, and the probability of the occurrence of high-amplitude events is transformed from a long-tailed statistics to the localized structure, as a function of the feedback strength corroborates our results. Further, we show that the autonomous Li\'enard system loses its conservative nature when delay feedback is added and only a dissipative nature remains in the entire phase space, which is the underlying mechanism behind the elimination of such large events. Further, we have revealed a type of delay-induced damping behavior, named anomalous damping, in which the amplitude of the oscillations suddenly vanishes when the total energy of the system becomes zero.

Basal melting driven by turbulent thermal convection

Melting and, conversely, solidification processes in the presence of convection are key to many geophysical problems. An essential question related to these phenomena concerns the estimation of the (time-evolving) melting rate, which is tightly connected to the turbulent convective dynamics in the bulk of the melt fluid and the heat transfer at the liquid-solid interface. In this work, we consider a convective-melting model, constructed as a generalization of the Rayleigh-B\'enard system, accounting for the basal melting of a solid. As the change of phase proceeds, a fluid layer grows at the heated bottom of the system and eventually reaches a turbulent convection state. By means of extensive Lattice-Boltzmann numerical simulations employing an enthalpy formulation of the governing equations, we explore the model dynamics in two and three-dimensional configurations. The focus of the analysis is on the scaling of global quantities like the heat flux and the kinetic energy with the Rayleigh number, as well as on the interface morphology and the effects of space dimensionality. Independently of dimensionality, we find that the convective-melting system behavior shares strong resemblances with that of the Rayleigh-B\'enard one, and that the heat flux is only weakly enhanced with respect to that case. Such similarities are understood, at least to some extent, considering the resulting slow motion of the melting front (with respect to the turbulent fluid velocity fluctuations) and its generally little roughness (compared to the height of the fluid layer). Varying the Stefan number, accounting for the thermodynamical properties of the material, also seems to have only a mild effect, which implies the possibility to extrapolate results in numerically delicate low-Stefan setups from more convenient high-Stefan ones.

Conservative perturbation theory for nonconservative systems.

In this paper, we show how to use canonical perturbation theory for dissipative dynamical systems capable of showing limit-cycle oscillations. Thus, our work surmounts the hitherto perceived barrier for canonical perturbation theory that it can be applied only to a class of conservative systems, viz., Hamiltonian systems. In the process, we also find Hamiltonian structure for an important subset of Liénard system-a paradigmatic system for modeling isolated and asymptotic oscillatory state. We discuss the possibility of extending our method to encompass an even wider range of nonconservative systems.

Effect of shear and magnetic field on the heat-transfer efficiency of convection in rotating spherical shells

We study rotating thermal convection in spherical shells. We base our analysis on a set of about 450 direct numerical simulations of the (magneto)hydrodynamic equations under the Boussinesq approximation. The Ekman number ranges from $10^{-3}$ to $10^{-5}$. The supercriticality of the convection reaches about 1000 in some models. Four sets of simulations are considered: non-magnetic simulations and dynamo simulations with either free-slip or no-slip flow boundary conditions. The non-magnetic setup with free-slip boundaries generates the strongest zonal flows. Both non-magnetic simulations with no-slip flow boundary conditions and self-consistent dynamos with free-slip boundaries have drastically reduced zonal-flows. Suppression of shear leads to a substantial gain in heat-transfer efficiency, increasing by a factor of 3 in some cases. Such efficiency enhancement occurs as long as the convection is significantly influenced by rotation. At higher convective driving the heat-transfer efficiency tends towards that of the classical non-rotating Rayleigh-B\'enard system. Analysis of the latitudinal distribution of heat flow at the outer boundary reveals that the shear is most effective at suppressing heat-transfer in the equatorial regions. Furthermore, we explore the influence of the magnetic field on the {\em non-zonal} flow components of the convection. For this we compare the heat-transfer efficiency of no-slip non-magnetic cases with that of the no-slip dynamo simulations. We find that at $E=10^{-5}$ magnetic field significantly affects the convection and a maximum gain of about 30\% (as compared to the non-magnetic case) in heat-transfer efficiency is obtained for an Elsasser number of about 3. Our analysis motivates us to speculate that convection in the polar regions in dynamos at $E=10^{-5}$ is probably in a `magnetostrophic' regime.

