Externally positive linear systems from transfer function properties

The characterisation of single-input-single-output externally positive linear systems is considered. A complete characterisation of the class of externally positive second-order and a class of underdamped third-order systems is given and connections to negative-imaginary systems are highlighted. It is shown that negative-imaginary systems have non-negative step responses, leading to a condition for external positivity based on negative imaginary systems theory. Finally, a class of externally positive systems which can be verified using the developed results but which fail a recently developed numerical test for external positivity based upon linear matrix inequalities are introduced. These results extend the class of system for which external positivity can be verified, facilitating large-scale control and less conservative absolute stability analysis

Study on stability and rotating speed stable region of magnetically suspended rigid rotors using extended Nyquist criterion and gain-stable region theory

This paper presents a novel and simple method to analyze the absolute stability and the rotor speed stable region of a magnetically suspended rotor (MSR). At the beginning of the paper, a complex variable is introduced to describe the movement of the MSR and a complex coefficient transfer function is obtained accordingly. The equivalent stability relationship between this new variable and the two traditional deflection angles is also demonstrated in a simple way. The detailed characteristics of the open-loop MSR system with time delay are studied carefully based on the characteristics of its Nyquist curve. A sufficient and necessary condition of absolute stability is then deduced by using an extended complex Nyquist stability criterion for MSRs. Based on the relationship between the rotor speed and gain-stable region proposed in this paper, the rotor speed stable region can be solved simply and directly. The usefulness and effectiveness of the proposed approaches are validated by examples and simulations

Optical solitons in PT\mathcal{PT}-symmetric nonlinear couplers with gain and loss

We study spatial and temporal solitons in the $\mathcal{PT}$ symmetric coupler with gain in one waveguide and loss in the other. Stability properties of the high- and low-frequency solitons are found to be completely determined by a single combination of the soliton's amplitude and the gain/loss coefficient of the waveguides. The unstable perturbations of the high-frequency soliton break the symmetry between its active and lossy components which results in a blowup of the soliton or a formation of a long-lived breather state. The unstable perturbations of the low-frequency soliton separate its two components in space blocking the power drainage of the active component and cutting the power supply to the lossy one. Eventually this also leads to the blowup or breathing.Comment: 14 pages, 11 figure

Numerical computation of an Evans function for travelling waves

We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller-Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed $c \geq 2 \sqrt{\delta}$ in the F-KPP equation.Comment: 37 pages, 11 figure

Absolute instabilities of travelling wave solutions in a Keller-Segel model

We investigate the spectral stability of travelling wave solutions in a Keller-Segel model of bacterial chemotaxis with a logarithmic chemosensitivity function and a constant, sublinear, and linear consumption rate. Linearising around the travelling wave solutions, we locate the essential and absolute spectrum of the associated linear operators and find that all travelling wave solutions have essential spectrum in the right half plane. However, we show that in the case of constant or sublinear consumption there exists a range of parameters such that the absolute spectrum is contained in the open left half plane and the essential spectrum can thus be weighted into the open left half plane. For the constant and sublinear consumption rate models we also determine critical parameter values for which the absolute spectrum crosses into the right half plane, indicating the onset of an absolute instability of the travelling wave solution. We observe that this crossing always occurs off of the real axis

Fast flavor conversions of supernova neutrinos: Classifying instabilities via dispersion relations

Supernova neutrinos can exhibit a rich variety of flavor conversion mechanisms. In particular, they can experience "fast" self-induced flavor conversions almost immediately above the core. Very recently, a novel method has been proposed to investigate these phenomena, in terms of the dispersion relation for the complex frequency and wave number ($\omega$,$k$) of disturbances in the mean field of the $\nu_e\nu_x$ flavor coherence. We discuss a systematic approach to such instabilities, originally developed in the context of plasma physics, and based of the time-asymptotic behavior of the Green's function of the system. Instabilities are typically seen to emerge for complex $\omega$, and can be further characterized as convective (moving away faster than they spread) and absolute (growing locally), depending on $k$-dependent features. Stable cases emerge when $k$ (but not $\omega$) is complex, leading to disturbances damped in space, or when both $\omega$ and $k$ are real, corresponding to complete stability. The analytical classification of both unstable and stable modes leads not only to qualitative insights about their features but also to quantitative predictions about the growth rates of instabilities. Representative numerical solutions are discussed in a simple two-beam model of interacting neutrinos. As an application, we argue that supernova and binary neutron star mergers exhibiting a "crossing" in the electron lepton number would lead to an absolute instability in the flavor content of the neutrino gas.Comment: (v2, revised version: 25 pages, 15 pdf figures. Minor changes. Figures improved. Matches the version published on PRD

Convective instability induced by nonlocality in nonlinear diffusive systems

We consider a large class of nonlinear diffusive systems with nonlocal coupling. By using a non-perturbative analytical approach we are able to determine the convective and absolute instabilities of all the uniform states of these systems. We find a huge window of convective instability that should provide a great opportunity to study experimentally and theoretically noise sustained patterns.Comment: 5 pages, accepted for publication in PR

An analytical connection between temporal and spatio-temporal growth rates in linear stability analysis

We derive an exact formula for the complex frequency in spatio-temporal stability analysis that is valid for arbitrary complex wave numbers. The usefulness of the formula lies in the fact that it depends only on purely temporal quantities, which are easily calculated. We apply the formula to two model dispersion relations: the linearized complex Ginzburg--Landau equation, and a model of wake instability. In the first case, a quadratic truncation of the exact formula applies; in the second, the same quadratic truncation yields an estimate of the parameter values at which the transition to absolute instability occurs; the error in the estimate decreases upon increasing the order of the truncation. We outline ways in which the formula can be used to characterize stability results obtained from purely numerical calculations, and point to a further application in global stability analyses.Comment: 36 pages, 16 figures; Article has been tweaked and reduced in size but essential features remain the same; Supplementary material (16 pages) is also include