A new type of steady steep two-dimensional irrotational symmetric periodic gravity waves on inviscid incompressible fluid of infinite depth is revealed. We demonstrate that these waves have sharper crests in comparison with the Stokes waves of the same wavelength and steepness. The speed of a fluid particle at the crest of new waves is greater than their phase speed. [5,6], requires knowledge of a form and dynamics of steep water waves. For the first time surface waves of finite amplitude were considered by Stokes [7]. Stokes conjectured that such waves must have a maximal amplitude (the limiting wave) and suggested that a free surface of the limiting wave near the crest forms a sharp corner with the 120• internal angle (the Stokes corner flow). A strict mathematical proof of the existence of small amplitude Stokes waves was given by Nekrasov. Toland [8] proved that Nekrasov's equation has a limiting solution describing a progressive periodic wave train which is such that the flow speed at the crest equals to the train phase speed, in a reference frame where fluid is motionless at infinite depth. Longuet-Higgins and Fox [9] constructed asymptotic expansions for waves close to the 120• -cusped wave (almost highest waves) and showed that the wave profile oscillates infinitely as the limiting wave is approached. Later, in [10], the crest of a steep, irrotational gravity wave was theoretically shown to be unstable.The purpose of the present work is to give evidence that a second branch of two-dimensional irrotational symmetric periodic gravity waves of permanent form exists besides the Stokes waves of the same wavelength. The original motivation is as follows: the Bernoulli equation is quadratic in velocity and admits two values of the particle speed at the crest. The first one corresponds to the Stokes branch of symmetric waves for which the particle speed at the crest is smaller than the wave phase speed. The opposite inequality takes place for the second branch which might correspond to a new type of waves. In the second part of the Letter, we prove this numerically by using two different methods.Consider a symmetric two-dimensional periodic train of waves which propagates without changing a form from left to right along the x-axis with the constant speed c relative to the motionless fluid at infinite depth. The set of equations governing steady potential gravity waves on * Electronic address: lukom@iop.kiev.ua a surface of irrotational, inviscid, incompressible fluid iswhere Φ(θ, y) is the velocity potential, η(θ) is the elevation of a free surface, and y is the upward vertical axis such that y = 0 is the still water level. We have chosen the units of time and length such that the acceleration due to gravity and wavenumber are equal to unity. As it follows from the Bernoulli equation (2), a solution may be not single-valued in the vicinity of the limiting point. Indeed, the particle speed at the crest q(0) is horizontal and is defined as follows:η(0) being the height of the crest above the still water level. Th...

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A highly accurate technique for calculating the parameters of the Stokes 120° gravity wave on deep water is proposed. Tanaka's nonlinear transformation of the inverse plane is used to stretch the region near the wave crest for accelerating the convergence of the classical Michell series. Several families of new irregular self-intersecting profiles with a 120° angle at the crest are presented as well. They all degenerate into the Stokes 120° profile as numerical accuracy increases. These solutions seem to represent some kind of the so-called parasitic (or ghost) solutions, which emerge due to discretization of the original continuous problem. Probable physical relevance of such irregular solutions is discussed. The validity of the Stokes theorem about a 120° angle is tested numerically. No solutions with crest angles different from 120° were found. Hence, the Stokes 120° wave seems to be the unique gravity wave with a corner at the crest.

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The dynamics of a one-degree of freedom oscillator with arbitrary polynomial non-linearity subjected to an external periodic excitation is studied. The sequences (cascades) of harmonic and subharmonic stationary solutions to the equation of motion are obtained by using the harmonic balance approximation adapted for arbitrary truncation numbers, powers of non-linearity, and orders of subharmonics. A scheme for investigating the stability of the harmonic balance stationary solutions of such a general form is developed on the basis of the Floquet theorem. Besides establishing the stable/unstable nature of a stationary solution, its stability analysis allows obtaining the regions of parameters, where symmetry-breaking and period-doubling bifurcations occur. Thus, for perioddoubling cascades, each unstable stationary solution is used as a base solution for finding a subsequent stationary state in a cascade. The procedure is repeated until this stationary state becomes stable provided that a stable solution can finally be achieved. The proposed technique is applied to calculate the sequences of subharmonic stationary states in driven hardening Duffing's oscillator. The existence of stable subharmonic motions found is confirmed by solving the differential equation of motion numerically by means of a time-difference method, with initial conditions being supplied by the harmonic balance approximation.