Nonlinear variations in axisymmetric accretion

We subject the stationary solutions of inviscid and axially symmetric rotational accretion to a time-dependent radial perturbation, which includes nonlinearity to any arbitrary order. Regardless of the order of nonlinearity, the equation of the perturbation bears a form that is similar to the metric equation of an analogue acoustic black hole. We bring out the time dependence of the perturbation in the form of a Li\'enard system, by requiring the perturbation to be a standing wave under the second order of nonlinearity. We perform a dynamical systems analysis of the Li\'enard system to reveal a saddle point in real time, whose implication is that instabilities will develop in the accreting system when the perturbation is extended into the nonlinear regime. We also model the perturbation as a high-frequency travelling wave, and carry out a Wentzel-Kramers-Brillouin analysis, treating nonlinearity iteratively as a very feeble effect. Under this approach both the amplitude and the energy flux of the perturbation exhibit growth, with the acoustic horizon segregating the regions of stability and instability.

Implications of nonlinearity for spherically symmetric accretion

We subject the steady solutions of a spherically symmetric accretion flow to a time-dependent radial perturbation. The equation of the perturbation includes nonlinearity up to any arbitrary order, and bears a form that is very similar to the metric equation of an analogue acoustic black hole. Casting the perturbation as a standing wave on subsonic solutions, and maintaining nonlinearity in it up to the second order, we get the time-dependence of the perturbation in the form of a Li\'enard system. A dynamical systems analysis of the Li\'enard system reveals a saddle point in real time, with the implication that instabilities will develop in the accreting system when the perturbation is extended into the nonlinear regime. The instability of initial subsonic states also adversely affects the temporal evolution of the flow towards a final and stable transonic state.

Stationary and oscillatory convection of binary fluids in a porous medium

We investigate numerically stationary convection and traveling wave structures of binary fluid mixtures with negative separation ratio in the Rayleigh-Bénard system filled with a porous medium. The bifurcation behavior of these roll structures is elucidated as well as the properties of the velocity, temperature, and concentration fields. Moreover, we discuss lateral averaged currents of temperature and concentration. Finally, we investigate the influence of the Lewis number, of the separation ratio, and of the normalized porosity on the bifurcation branches.

Pattern formation in spatially ramped Rayleigh-B'enard systems

We study pattern formation and selection in Rayleigh-Bsystems confined between well conducting horizontal boundaries and subjected to a weak horizontal gradient of the Rayleigh number. The study is based on the numerical integration of the Swift-Hohenberg equation and addresses the questions of the preferred orientation of the patterns with respect to the gradient of the Rayleigh number, boundary effects observed at subcritial sidewalls, the characteristics of long-term evolution of the patterns with emphasis on the wa- velength selection, and the effect of non-potential modifications of the Swift-Hohenberg equation. It is shown that, contrary to common belief, the rolls do not align with the direction of the horizontal temperature gradient (Dewel, 1989; Malomed, 1993), due to the influence of the walls. Rather, rolls approach a sidewall perpendicularly when the lo- cal bifurcation parameter is sufficiently beyond the threshold but tend to be parallel to a subcritical or critical sidewall. Simulati- ons performed with non-potential modifications of the Swift-Hohenberg equation lead in most cases, to asymptotic time-dependent behaviours.

A survey of isochronous centers

In this survey we give an overview of the results obtained in the study of isochronous centers of vector fields in the plane. This paper consists of two parts. In the first one (sections 2--8), we review some general techniques that proved to be useful in the study of isochronicity. In the second one (sections 9--16), we try to give a picture of the state of the art at the moment this review was written. In section 2, we give some basic definitions about centers, isochronous centers, first integrals, integrating factors, particular algebraic solutions, and other related concepts. In this sections we also give some general theorems about centers and isochronous centers, and we give a brief account of the evolution of the researches in this field. In the successive sections we focus on various methods that have been used in attacking the isochonicity problem. We start with linearizations in section 3, stating Poincar\'e's classical theorem and some of its consequences. Section 4 is devoted to describe the procedure that leads to define and compute isochronous constants. In section 5, commutators are introduced, and basic facts about couples of commuting systems are described. Classical theorems about systems obtained from complex ordinary differential equations are collected in section 6. Hamiltonian systems are considered in section 7, where their connection to the study of the Jacobian Conjecture is showed, too. Section 8 is concerned with systems having constant angular speed with respect to some coordinate system. The second part starts with section 9, that is devoted to recent results about second order differential equations not immediately reducible to hamiltonian systems. This section also contains the characterization of isochronous centers of reversible Li\'enard systems. In section 10 we list all fundamental results about isochronous centers of quadratic systems. Next section contains results about cubic systems with homogeneous nonlinearities. Sections 12 is devoted to cubic reversible systems. In section 13 we collect results about quartic and quintic systems with homogeneous nonlinearities. A class of particular cubic systems, with degenerate infinity is considered in section 14. Finally, section 15 is devoted to Kukles system. All the sections of the second part, and some of the first part, contain tables, where the main features of the considered systems are collected. When possible, for every class of systems we have written the system in rectangular and polar coordinates, and we have reported a first integral, a commutator, a linearization and a reciprocal integrating factor. The bibliography contains references both to papers devoted to the study of isochronicity and to papers concerned with integrability of plane systems and the study of the period function of centers. We have tried to make the bibliography so complete as possible for what is concerned with isochronicity. We have made no effort to make it complete for papers about integrability and the study of the period function. We apologize for possible mistakes and encourage the readers to communicate us any corrections.

Limit cycles of polynomial Liénard systems

Recently [H. Giacomini and S. Neukirch, Phys. Rev. E 56, 3809 (1997)], an algorithm to obtain the number of limit cycles of Li\'enard systems has been proposed. The quoted paper also includes a method to approximate the eventual limit cycles and a conjecture on the behavior of the algorithm. The algorithm is reviewed and some examples, which show that the algorithm is really efficient, are given. However, these examples indicate that the aforementioned conjecture may have been incorrectly stated. A different conjecture is proposed and some open questions are formulated.

Time-dependent patterns in the two-layer Rayleigh-Bénard system

We report experiments on the two-layer Rayleigh-B\'enard system with two different pairs of fluids and with both rectangular and annular geometries. Theoretical studies have shown that in addition to two distinct stationary states, it is possible to have time-dependent patterns at or near convective onset. We have found a traveling wave pattern at the onset to convection using silicone oil over water. Standing and traveling wave states have been found slightly above the onset to convection in a system of silicone oil over Fluorinert™. The basic form of these time-dependent states is consistent with the predictions, while some of the quantitative properties of the states depend on the system geometry.

Improving a method for the study of limit cycles of the Liénard equation

In recent papers we have introduced a method for the study of limit cycles of the Li\'enard system, $\mathrm{x\ifmmode \dot{}\else \.{}\fi{}}=y\ensuremath{-}F(x)$, $\mathrm{y\ifmmode \dot{}\else \.{}\fi{}}=\ensuremath{-}x$, where $F(x)$ is an odd polynomial. The method gives a sequence of polynomials ${R}_{n}(x)$, whose roots are related to the number and location of the limit cycles, and a sequence of algebraic approximations to the bifurcation set of the system. In this paper, we present a variant of the method that gives very important qualitative and quantitative improvements.

Liénard systems, limit cycles, and Melnikov theory

Li\'enard systems constitute a general class of two-dimensional autonomous systems, among which the van der Pol equation is found. Recently Giacomini and Neukirch [Phys. Rev. E 57, 3809 (1997)] introduced a sequence of polynomials whose roots are related to the number and location of limit cycles of Li\'enard systems. We show that in the limit of these sequences, the same information is given by a polynomial which Melnikov theory associates with a given Li\'enard system, and discuss the relationship existing among them.

Increase in chaotic motions of atoms in a large-scale self-organized motion.

The macroscopic flow transition between heat conduction and convection in the two-dimensional Rayleigh-B\'enard system is simulated using the molecular-dynamics method. The heat conduction state, in which motions of atoms are random and a large-scale flow does not appear, is established when the temperature difference between the top and bottom walls is small. It is shown, when the temperature difference is large, that macroscopic convection rolls appear and the trajectories of atoms are along the rolls. The degree of the chaotic motions of atoms is observed in terms of the Lyapunov exponent. In the heat conduction state, the Lyapunov exponent is shown to increase with the average temperature and the temperature difference. It is found that the Lyapunov exponent depends not only on the temperatures but also on the macroscopic flow velocity when the flow field shows the convecting state. The chaotic motions of atoms are shown to increase in the large-scale self-organized motion. \textcopyright{} 1996 The American Physical Society.

