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2 edition of stability of codimension one bifurcations of the planar replicator equations found in the catalog.

stability of codimension one bifurcations of the planar replicator equations

Abbas Edalat

stability of codimension one bifurcations of the planar replicator equations

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Published by typescript in [s.l.] .
Written in English


Edition Notes

Thesis (Ph.D.) - University of Warwick, 1985.

Statementby Abbas Edalat.
ID Numbers
Open LibraryOL13863384M

  As parameters are varied, a boundary equilibrium bifurcation (BEB) occurs when an equilibrium collides with a discontinuity surface in a piecewise-smooth system of ordinary differential equations.


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stability of codimension one bifurcations of the planar replicator equations by Abbas Edalat Download PDF EPUB FB2

In this thesis three main original results are obtained: (i) The codimension one bifurcations of the planar replicatorsystem have been determined and classified by proving stability (miniversality) [proposition and theorems, ,and ].

(ii) The codimension two bifurcations (of the planar replicator system) have been determined, without however proving the stability Cited by: 2. THE STABILITY OF CODIMENSION ONE BIFURCATIONS OF THE PLANAR REPLICATOR EQUATIONS by Abbas Edalat Thesis submitted to the University of Warwick for the degree of Doctor of Philosophy.

December Mathematics Institute University of Warwick Coventry, England. The stability of codimension one bifurcations of the. (ii) The codimension two bifurcations (of the planar replicator system) have been determined, without however proving the stability (miniversality) [proposition ].

(iii) The conjugacy classes of certain families of maps of intervals have been determined [Theorem ], and the equivalence of certain families of vector fields has been by: 2.

Dissertation: The Stability of Codimension-One Bifurcations of stability of codimension one bifurcations of the planar replicator equations book Planar Replicator Equations. Advisor: E. Christopher (Eric) Zeeman. Student: Name School Year Descendants; Bilokon, Paul: Imperial College London: According to our current on-line database, Abbas Edalat has 1.

Hopf-like bifurcations in planar piecewise linear systems. surfaces of codimension one bifurcation points, namely, the quadrants of In each case a stationary solution changes stability and. In mathematics, the replicator equation is a deterministic monotone non-linear and non-innovative game dynamic used in evolutionary game theory.

The replicator equation differs from other equations used to model replication, such as the quasispecies equation, in that it allows the fitness function to incorporate the distribution of the population types rather than setting the fitness of a.

Edalat and E. Zeeman, The stable classes of the codimension-one bifurcations of the planar replicator system, Nonlinearity, 5 (), Google Scholar [12] J.

Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics,", Cambridge University Press. codimension-one equilibrium bifurcations of ODEs in the book complete. This chapter also includes an example of the Hopf bifurcation analysis in a planar system using MAPLE,a symbolic manipulation software.

Chapter 4 includes a detailed normal form analysis of the Neimark-Sacker bifur-cation in the delayed logistic map. Our numerical results show very interesting dynamics resulting from codimension one bifurcations including: Hopf, fold, transcritical, cyclic-fold, and homoclinic bifurcations as well as codimension two bifurcations including: Bautin and Bogdanov-Takens bifurcations, and a codimension three Bogdanov-Takens bifurcation.

solutions to ODEs. Topological equivalence, hyperbolicity and structural stability of ows. Stable and unstable manifolds.

Codimension{one local bifurcations in ows and maps. Centre manifolds. Reduction to normal forms; normal form symmetries. Global bifurcations: Lorenz and Shil’nikov mechanisms. Codimension-two bifurcations: degenerate.

2 A prototypical equation to study boundary equilibrium col-lisions In Fig. 1 we show two bifurcations, each a local codimension one collision of an equilibrium with a switching surface in the plane, under typical conditions.

Both appear in Filippov’s book. In fact case (ii) is actually one of the few that is fully and graphically unfolded, yet. We present two codimension-one bifurcations that occur when an equilibrium collides with a discontinuity in a piecewise smooth dynamical system.

from Filippov’s book Differential Equations. () Generic bifurcations of low codimension of planar Filippov Systems. Journal of Differential Equations() Coexisting tori and torus bubbling in non-smooth systems.

The Stable Classes and the Codimension One Bifurcations of the Planar Replicator System (with E. Zeeman) Nonlinearity 5(4), (), The Stability of the Unfolding of the Predator-Prey Model. bifurcation by the authors, while only its codimension-one cases are treated.

Indeed, codimension-two bifurcations receive hardly any attention at all in the book. Chapter 8 presents results of numerical simulations of some low-dimensional continuous and dis.

The term bifurcation in this paper refers to changes in the qualitative structure of any solutions to a system of ordinary differential equations with a varying parameter. This paper is about multiple bifurcations for which there is a multiple degeneracy in some feature of the system and a multi-dimensional parameter in its definition.

Regarding the present work, on the one hand, the paper stands for a review of the analytical procedure concerning the restriction of an -dimensional nonlinear three-parameter system on the center manifold, as well as the normal forms of a planar system with respect to degenerate Hopf bifurcations of higher codimension.

This textbook presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems.

Although the main topic of the book is the local and global behavior of nonlinear systems and their bifurcations, a thorough treatment of linear systems is given at the beginning of the text.

This book provides a crash course on various methods from the bifurcation theory of Functional Differential Equations (FDEs).

FDEs arise very naturally in economics, life sciences and engineering and the study of FDEs has been a major source of inspiration for advancement in nonlinear analysis and infinite dimensional dynamical systems.

However, this is but one of the possible codimension-one cases of bifurcations toward a dissipative ise we can have: • uniform and stationary instabilities (k c = ω c = 0) breaking only an internal symmetry; this rather infrequent and relatively trivial case will not be considered further;uniform and time periodic modes with k c = 0 but ω c ≠ 0, breaking time.

The replicator equation is the first and most important game dynamics studied in connection with evolutionary game theory.

It was originally developed for symmetric games with finitely many strategies. Properties of thesedynamics are briefly summarized for this case, including the convergence to and stability of the Nash.

From the reviews: "This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review of basic results in the theory of dynamical systems, covering both ordinary differential equations and discrete mappings.

The proposed normal form shows several bifurcations of codimension-1, see Fig. we consider the existence of local bifurcations. At the line μ = 0 the focus collides with the sliding segment at the origin.

Then, depending on the sign of ε two different boundary focus bifurcations, which are described in, appear by moving the parameter precisely, for ε. Edalat, E.C. Zeeman, The stable classes and the codimension-one bifurcations of the planar replicator system, Nonlinearity () E.C.

Zeeman, Banquet address at the Smalefest, From Topology to Computation: Proceedings of the Smalefest (Berkeley, ),Springer-Verlag, CHAPTER BIFURCATION THEORY 2 Since U0 is a time independent state, Kij is a constant matrix, and its eigenvalues ˙ (ordered so that Re˙1 Re˙) give the growth rates of perturbations: U/ X A e ˙ tu.

/, () with A a set of initial u. /are the eigenvectors, and tell us the character of the exponentially growing or decaying solutions. to planar Filippov systems, one can see [KRG03], [GST11] and [Der+11] and for examples on higher dimensions, look at [CJ11] and [DRD12].

Concerning the study of the generic behavior for planar Filippov systems, it was firstly made by Kozlova ([Koz84]). In [KRG03], Kuznetsov at al., classified and studied all the codimension one bifurcations and.

One-parameter bifurcations in planar Filippov systems. Internat. Bifur. Chaos Appl. Sci. Engrg 13 (), 3. "Continuation of homoclinic bifurcations of equilibria" A.R.

Champneys (Bristol) The course will focus on codim 1 and 2 bifurcations of homoclinic orbits to. The argument to prove stability of a random differential equation in the proof of Theorem applies at and near such a parameter value.

Violation of conditions (1), (4) or (5) leads to bifurcations in generic one parameter families, as discussed below. The bifurcation analysis of codimension one and two linear singularities has been investigated by many researchers (e.g.

see [14,15,21] and references therein). In contrast, there are few studies of codimension three or higher problems in the literature. This is perhaps due to the relative rarity in ODE models of higher codimension singularities. Abstract We present two codimension-one bifurcations that occur when an equilib-rium collides with a discontinuity in a piecewise smooth dynamical system.

These simple cases appear to have escaped recent classifications. We present them here to highlight some of the powerful results from Filippov’s book Differential Equations. We study the stability and possible bifurcations of MFI sets. In dimensions 1 and 2 we classify all minimal forward invariant sets and their codimension one bifurcations in bounded noise random differential equations.

1 Introduction We will consider bifurcations in a class of random differential equations (RDEs) x˙ = fλ(x,ξt), (1). 1. Yudovich, “ Cosymmetry, degeneracy of the solutions of operator equations, appearance of filtrational convection,” i 49, – ().

Google Scholar; 2. Yudovich, “ Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation, and impossibility of symmetric treatment of it,” Chaos 5, – (). The replicator equation is the first and most important game dynamics studied in connection with evolutionary game theory.

It was originally developed for symmetric games with finitely many strategies. Properties of these dynamics are briefly summarized for this case, including the convergence to and stability of the Nash equilibria and evolutionarily stable strategies. The bifurcation analysis of codimension one and two linear singularities has been investigated by many researchers (e.g., see [14, 15, 21] and references therein).

By contrast, there are few studies of codimension three or higher problems in the literature. This is perhaps due to the relative rarity in ODE models of higher codimension. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations.

This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system.

One approach is the amplitude equation is based on the linear instability of a homogeneous state and leads naturally to a classification of patterns in terms of characteristic wave numbers and frequencies. [from Kuramoto book ] Codimension-one, codimension-two bifurcations Lyapunov exponents v.s.

Floquet exponents. differential equations is contained in Chapters of this book and in or-der to cover the main ideas in those chapters in a one semester course, it is necessary to cover Chapter 1 as quickly as possible.

In addition to the new sections on center manifold and normal form theory, higher codimension bifurcations, higher order Melnikov theory, the. This textbook presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems.

Although the main topic of the book is the local and global behavior of nonlinear systems and their bifurcations, a thorough treatment of linear systems is given at the beginning of the s: respect to stability and local bifurcations generated by the variation of the involved parameters. We refer, for example, forms for planar systems, applied to -dimensional ones, is for degenerate Hopf bifurcations up to codimension.

e same authors deal with a three-parameter system, obtaining. The replicator equation is defined on the simplex S n, which is given by the set of all points (x 1,x n) with the property ∑ i = 1 n x i = 1.

The simplex S n is invariant under replicator dynamics: a trajectory which begins in the simplex, never leaves the simplex. Each face of the simplex, defined by one or several startegies being.Nonlinear systems: Bifurcation theory Structural stability and Peixoto's theorem Bifurcations at nonhyperbolic equilibrium points Higher codimension bifurcations at nonhyperbolic equilibrium points Hopf bifurcations and bifurcations of limit cycles from a multiple focus Bifurcations at nonhyperbolic periodic orbits.The use of delayed fitness functions makes the replicator equa-tion into the delay differential equation (DDE) x˙ =x (f f) (20) where f =å i x i f i =å i x i(Ax¯i) i: (21) As a system of ODEs, the standard replicator equation is an (n 1)- dimensional problem, since n 1 of the x i are required to specify a point in phase space.

The delayed.