Numerical Continuation Methods for Dynamical Systems

Abstract

Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems -cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science.Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse "real-life" situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications.Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence.The two major book publication platforms of the Springer Complexity program are the monograph series "Understanding Complex Systems" focusing on the various applications of complexity, and the "Springer Series in Synergetics", which is devoted to the quantitative theoretical and methodological foundations. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.

Pattern localization to a domain edge

“…A more detailed explanation of continuation methods can be found in Ref. [42]). We find that the flux-balance construction gives a estimate of the steady states profiles for sufficiently large system sizes (see Appendix E).…”

“…The basic idea of numerical continuation is to follow a solution branch through parameter space (see for instance, Ref. [42] for an excellent overview over continuation methods). This "path-following" is often performed by emplying a predictor-corrector scheme: Starting from one solution, the next solution along the branch is predicted from the tangent space of the solution branch which can be obtained from the Jacobian.…”

“…See Appendix E for a brief description of the core idea behind numerical continuation. An excellent overview is provided in Ref [42]…”

Experimental bifurcation analysis—Continuation for noise-contaminated zero problems

“…An algorithm for computing branches is the so-called method of pseudo arclength continuation [15,16]. This is a predictor-corrector method that is based on augmenting the zero problem F : R n+1 → R n with the so-called pseudo arc-length condition g(u) := t T (u − u 0 ) − h = 0, where t is a unit vector tangent to the branch at u 0 , u 0 is a point on the branch, and h is the continuation step-size.…”

“…In the context of Coco [13,12], an algorithm for computing a branch is referred to as a covering algorithm; see also Chapters 1 and 3 in [16]. Covering algorithms follow the concept of predictor-corrector methods and pseudo arc-length continuation is an example of a covering algorithm for differentiable branches.…”