On the use of stabilising transformations for detecting unstable periodic orbits in the Kuramoto-Sivashinsky equation

In this paper we develop further a method for detecting unstable periodic orbits (UPOs) by stabilising transformations, where the strategy is to transform the system of interest in such a way that the orbits become stable. The main difficulty of using this method is that the number of transformations, which were used in the past, becomes overwhelming as we move to higher dimensions (Davidchack and Lai 1999; Schmelcher et al. 1997, 1998). We have recently proposed a set of stabilising transformations which is constructed from a small set of already found UPOs (Crofts and Davidchack 2006). The main benefit of using the proposed set is that its cardinality depends on the dimension of the unstable manifold at the UPO rather than the dimension of the system. In a typical situation the dimension of the unstable manifold is much smaller than the dimension of the system so the number of transformations is much smaller. Here we extend this approach to high-dimensional systems of ODEs and apply it to the model example of a chaotic spatially extended system -- the Kuramoto-Sivashinsky equation. A comparison is made between the performance of this new method against the competing methods of Newton-Armijo (NA) and Levernberg-Marquardt (LM). In the latter case, we take advantage of the fact that the LM algorithm is able to solve under-determined systems of equations, thus eliminating the need for any additional constraints

Global Asymptotic Behavior of Iterative Implicit Schemes

The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics

Exploring Critical Points of Energy Landscapes: From Low-Dimensional Examples to Phase Field Crystal PDEs

In the present work we explore the application of a few root-finding methods to a series of prototypical examples. The methods we consider include: (a) the so-called continuous-time Nesterov (CTN) flow method; (b) a variant thereof referred to as the squared-operator method (SOM); and (c) the the joint action of each of the above two methods with the so-called deflation method. More traditional methods such as Newton's method (and its variant with deflation) are also brought to bear. Our toy examples start with a naive one degree-of-freedom (dof) system to provide the lay of the land. Subsequently, we turn to a 2-dof system that is motivated by the reduction of an infinite-dimensional, phase field crystal (PFC) model of soft matter crystallisation. Once the landscape of the 2-dof system has been elucidated, we turn to the full PDE model and illustrate how the insights of the low-dimensional examples lead to novel solutions at the PDE level that are of relevance and interest to the full framework of soft matter crystallization.Comment: 17 pages, 16 figure

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations

The global asymptotic nonlinear behavior of 1 1 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODES) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDES