A fast technique for calculating master stability function

Investigating the stability of the synchronization manifold is a critical topic in the field of complex dynamical networks. Master stability function (MSF) is known as a powerful and efficient tool for the study of synchronization in complex identical networks. The network can be synchronized whenever the MSF is negative. MSF uses the Lyapunov or Floquet exponent theory to determine the stability of the synchronization state. Both of these methods need extensive numerical simulation and a long computational time. In this paper, a new approach to calculate MSF is proposed. The accuracy of the results and time of simulations are tested on seven different known oscillators and also compared with the conventional methods of MSF. The results show that the proposed technique is faster and more efficient than the existing methods.

Sensitivity analysis and passive control of the secondary instability in the wake of a cylinder

The stability properties of selected flow configurations, usually denoted as base flows, can be significantly altered by small modifications of the flow, which can be caused, for instance, by a non-intrusive passive control. This aspect is amply demonstrated in the literature by ad hoc sensitivity studies which, however, focus on configurations characterised by a steady base flow. Nevertheless, several flow configurations of interest are characterised by a time-periodic base flow. To this purpose, we propose here an original theoretical framework suitable to quantify the effects of base-flow variations in the stability properties of saturated time-periodic limit cycles. In particular, starting from a Floquet analysis of the linearised Navier–Stokes equations and using adjoint methods, it is possible to estimate the variation of a selected Floquet exponent caused by a generic structural perturbation of the base-flow equations. This link is expressed concisely using the adjoint operators coming from the analysis, and the final result, when applied to spatially localised disturbances, is used to build spatial sensitivity and control maps. These maps identify the regions of the flow where the placement of a infinitesimal small object produces the largest effect on the Floquet exponent and may also provide a quantification of this effect. Such analysis brings useful insights both for passive control strategies and for further characterising the investigated instability. As an example of application, the proposed analysis is applied here to the three-dimensional flow instabilities in the wake past a circular cylinder. This is a classical problem which has been widely studied in the literature. Nevertheless, by applying the proposed analysis we derive original results comprising a further characterisation of the instability and related control maps. We finally show that the control maps obtained here are in very good agreement with control experiments documented in the literature.

Analysis of Effects of Delays and Diffusion on a Predator-Prey System

A reaction-diffusion predator-prey system with two delays is investigated. It is found that the spatially homogeneous periodic solution will occur when the sum of two delays crosses some critical values and Hopf bifurcation takes place. For the fixed domain and diffusion, some numerical simulations are also given to illustrate the theoretical analysis. In addition, special attention is paid to effects of diffusion on the bifurcating periodic solution. It is found that the diffusion would lead to the bifurcating period solution to destabilize by calculating the relevant expression of the Floquet exponent.

Stability Analysis of Parametrically Excited Systems Using the Energy-Growth Exponent/Coefficient

In this paper, the energy exponent is used to study the instability of parametrically excited systems governed by the damped Mathieu equation. Through the numerical tests, an energy-growth exponent (EGE) is adopted to evaluate the instability intensity and instability boundary of the system. The EGE can be expressed as a product of the modal frequency and a dimensionless coefficient, defined as the energy-growth coefficient (EGC). Based on the Floquet theory, the relationship between the EGE, Floquet exponent and Lyapunov exponent are derived. An energy criterion of parametric instability is proposed by using the EGE. Using a simple pendulum as an example, the geometric characteristics of the EGC are investigated, and approximate analytical formulae of the EGC/EGE for three different unstable patterns are developed. The EGC/EGE formulae are applicable to the parametrically excited systems governed by the damped Mathieu equation. The unstable behaviors and properties of parametric vibrations are analyzed and discussed in details with EGE/EGC for three systems including a triple pendulum, two-dimensional sloshing fluid, and a two-span continuous beam. The stability boundaries established by using EGE/EGC agree well with those by the conventional theory and experiment. As a practical tool, the EGE/EGC formulae can be easily applied to analyzing the unstable intensities and boundaries of the parametrically excited systems.

Preheating with fractional powers

We consider preheating in models in which the potential for the inflaton is given by a fractional power, as is the case in axion monodromy inflation. We assume a standard coupling between the inflaton field and a scalar matter field. We find that in spite of the fact that the oscillation of the inflaton about the field value which minimizes the potential is anharmonic, there is nevertheless a parametric resonance instability, and we determine the Floquet exponent which describes this instability as a function of the parameters of the inflaton potential.

Linear waves in two-layer fluids over periodic bottoms

A new, exact Floquet theory is presented for linear waves in two-layer fluids over a periodic bottom of arbitrary shape and amplitude. A method of conformal transformation is adapted. The solutions are given, in essentially analytical form, for the dispersion relation between wave frequency and generalized wavenumber (Floquet exponent), and for the waveforms of free wave modes. These are the analogues of the classical Lamb’s solutions for two-layer fluids over a flat bottom. For internal modes the interfacial wave shows rapid modulation at the scale of its own wavelength that is comparable to the bottom wavelength, whereas for surface modes it becomes a long wave carrier for modulating short waves of the bottom wavelength. The approximation using a rigid lid is given. Sample calculations are shown, including the solutions that are inside the forbidden bands (i.e. Bragg resonated).

Non-Linear Dynamic Characteristics of Single-Layer Shallow Reticulated Spherical Shells

The nonlinear dynamical equations are established by using the method of quasi-shells for three-dimensional shallow spherical shells with circular bottom. Displacement mode that meets the boundary conditions of fixed edges is given by using the method of the separate variable, A nonlinear forced vibration equation containing the second and the third order is derived by using the method of Galerkin. The stability of the equilibrium point is studied by using the Floquet exponent.

Floquet exponents for Jacobi fields

This paper introduces a Riemannian invariant of a compact Riemannian manifold based on the spectral theory for the Jacobi field operator. It is the Floquet exponent for this operator, a purely dynamical quantity computable directly from the asymptotic behavior of Jacobi fields. We show that it is related to certain traces of the Green's function, and we derive further regularity and analyticity properties for the Green's function. In case the geodesic flow is ergodic, the Floquet exponent generalizes the measure entropy, and several entropy estimates follow. An asymptotic expansion of the Floquet exponent gives rise to a sequence of ‘Jacobi invariants’, which are related to the polynomial invariants of the K dV equation.

The algebraic-geometric AKNS potentials

We characterize the algebraic-geometric potentials for the Schrödinger and AKNS operators using the Weyl m-functions and the Floquet exponent for these operators. The characterization is this: among random ergodic Schrödinger operators, the alebraic-geometric potentials are those for which (i) the spectrum is a union of finitely many intervals (or bands); (ii) the Lyapounov exponent vanishes on the spectrum.