Pendulum-like Equations Research Articles

On the number of limit cycles in piecewise smooth generalized Abel equations with two asymmetric zones

This paper devotes to the study of a kind of generalized Abel equation dxdθ=ap(θ)xp+aq(θ)xq,where p,q∈Z∖{1}, q−1p−1∉Z≤1, and ap, aq are piecewise trigonometrical polynomials of degree m with two zones 0≤θ<θ1 and θ1≤θ≤2π. We focus on the maximum number of positive and negative limit cycles (i.e., positive and negative isolated periodic solutions) that the equation can have, and the problem that how this maximum number, denoted by Hθ1(m), is affected by the location of the separation line θ=θ1. Then, by virtue of arbitrary higher order analysis using the theories of Melnikov functions and Chebyshev systems, we obtain lower bounds for Hθ1(m) that, Hθ1(m)≥2(3m+1) (resp. Hθ1(m)≥3m+1) when θ1∈0,π∪π,2π and either p or q is odd (resp. both p and q are even), and Hθ1(m)≥4m (resp. Hθ1(m)≥2m) when θ1=π and either p or q is odd (resp. both p and q are even). This result includes the estimates for not only the classical Abel equations but also some other equations from the real problems, such as the pendulum-like equations. In general, the asymmetry of the two zones increases the number of limit cycles in comparison with the case where the two zones are symmetric.

On the number of limit cycles of a pendulum-like equation with two switching lines

This paper is devoted to study the limit cycle bifurcations of a pendulum equation x˙=y,y˙=−sinx under non-smooth perturbations of polynomials of cosx, sinx and y of degree n with switching lines x=0 and y=0. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained by expressing the corresponding first order Melnikov functions as several generating functions, some of which are complete elliptic integrals of the first and second kind.

On the number of limit cycles for perturbed pendulum equations

We consider perturbed pendulum-like equations on the cylinder of the form x¨+sin⁡(x)=ε∑s=0mQn,s(x)x˙s where Qn,s are trigonometric polynomials of degree n, and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case ε=0 in terms of m and n. Our first result gives upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind. Some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be Chebyshev systems.

Dynamics of generalized PT-symmetric dimers with time-periodic gain–loss

A parity-time (PT)-symmetric system with periodically varying-in-time gain and loss modeled by two coupled Schrödinger equations (dimer) is studied. It is shown that the problem can be reduced to a perturbed pendulum-like equation. This is done by finding two constants of motion. Firstly, a generalized problem using Melnikov-type analysis and topological degree arguments is studied for showing the existence of periodic (libration), shift- periodic (rotation), and chaotic solutions. Then these general results are applied to the PT-symmetric dimer. It is interestingly shown that if a sufficient condition is satisfied, then rotation modes, which do not exist in the dimer with constant gain–loss, will persist. An approximate threshold for PT-broken phase corresponding to the disappearance of bounded solutions is also presented. Numerical study is presented accompanying the analytical results.

A Global Solution Curve for a Class of Free Boundary Value Problems Arising in Plasma Physics

We study the existence and multiplicity of solutions and the global solution curve of the following free boundary value problem, arising in plasma physics, see Temam (Arch Ration Mech Anal 60(1):51---73, 1975---1976), and Berestycki and Brezis (Nonlinear Anal. 4(3):415---436, 1980): find a function $$u(x)$$ u ( x ) and a constant $$b$$ b , satisfying $$\begin{aligned}&\Delta u+g(x,u)=p(x) \; \;\text{ in }\, D \\u \;\; \;\int \limits _{\partial D} \frac{\partial u}{\partial n} \, ds=0. \end{aligned}$$ Δ u + g ( x , u ) = p ( x ) in D u | ? D = b , ? ? D ? u ? n d s = 0 . Here $$D \subset R^n$$ D ? R n , is a bounded domain, with a smooth boundary. This problem can be seen as a PDE generalization of the periodic problem for one-dimensional pendulum-like equations. We use continuation techniques. Our approach is suitable for numerical computations.

On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian

In this paper we study the existence, multiplicity and stability of T-periodic solutions for the equation $\left(\phi(x')\right)'+c\, x'+g(x)=e(t)+s.$

Relation between single-particle kinetic energy and exchange energy in DFT for the inhomogeneous electron liquid in the Be atom

In a recent study, the authors have used the semi empirical fine-tuned Hartree–Fock ground-state electron density n(r) of Cordero et al. [Phys. Rev. A 75, 052502 (2007)] for the Be atom to calculate the phase θ(r) from a non-linear pendulum-like equation. Since the density amplitude n(r)1/2 plus θ(r) determine, in turn, the idempotent Dirac density matrix γ(r, r′), we use n(r) and θ(r) first of all to calculate the exchange energy density e X (r) of the density functional theory (DFT). This enables us to obtain the Slater (Sl) approximation to the exchange-only potential. A comparison can then be made, by integrating the earlier predicted exchange-correlation force −∂V XC (r)/∂r, of V XC (r) with . Relationship to the Becke semiempirical density gradient approximation for exchange is also established. Some brief discussion of the Perdew–Burke–Ernzerhof density functional is added.

