Numerical continuation methods for large-scale dissipative dynamical systems

Abstract

Abstract. A tutorial on continuation and bifurcation methods for the analysis of truncated dissipative partial differential equations is presented. It focuses on the computation of equilibria, periodic orbits, their loci of codimension-one bifurcations, and invariant tori. To make it more self-contained, it includes some definitions of basic concepts of dynamical systems, and some preliminaries on the general underlying techniques used to solve non-linear systems of equations by inexact Newton methods, and eigenvalue problems by means of subspace or Arnoldi iterations.

Continuation and stability of rotating waves in the magnetized spherical Couette system: secondary transitions and multistability

“…Our study represents the first application of Newton-Krylov continuation and stability analysis of the solutions to the MSC problem. For such high dimensional systems, modelling three-dimensional flows, there exist very few studies based on continuation methods, even in the more general context of fluid dynamics [28,35].…”

“…To study the dependence of RW, rotating at a frequency ω and with m-fold azimuthal wave number, on the parameter p = Ha, pseudo-arclength continuation methods for periodic orbits are used [35,50]. These methods obtain the curve of periodic solutions x(s) = (u(s), τ (s), p(s)) ∈ R n+2 , s being the arclength parameter and τ = 2π/(mω) the rotation period, by adding the pseudo-arclength condition m(u, τ , p) ≡ w, x − x 0 = 0, where x 0 = (u 0 , τ 0 , p 0 ) and w = (w u , w τ , w p ) are the predicted point and the tangent to the curve of solutions, respectively, obtained by extrapolation of the previous points along the curve.…”

“…The appearance of multiple states in experimental flows strongly depends on the initial state [16], and in DNS the type of perturbation applied determines the type of instability or the mode that will be selected among the bifurcated solutions [20]. To get a complete picture of the skeleton of the phase space, and thus to provide a better characterization of the instabilities and the physically realizable flows, a continuation method is necessary [22,30,33,34] (see [35] for a nice tutorial).…”

Torsional solutions of convection in rotating fluid spheres

“…The details on how to compute the cycles and to obtain their stability (Floquet multipliers) can be found in Ref. [22] or in the more recent and detailed review on continuation methods for partial differential equations [24]. Here the spatiotemporal symmetries of the torsional periodic orbits were used to halve the integration time.…”

“…Therefore to study the periodic solutions it is necessary to use time evolution codes to simulate their behavior or continuation methods for periodic orbits like those described in Refs. [22][23][24]. We adopt this latter approach, which, in addition to providing a complete description of the dependence of the properties of the solutions on the parameters, also allows us to obtain unstable states, which could be relevant in complex dynamics involving, for instance, mixed solutions or heteroclinic chains visiting several unstable invariant objects.…”