Mathematics
Invariant Points
Invariant points are points that remain fixed or unchanged under a given transformation. In other words, they are points that are mapped onto themselves by the transformation. Invariant points are important in mathematics as they help to identify symmetries and other properties of geometric shapes.
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eBook - PDF- Jean-marc Ginoux(Author)
- 2009(Publication Date)
- World Scientific(Publisher)
6 Four degree algebraic equations have no more discriminant (excepted bi-squared equation). Chapter 9 Invariant Sets “Invention consists in avoiding the constructing of useless contraptions and in constructing the useful combinations which are in infinite minority.” — H. Poincar´ e — As pointed out in the first part (Ch. 3) invariant manifolds play a very important role in stability and structure of dynamical systems and especially for singularly perturbed systems . It will be established in this chapter that flow curvature manifold enables to find out linear invariant manifolds 1 ( straight lines, planes, hyperplanes ), i.e. first degree algebraic invariant manifolds while extatic algebraic manifolds provide higher degree algebraic invariant manifolds . Thus, using Darboux theory of integrability, such manifolds invariant with respect to the flow of n -dimensional dynam-ical systems will be used in order to built the corresponding first integrals . 9.1 Invariant manifolds Let’s focus on two concepts introduced by Gaston Darboux in his memoir (1878 page 71 and next) with the meaning of algebraic particular integral (Darboux invariance theorem, Ch. 6), general integral (Corollary, Ch. 6). Theorem 9.1. An algebraic particular integral in the sense of Darboux is an invariant manifold. Proof. Let φ ( X ) = 0 be a manifold where φ is a C 1 in an open set U and such there exists a C 1 polynomial function of degree m − 1 denoted k ( X ) called cofactor satisfying for all X ∈ U : L V φ ( X ) = k ( X ) φ ( X ). 1 This terminology has been suggested by Professor Dana Schlomiuk. 145 146 Differential Geometry Applied to Dynamical Systems The Lie derivative of the vector field V is also called derivative along the flow Φ t generated by the vector field ( X ). Thus ∀ X ∈ φ , Φ t ( X ) ∈ φ for all t ∈ R . According to Def. 3.1 manifold φ is therefore an invariant set. Theorem 9.2. A general integral in the sense of Darboux is a first integral.
eBook - PDFSources in the Development of Mathematics
Series and Products from the Fifteenth to the Twenty-first Century
- Ranjan Roy(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
34 Invariant Theory: Cayley and Sylvester 34.1 Preliminary Remarks The invariant theory of forms, with forms defined as homogeneous polynomials in several variables, was developed extensively in the nineteenth century as an important branch of algebra but with very close connections to algebraic geometry. Several ideas and methods of invariant theory were influential in diverse areas of mathematics: topics as concrete as enumerative combinatorics and the theory of partitions and as general as twentieth-century abstract commutative algebra. George Boole, the highly original British mathematician, may be taken as the founder of invariant theory, though early examples of the use of invariance can be found in the works of Lagrange, Laplace, and Gauss. Boole had almost no formal training in math- ematics, but he carefully studied the work of great mathematicians, including Newton, Lagrange, and Laplace. In a paper on analytic geometry written in 1839, Boole took the first tentative steps toward the idea of invariance, but he gave a clearly formulated definition in his 1841 “Exposition of a General Theory of Linear Transformations.” He wrote that he found his inspiration in Lagrange’s researches on the rotation of rigid bodies, contained in the 1788 Mécanique analytique. Lagrange’s result is most eco- nomically described in terms of matrices, a concept developed in the 1850s by Cayley. In modern terms, Lagrange’s problem was to diagonalize a 3 × 3 symmetric matrix A; Lagrange expressed this in terms of binary quadratic forms. Given a quadratic form x t Ax , with x a three vector, the problem would be to find a matrix P such that P P t = I , the identity matrix, and P t AP is a diagonal matrix. This means that if x 1 ,x 2 ,x 3 are the components of x , y 1 ,y 2 ,y 3 of y = P t x , and λ 1 ,λ 2 ,λ 3 are the diagonal entries in the diagonal matrix, then x t Ax = λ 1 y 2 1 + λ 2 y 2 2 + λ 3 y 2 3 , (34.1) x 2 1 + x 2 2 + x 2 3 = y 2 1 + y 2 2 + y 2 3 .
eBook - PDF- Bernd Aulbach, Fritz Colonius(Authors)
- 1996(Publication Date)
- World Scientific(Publisher)
An up-to-date list of areas in science and engineering to which invariant manifolds permeate can be found, e.g., in Wiggins [26]. In general, a dynamical system has many invariant manifolds. Certain of them are of prime importance for the study of the dynamical evolution of a system. The most commonly encountered types of such invariant manifolds are: 1. Equilibrium points; Supported by the Russian Fund of Fundamental Investigations under Grant 94-01-00294 and in part by Grant NWJOOO from the International Science Foundation. 213 214 George Osipenko 8c Eugene Ershov 2. Periodic orbits; 3. Invariant tori, which are the closure of quasiperiodic orbits with two or more basic frequencies; 4. Stable, unstable and center manifolds associated with equilibrium points, pe-riodic and quasiperiodic orbits. Invariant manifold theory begins by assuming or by proving that a dynamical system has some invariant manifold. The question arises at once of whether or not this invariant manifold persists under perturbation of a dynamical system. The mat-ter is that in most cases a dynamical system, which is the mathematical model of a real phenomenon, is known with a certain degree of inaccuracy. Therefore, the existence of an invariant manifold is of moderate significance if this manifold can be destroyed by a perturbation of the initial dynamical system. The question of whether or not a persistent invariant manifold maintains or loses differentiability also arises. The problem of perturbation of invariant manifolds has a long history in dynamical systems theory. Historical descriptions can be found in Bogoliubov and Mitropolsky [2], Mitropolsky and Lykova [16], Hirsch, Pugh, and Shub [7]. In the present paper we consider the problem of perturbation of invariant mani-folds of smooth dynamical systems given by a general autonomous ordinary differen-tial equation defined on R n : x = ^ = F 0 {x), x e R n , (1) where F 0 : RJ 1 —► RJ 1 is a C 1 vector field.
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