In the last 30 days

• Open/close Tibetan singing bowls

  Denis Terwagne and John W M Bush 2011 Nonlinearity 24 R51 Tag this article

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  We present the results of an experimental investigation of the acoustics and fluid dynamics of Tibetan singing bowls. Their acoustic behaviour is rationalized in terms of the related dynamics of standing bells and wine glasses. Striking or rubbing a fluid-filled bowl excites wall vibrations, and concomitant waves at the fluid surface. Acoustic excitation of the bowl's natural vibrational modes allows for a controlled study in which the evolution of the surface waves with increasing forcing amplitude is detailed. Particular attention is given to rationalizing the observed criteria for the onset of edge-induced Faraday waves and droplet generation via surface fracture. Our study indicates that drops may be levitated on the fluid surface, induced to bounce on or skip across the vibrating fluid surface.

• Open/close Transition state geometry near higher-rank saddles in phase space

  George Haller et al 2011 Nonlinearity 24 527 Tag this article

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  We present a detailed analysis of invariant phase space structures near higher-rank saddles of Hamiltonian systems. Using the theory of pseudo-hyperbolic invariant surfaces, we show the existence of codimension-one normally hyperbolic invariant manifolds that govern transport near the higher-rank saddle points. Such saddles occur in a number of problems in celestial mechanics, chemical reactions, and atomic physics. As an example, we consider the problem of double ionization of helium in an external electric field, a basis of many modern ionization experiments. In this example, we illustrate our main results on the geometry and transport properties near a rank-two saddle.

• Open/close Nonlinear modelling of cancer: bridging the gap between cells and tumours

  J S Lowengrub et al 2010 Nonlinearity 23 R1 Tag this article

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  Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

• Open/close On the dynamics in the one-dimensional piston problem

  Igor Gorelyshev 2011 Nonlinearity 24 2119 Tag this article

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  In this paper we consider the dynamics in the one-dimensional piston problem. We give a description of the dynamics of the system in the second approximation of the perturbation theory. The first approximation of the perturbation theory was studied previously (Gorelyshev and Neishtadt 2006 J. Appl. Math. Mech. 70 4–17, Sinai 1999 Theor. Math. Phys. 125 1351–7, Wright 2006 Nonlinearity 19 2365–89). Given the small parameter of the system ε, the first approximation prescribes non-damping oscillations of the piston in time intervals of order ε ⁻¹. We prove that in time intervals of order ε ⁻² the oscillations of the piston are not damped either.

• Open/close Collapse in the nonlocal nonlinear Schrödinger equation

  F Maucher et al 2011 Nonlinearity 24 1987 Tag this article

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  We discuss spatial dynamics and collapse scenarios of localized waves governed by the nonlinear Schrödinger equation with nonlocal nonlinearity. Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimension collapse does not occur. Then we study in detail the effect of singular nonlocal kernels in arbitrary dimension using both Lyapunoff's method and virial identities. We find that in the one-dimensional case, i.e. for n = 1, collapse cannot happen for nonlocal nonlinearity. On the other hand, for spatial dimension n ≥ 2 and singular kernel ~1/ r ^α, no collapse takes place if α < 2, whereas collapse is possible if α ≥ 2. Self-similar solutions allow us to find an expression for the critical distance (or time) at which collapse should occur in the particular case of ~1/ r ² kernels for n = 3. Moreover, different evolution scenarios for the three-dimensional physically relevant case of Bose–Einstein condensates are studied numerically for both the ground state soliton and higher order toroidal states with, and without, an additional local repulsive nonlinear interaction. In particular, we show that the presence of local repulsive nonlinearity can prevent collapse in those cases.

• Open/close No elliptic islands for the universal area-preserving map

  Tomas Johnson 2011 Nonlinearity 24 2063 Tag this article

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  A renormalization approach has been used in Eckmann et al (1982) and Eckmann et al (1984) to prove the existence of a universal area-preserving map, a map with hyperbolic orbits of all binary periods. The existence of a horseshoe, with positive Hausdorff dimension, in its domain was demonstrated in Gaidashev and Johnson (2009a). In this paper the coexistence problem is studied, and a computer-aided proof is given that no elliptic islands with period less than 18 exist in the domain. It is also shown that less than 1.5% of the measure of the domain consists of elliptic islands. This is proven by showing that the measure of initial conditions that escape to infinity is at least 98.5% of the measure of the domain, and we conjecture that the escaping set has full measure. This is highly unexpected, since generically it is believed that for conservative systems hyperbolicity and ellipticity coexist.

• Open/close Nonlinear stochastic dynamics of mesoscopic homogeneous biochemical reaction systems—an analytical theory

  Hong Qian 2011 Nonlinearity 24 R19 Tag this article

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  The nonlinear dynamics of biochemical reactions in a small-sized system on the order of a cell are stochastic. Assuming spatial homogeneity, the populations of n molecular species follow a multi-dimensional birth-and-death process on . We introduce the Delbrück–Gillespie process, a continuous-time Markov jump process, whose Kolmogorov forward equation has been known as the chemical master equation, and whose stochastic trajectories can be computed via the Gillespie algorithm. Using simple models, we illustrate that a system of nonlinear ordinary differential equations on emerges in the infinite system size limit. For finite system size, transitions among multiple attractors of the nonlinear dynamical system are rare events with exponentially long transit times. There is a separation of time scales between the deterministic ODEs and the stochastic Markov jumps between attractors. No diffusion process can provide a global representation that is accurate on both short and long time scales for the nonlinear, stochastic population dynamics. On the short time scale and near deterministic stable fixed points, Ornstein–Uhlenbeck Gaussian processes give linear stochastic dynamics that exhibit time-irreversible circular motion for open, driven chemical systems. Extending this individual stochastic behaviour-based nonlinear population theory of molecular species to other biological systems is discussed.

• Open/close Dimension of non-conformal repellers: a survey

  Jianyu Chen and Yakov Pesin 2010 Nonlinearity 23 R93 Tag this article

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  This paper is a survey of recent results on the dimension of repellers for expanding maps and limit sets for iterated function systems. While the case of conformal repellers is well understood, the study of non-conformal repellers is in its early stages though a number of interesting phenomena have been discovered, some remarkable results obtained and several interesting examples constructed. We will describe contemporary state of the art in the area with emphasis on some new emerging ideas and open problems.

• Open/close Dimensional entropy over sets and fibres

  Piotr Oprocha and Guohua Zhang 2011 Nonlinearity 24 2325 Tag this article

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  This paper is devoted to a study of dimensional entropy, especially entropy on the fibres of factor maps. We show that dimensional entropy and topological entropy of sets are not usually equal, while dimensional entropy over fibres always corresponds to entropy of the factor. Then we provide an estimate of the dimensional entropy of the image of a set under a factor map. Finally we solve a few questions stated recently by Dai and Jiang on distance entropy (which is a modification of dimensional entropy).

• Open/close Which hole is leaking the most: a topological approach to study open systems

  V S Afraimovich and L A Bunimovich 2010 Nonlinearity 23 643 Tag this article

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  We study the process of escape of orbits through a hole in the phase space of a dynamical system generated by a chaotic map of an interval. If this hole is an element of Markov partition we are able to use the machinery of symbolic dynamics and estimate the probability of an orbit to escape at the instant of time n in terms of the topological pressure over corresponding symbolic dynamical systems. We obtain exact formulae allowing us to compare different holes according to their ability to support escaping flow of orbits. These results are applicable to the classification of vertices of dynamical networks in terms of the loads on nodes and edges of a network.