BOUNDEDNESS OF SOLUTIONS FOR SUPERLINEAR REVERSIBLE SYSTEMS

51. IntroductionThe boundedness problem of solutions of the following nonlinear scalar differelltial equa-nons of second order..x f(x, t)ic g(x, t) = 0 (1'1)has been widely investigated by many authors since the 1940's, where f(x, t 1) = j(x, t),g(x, t 1) = g(x, t).When j(x, t) = 0, Equation (1.1) is a conservative system and takes the following form..x g(x, t) ~ 0. (1'2)The first contribution of the boundedness of all the solutions of Equation (1.2) is due toMorrislll, whO proved …     

A remark on stable subharmonic solutions of time-periodic reaction-diffusion equations

This paper is concerned with a time-periodic reaction-diffusion equation. It is known that typical trajectories approach periodic solutions with possibly longer period than that of the equation. Such solutions are called subharmonic solutions. In this paper, for any domain Ω, time-period τ>0 and integer n?2, we construct an example of a time-periodic reaction-diffusion equation on Ω with a minimal period τ which possesses a stable solution of minimal period nτ.     

Axiomatic Potential Theory,Asymptotic Values and Elliptic Differential Operators

In [14] Fernández, Heinonen and Llorente extend the Hornblower's results, about boundary behaviour of subharmonic functions in the unit disc of the complex plane, to subharmonic functions in the unit ball or the upper half space in higher dimensions. In this paper we establish that those results are also valid in the much more general setting of linear axiomatic potential theory. The interest of our general formulation relies on the applications to differential operators. We apply our result to Laplace–Beltrami operator and some uniformly elliptic second order operators in divergence form.     

The heat flow for subharmonic orbits

LetV: M be a smooth (potential) function on a compact Riemannian manifold. This gives rise to a second order Hamiltonian system. Assuming that the corresponding action functional is a Morse function, we will prove that the heat flow for subharmonics exists globally and converges to a critical point of the energy. As a Corollary, this shows the convergence of the geodesic heat flow (to a geodesic) without any curvature assumptions onM.     

Nonlinear vibration analysis by an extended averaged equation approach

The paper presents an extended averaged equation approach to the investigation of nonlinear vibration problems. The proposed method is applied to some free and self-excited oscillators, the Duffing's forced oscillators including main resonance, subharmonic resonance and super harmonic resonance. The results in analyzing the vibration systems with arbitrary non-linearity show advantages of the method.     

A Complement to the Valiron-Titchmarsh Theorem for Subharmonic Functions

The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in Rn, n ≥ 3. Here we extend the Drasin complement to the Valiron-Titchmarsh theorem and show that if u is a subharmonic function of this class and of order 0 ρ 1, then the existence of the limit limr →∞ logu(r)/N(r),where N(r) is the integrated counting function of the masses of u, implies the regular asymptotic behavior for both u and its associated measure.     

Characterising the cavitation activity generated by an ultrasonic horn at varying tip-vibration amplitudes

Dual-perspective high-speed imaging and acoustic detection is used to characterise cavitation activity at the tip of a commercial 20 kHz (f₀) ultrasonic horn, over 2 s sonications across the range of input powers available (20 – 100%). Imaging at 1 × 10⁵ frames per second (fps) captures cavitation-bubble cluster oscillation at the horn-tip for the duration of the sonication. Shadowgraphic imaging at 2 Mfps, from an orthogonal perspective, probes cluster collapse and shock wave generation at higher temporal resolution, facilitating direct correlation of features within the acoustic emission data generated by the bubble activity. f₀/m subharmonic collapses of the primary cavitation cluster directly beneath the tip, with m increasing through integer values at increasing input powers, are studied. Shock waves generated by periodic primary cluster collapses dominate the non-linear emissions of the cavitation noise spectra. Transitional input powers for which the value of m is indistinct, are identified. Overall shock wave content within the emission signals collected during sonications at transitional input powers are reduced, relative to input powers with distinct m. The findings are relevant for the optimisation of applications such as sonochemistry, known to be mediated by bubble collapse phenomena.     

Subharmonic solutions of Hamiltonian systems and the Maslov-type index theory

This paper deals with the subharmonic solutions of Hamiltonian systems

(H)