Numerical Continuation, and Computation of Normal Forms

a b s t r a c tThe Continuation of Invariant Subspaces (CIS) algorithm produces a smoothly-varying basis for an invariant subspace R(s) of a parameter-dependent matrix A(s). We have incorporated the CIS algorithm into Cl_matcont, a Matlab package for the study of dynamical systems and their bifurcations. Using subspace reduction, we extend the functionality of Cl_matcont to large-scale computations of bifurcations of equilibria. In this paper, we describe the algorithms and functionality of the resulting Matlab bifurcation package Cl_matcontL. The novel features include: new CIS-based, continuous, well-scaled test functions for codimension 1 and 2 bifurcations; detailed description of locators for large problems; and examples of bifurcation analysis in large sparse problems.

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“…[26]), we have a simple branch point (BP); that is, two distinct solution branches of (2) pass through x 0 .…”

Section: Switching Branches At Simple Branch Pointsmentioning

confidence: 99%

“…A branch switching method based on the ABE (see e.g. [26] and references there), which requires the computation of second order derivatives using (55), (56). 2.…”

Section: Switching Branches At Simple Branch Pointsmentioning

confidence: 99%

Numerical computation of bifurcations in large equilibrium systems in matlab

Bindel

Friedman

Govaerts

et al. 2014

Journal of Computational and Applied Mathematics

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“…This result considerably speeds up the annealing procedure. Standard continuation algorithms [20,16] can then be used to trace this solution as β increases. Consecutive phase transitions then produce approximations of the normalized K-cut problem for all 2 < K ≤ N.…”

Section: From This We Getmentioning

confidence: 99%

“…Recall that the vector of conditional probabilities q = q(t|y) satisfies (19) These equations form an equality constraint on the maximization problem (6) giving the Lagrangian (20) which incorporates the vector of Lagrange multipliers ξ, imposed by the equality constraints (19). Maxima of (6) are critical points of the Lagrangian i.e.…”

Section: Letmentioning

confidence: 99%

“…The algorithm consists of incrementing the value of β, initializing a fixed point search at the solution value at the previous β and iterating until a solution at the new β is found. Various improvements of this basic algorithm are possible [20,16,13]. An alternative approach is to use agglomeration [21,22] starting at β = ∞ and lowering the value of β.…”

Section: Introductionmentioning

confidence: 99%

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Annealing and the normalized N-cut

Gedeon

Parker

Campion

et al. 2008

Pattern Recognition

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We describe an annealing procedure that computes the normalized N-cut of a weighted graph G. The first phase transition computes the solution of the approximate normalized 2-cut problem, while the low temperature solution computes the normalized N-cut. The intermediate solutions provide a sequence of refinements of the 2-cut that can be used to split the data to K clusters with 2 ≤ K ≤ N. This approach only requires specification of the upper limit on the number of expected clusters N, since by controlling the annealing parameter we can obtain any number of clusters K with 2 ≤ K ≤ N. We test the algorithm on an image segmentation problem and apply it to a problem of clustering high dimensional data from the sensory system of a cricket.

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Dynamical Systems and Their Bifurcations

Cerutti¹,

Marchesi²

2011

Advanced Methods of Biomedical Signal Processing

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In this chapter we summarize the basic definitions and tools of analysis of dynamical systems, with particular\ud emphasis on the asymptotic behavior of continuous-time autonomous systems. In particular, the possible\ud structural changes of the asymptotic behavior of the system under parameter variation, called bifurcations,\ud are presented together with their analytical characterization and hints on their numerical analysis. The\ud literature on dynamical systems is huge and we do not attempt to survey it here. Most of the results on\ud bifurcations of continuous-time systems are due to Andronov and Leontovich [see Andronov et al., 1973].\ud More recent expositions can be found in Guckenheimer & Holmes [1997] and Kuznetsov [2004], while less\ud formal but didactically very effective treatments, rich in interesting examples and applications, are given\ud in Strogatz [1994] and Alligood et al. [1996]. Numerical aspects are well described in Allgower & Georg\ud [1990] and in the fundamental papers by Keller [1977] and Doedel et al. [1991a,b], but see also Beyn et al.\ud [2002] and Kuznetsov [2004]. This chapter mainly combines material from two previous contributions of the\ud authors, the first part of the book Biosystems and Complexity [Rinaldi, 1993, in Italian] and the Appendix A\ud of a recent book on evolutionary dynamics [Dercole & Rinaldi, 2008]

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Numerical Continuation, and Computation of Normal Forms

Cited by 123 publications

References 41 publications

Numerical computation of bifurcations in large equilibrium systems in matlab

Numerical computation of bifurcations in large equilibrium systems in matlab

Annealing and the normalized N-cut

Dynamical Systems and Their Bifurcations

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