Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?

The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler's paradox (1952), has stimulated an ever growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary. What is less known is that the first complete explanation of Ziegler's paradox by means of the Whitney umbrella singularity dates back to 1956. We revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. We discuss subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period, involving structural stability of matrices, Krein collision, Hamilton-Hopf bifurcation and related bifurcations.Comment: 35 pages, 11 figure

Recent Developments in the Dynamic Stability of Elastic Structures

Dynamic instability in the mechanics of elastic structures is a fascinating topic, with many issues still unsettled. Accordingly, there is a wealth of literature examining the problems from different perspectives (analytical, numerical, experimental etc.), and coverings a wide variety of topics (bifurcations, chaos, strange attractors, imperfection sensitivity, tailor-ability, parametric resonance, conservative or non-conservative systems, linear or nonlinear systems, fluid-solid interaction, follower forces etc.). This paper provides a survey of selected topics of current research interest. It aims to collate the key recent developments and international trends, as well as describe any possible future challenges. A paradigmatic example of Ziegler's paradox on the destabilizing effect of small damping is also included

On periodic differential equations with dissipation

In this work we present an unexpected relation between the discriminant associated to a Hill equation with and without dissipation. We prove that by knowing the discriminant associated to a periodic differential equation, which is the summation of the monodromy matrix main diagonal entries, we are able to obtain the stability properties of damped periodic differential equation solutions. We propose to conceive the discriminant as a manifold, by doing this one can observe that the stability properties of periodic differential equations are closely related to the growing rate of unstable solutions of periodic differential equations without dissipation. We show the appearance of the Ziegler destabilization paradox in systems of one degree of freedom. This work may be of interest for scientists and engineers dealing with parametric resonance applications or physicist working on the motion of a damped wave in a periodic media

Shot noise in a harmonically driven ballistic graphene transistor

We study time-dependent electron transport and quantum noise in a ballistic graphene field effect transistor driven by an ac gate potential. The non-linear response to the ac signal is computed through Floquet theory for scattering states and Landauer-B\"uttiker theory for charge current and its fluctuations. Photon-assisted excitation of a quasibound state in the top-gate barrier leads to resonances in transmission that strongly influence the noise properties. For strong doping of graphene under source and drain contacts, when electrons are transmitted through the channel via evanescent waves, the resonance leads to a substantial suppression of noise. The Fano factor is then reduced well below the pseudo-diffusive value, $F<1/3$, also for strong ac drive. The good signal-to-noise ratio (small Fano factor) on resonance suggests that the device is a good candidate for high-frequency (THz) radiation detection. We show analytically that Klein tunneling (total suppression of back-reflection) persists for perpendicular incidence also when the barrier is driven harmonically. Although the transmission is inelastic and distributed among sideband energies, a sum rule leads to total suppression of shot noise.Comment: 12 pages, 7 figure

The paradox of Vito Volterra's predator-prey model

This article is dedicated to the late Giorgio Israel. R{\'e}sum{\'e}. The aim of this article is to propose on the one hand a brief history of modeling starting from the works of Fibonacci, Robert Malthus, Pierre Francis Verhulst and then Vito Volterra and, on the other hand, to present the main hypotheses of the very famous but very little known predator-prey model elaborated in the 1920s by Volterra in order to solve a problem posed by his son-in-law, Umberto D'Ancona. It is thus shown that, contrary to a widely-held notion, Volterra's model is realistic and his seminal work laid the groundwork for modern population dynamics and mathematical ecology, including seasonality, migration, pollution and more. 1. A short history of modeling 1.1. The Malthusian model. If the rst scientic view of population growth seems to be that of Leonardo Fibonacci [2], also called Leonardo of Pisa, whose famous sequence of numbers was presented in his Liber abaci (1202) as a solution to a population growth problem, the modern foundations of population dynamics clearly date from Thomas Robert Malthus [20]. Considering an ideal population consisting of a single homogeneous animal species, that is, neglecting the variations in age, size and any periodicity for birth or mortality, and which lives alone in an invariable environment or coexists with other species without any direct or indirect inuence, he founded in 1798, with his celebrated claim Population, when unchecked, increases in a geometrical ratio, the paradigm of exponential growth. This consists in assuming that the increase of the number N (t) of individuals of this population, during a short interval of time, is proportional to N (t). This translates to the following dierential equation : (1) dN (t) dt = $\epsilon$N (t) where $\epsilon$ is a constant factor of proportionality that represents the growth coe-cient or growth rate. By integrating (1) we obtain the law of exponential growth or law of Malthusian growth (see Fig. 1). This law, which does not take into account the limits imposed by the environment on growth and which is in disagreement with the actual facts, had a profound inuence on Charles Darwin's work on natural selection. Indeed, Darwin [1] founded the idea of survival of the ttest on the 1. According to Frontier and Pichod-Viale [3] the correct terminology should be population kinetics, since the interaction between species cannot be represented by forces. 2. A population is dened as the set of individuals of the same species living on the same territory and able to reproduce among themselves