Use of the differential virial theorem to estimate the spatial variation of the exchange-correlation force−∂VXC(r)∕∂rin the ground states of the spherical atoms He and Be

We use the differential virial theorem (DVT) directly to display the approximate spatial dependence of the exchange-correlation (XC) force in He and Be, applying an exact integral constraint on the XC force, recently established by March and Nagy. In He, an analytic ground-state density $n(r)$, combined with the DVT plus the von Weizs\"acker single-particle kinetic energy, suffices to determine an approximate XC force. For Be, the XC force is calculated for the semiempirical fine-tuned Hartree-Fock density, as proposed by Cordero et al. [Phys. Rev. A 75, 052502 (2007)]. However, for the single-particle kinetic energy, following Dawson and March, a phase $\ensuremath{\theta}(r)$ must be obtained by solving numerically a nonlinear pendulumlike equation.

A global solution curve for a class of periodic problems, including the pendulum equation

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for a class of periodically forced pendulum-like equations. Our results apply also to the first order equations. We also show that by choosing a forcing term, one can produce periodic solutions with any number of Fourier coefficients arbitrarily prescribed.

An nn-dimensional pendulum-like equation via topological methods

We study the elliptic boundary value problem Δu+g(u)=p(x) with nonlinear periodic-type boundary conditions. We prove the existence of a solution under a Landesman–Lazer-type condition. Moreover, using the method of upper and lower solutions we study a problem which concerns the range of the semilinear operator Su=Δu+g(u).

Using closed orbits to bifurcate many periodic solutions for pendulum-type equations

For pendulum-like equations, perturbation-type arguments and topological tools provide the existence of external forces with many associated periodic solutions.

A Generalization of the Lagrangian Points: Studies of Resonance for Highly Eccentric Orbits

We develop a framework based on energy kicks for the evolution of high-eccentricity long-period orbits in the circular planar restricted three-body problem with Jacobi constant close to 3 and with secondary-to-primary mass ratio μ 1. We use this framework to explore mean motion resonances between the test particle and the massive bodies. This approach leads to a redefinition of resonance orders for the high-eccentricity regime, in which a p : (p + q) resonance is called pth order instead of the usual qth order to reflect the importance of interactions at periapse. This approach also produces a pendulum-like equation describing the librations of resonance orbits about fixed points that correspond to periodic trajectories in the rotating frame. A striking analogy exists between these new fixed points and the Lagrangian points, as well as between librations around the fixed points and the well-known tadpole and horseshoe orbits; we call the new fixed points the generalized Lagrangian points. Finally, our approach gives a condition a ~ μ-2/5 for the onset of chaos at large semimajor axis a; a range μ <~ 5 × 10-6 in secondary mass for which a test particle initially close to the secondary cannot escape from the system, at least in the planar problem; and a simple explanation for the presence of asymmetric librations in exterior 1 : N resonances and the absence of these librations in other exterior resonances.

Convergent Solutions of Ordinary and Functional-Differential Pendulum-Like Equations

Sufficient frequency- domain conditions for complete stability of ordinary and functional-differential equations on the cylinder are given. For a class of phase-controlled systems, periodic Lyapunov functions and Popov functionals are constructed.

Multiple critical points of periodic functional and some applications

In this paper we will study some existence theorems of critical points for periodic functional. As applications, we consider the multiple solutions of conservative pendulum-like equation with periodic boundary conditions and forcing term of mean value zero.

A single-particle calculation of the FEL-Čerenkov gain

In this work we show that the basic FEL-Cerenkov dynamics can be modelled using a pendulumlike equation in close analogy with the FEL undulator case. The analysis, including the inhomogeneous broadening effects, is worked out in the hypothesis of single-slab geometry. Two-dimensional motion dynamics effects are also included.

Periodic solutions for systems of forced coupled pendulum-like equations

The existence of periodic solutions for systems of forced pendulum-like equations was studied in the papers by J. A. Marlin ( Internat. J. Nonlinear Mech. 3 (1968), 439–447) and J. Mawhin ( Internat. J. Nonlinear Mech. 5 (1970), 335–339). In both works some symmetry hypotheses on the forcing terms were considered. This paper discusses the existence and multiplicity of periodic solutions of systems under consideration without any requirement on the symmetry of the forcing terms. Note that as a model example it is possible to consider the motion of N coupled pendulums (see the already mentioned paper by J. A. Marlin) or the oscillations of an N-coupled point Josephson junction with external time-dependent disturbances studied in the autonomous case by M. Levi, F. C. Hoppensteadt, and W. L. Miranker ( Quart. Appl. Math. 36 (1978), 167–198).

On periodic solutions of forced pendulum-like equations

The existence and multiplicity of T-periodic solutions of the forced pen- dulum equation y”(t) + q’(t) + A sin J(t) = e(t) (1) have been considered recently by a number of authors and we refer to [2,3] for the history and a survey of this problem. The main techniques used up to now have been upper and lower solutions, topological degree and variational methods. A precise and explicit characterizarion of the set of e E