Spatially quasiperiodic convection and temporal chaos in two-layer thermocapillary instabilities.

This paper describes an amplitude equation analysis of the interactions between waves with wave number ${\mathit{k}}_{1}$ (and phase speed ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/${\mathit{k}}_{1}$) and stationary convection with wave number ${\mathit{k}}_{2}$. These two modes may bifurcate almost simultaneously from the conductive state of a two-layer B\'enard system, when the ratio of layer thicknesses is near a particular value (codimension-2 singularity). When ${\mathit{k}}_{2}$\ensuremath{\ne}2${\mathit{k}}_{1}$ (nonresonant case) and the first bifurcation occurs for steady convection, a secondary bifurcation to a spatially quasiperiodic and time-periodic mixed mode is obtained when increasing the driving gradient. No stable small-amplitude solution exists when the Hopf bifurcation is the first one. The occurrence of either of these two possibilities depends on the thickness ratio. When ${\mathit{k}}_{2}$=2${\mathit{k}}_{1}$ (resonant case), the system presents a much wider variety of dynamical behaviors, including quasiperiodic relaxation oscillations and temporal chaos. The discussion of the resonant system concentrates on a scenario of transition to chaos consisting of an infinite sequence of ``period-doubling'' homoclinic bifurcations of stable periodic orbits, for which the left-right symmetry of the convective system plays an essential role. For increasing constraint, a reverse cascade is observed, for which quadratic nonlinearities in the Ginzburg-Landau equations are shown to entirely determine the dynamics (cubic and higher-order terms may be neglected near the codimension-2 point). \textcopyright{} 1996 The American Physical Society.

Growth of long-range correlations in a transition between heat conduction and convection.

The transition between heat conduction and convection in the two-dimensional Rayleigh-B\'enard system is simulated using the direct simulation Monte Carlo method. Long-range correlations of temperature fluctuations are found to grow in the transition.

Transition to spiral-defect chaos in low Prandtl number convection.

We report a distinct transition at $\ensuremath{\epsilon}\ensuremath{\approx}0.55$ from a state with dislocation defects to spiral-defect chaos in a Rayleigh-B\'enard system of aspect ratio $\ensuremath{\Gamma}=40$. The transition occurred over a narrow range in $\ensuremath{\epsilon}$ and was seen in angle-time representations of azimuthal power spectra, in the spatially averaged roll curvature, in the skewness of the structure function, and in the roll-nucleation frequency of sidewall foci. Comparison with work by others shows that the onset of spiral-defect chaos moves to smaller $\ensuremath{\epsilon}$ as $\ensuremath{\Gamma}$ increases.

Taylor vortex flow under harmonic modulation of the driving force.

Time-periodic Taylor vortex flow between two concentric cylinders is studied. With the outer cylinder at rest and the inner one rotating with angular velocity \ensuremath{\Omega}(t)=\ensuremath{\Omega}(1+\ensuremath{\Delta} cos\ensuremath{\omega}t), the stability boundary of circular Couette flow and the fully nonlinear Taylor vortices are investigated. Results are obtained by a finite-difference numerical simulation of the full Navier-Stokes equations for a radius ratio \ensuremath{\eta}=0.65 and by an analytical Galerkin approximation with four modes for arbitrary gap width. Data obtained by both methods agree very well. Modulation is found to destabilize the basic flow in agreement with earlier theoretical results. The time dependence of Taylor vortices is determined in detail and compared with recent experiments. Their response to the modulated driving is elucidated and explained by investigating various limiting behaviors and by comparison with the amplitude equation. Subharmonic response is found for large modulation amplitudes when the driving \ensuremath{\Omega}(t) becomes supercritical in both rotation directions during one period. Differences and similarities of linear and nonlinear flow properties with modulated convection in Rayleigh-B\'enard systems are discussed in detail. That low-frequency small-amplitude modulation stabilizes the basic conductive state in the latter system while it destabilizes circular Couette flow is shown to be caused by the different coupling of the driving to the secondary flow field.