The 10th Seminar on Differential Equations and Dynamic Systems 6-7 November 2013, University of Mazandaran, Babolsar, Iran, pp xx-xx A brief survey on normal forms of singularities Majid Gazor1 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran E-mail: [email protected] Abstract. Normal form theory has been applied in different branches of mathematical sciences. Here, we provide a brief survey on a few from the many of its applications. Due to the broad applications' variety, we may hardly scratch the surface. Hence, our best goal here may perhaps be to partially demonstrate the theory's strength and to enlist a descriptive (of course not a complete) set of applications along with few appropriate references while our talk presentation will be only limited to the interesting and applied theory of singularities. This short reference list may only serve as an initial clue for tracking the extensive literature and this is by no means an introduction into the many applications from different disciplines. Keywords: Normal forms; vector fields; ordinary and Hamiltonian partial differential equations; unfolding theory; symmetry groups. 2010 MSC: 34C20, 34A34 1Phone: (98-311) 3913634; Fax: (98-311) 3912602; Email: [email protected] 1 2 1. Normal forms and unfolding theory Consider a family of objects in which their certain properties is up for the analysis. Then, one may consider an equivalence relation on the family such that the desired properties (up for the analysis) are preserved by the equivalence relation. Therefore, the analysis of the object may alternatively be performed by the investigation of any other representatives from its equivalence class. Of course, this is helpful when one can recognize the more accessible and simpler representatives from each class. The first two elementary examples of normal forms may be stated as integration by parts and Jordan canonical forms in the undergraduate calculus and Linear algebra courses. To describe the Jordan canonical forms, consider the space of all n by n matrices where the similarity of matrices constitutes the equivalence relation. Next, the Jordan canonical form of each matrix provides its normal form. Hence, any property of a linear transformation can be investigated by only considering its Jordan canonical matrix form representation. An equivalence relation is the intrinsic part of any normal forms and are mostly described by permissible changes of coordinates (usually invertible); however see re- sults from Dumortier et. al.. Once the equivalence relation is determined, certain representatives (determined by some rules) from each equivalent class is called a nor- mal form. In order to introduce the rules, one must have a deep understanding of the subject's purposes and any possible challenges. Such rules are usually called styles; see [10, 25, 26]. Therefore, different equivalence relations or different styles lead to essentially different results; this establishes the reason for parallel practices in different branches of mathematics. For instance, normal forms have been defined in non-autonomous ODEs [1, Subsections 2.3{2.5] and [35], piecewise smooth vector fields (Filippov systems) [19], stochastic dynamical systems [28], Hamiltonian partial differential equations (PDEs) [13], discrete dynamical systems [1], linear and nonlin- ear unfolding theory [1,5,10,15,16], singularity and catastrophe theories [14,18]. The most well noticed application of normal forms is due to the local bifurcation analysis of non-hyperbolic ordinary differential systems around an equilibrium or a periodic orbit; see [1,25,33]. Henri Poincar´ein 1879 was the first to raise the notion. He used near-identity transformations to linearize nonlinear hyperbolic autonomous ordinary differential equations and integrate the linearized system. Of course he was successful to linearize the systems when certain conditions (non-resonant) were as- sumed. Indeed, his results was a weaker version of Hartman-Grobman Theorem. Similar results were later obtained by S. Sternberg in 1957-8. Hyper-normalization of classical normal forms are feasible and the simplest normal forms associated with germs of smooth vector fields have yet been unanswered for many important singu- larities; [27]. The reader should be skeptical about some statements and the context that they are valid. Indeed, we have normal forms of vector fields in our mind throughout the paper (if we do not specifically include or exclude others), while they actually work well in many other contexts and perhaps they may not be applicable to some other disciplines. 3 These points are left at the reader's discretion. In addition, recall that normal form theory are mostly considered as a local theory and many normal forms are expected to diverge. Thus, most of our claims here have a local nature and/or refer to formal results; \formal results" here stand versus analytical results. (Convergent discussion of normal forms is beyond the scope of this article; see [8] for some appropriate references.) For example first integrals here mean formal first integrals and they may actually help via an approximation for the constant of the motion; this may be achieved via an optimal truncation of normal forms; see [21,22]. Despite the assumed local nature of the theory, the simplest normal forms provide a classification scheme and they have had substantially contributed to the understanding of both local and global analysis of vector fields; see [36]. For local bifurcation and stability analysis around an equilibrium, normal forms aim to find appropriate coordinates in which the logistic function assumes a simple representation and the analysis can be feasibly performed. The appropriate coordi- nates are usually found by first transforming the linear part of the system (linear part in terms of Taylor expansion at the equilibrium) into its Jordan canonical form and then, by consecutively applying a sequence of changes of near-identity coordinates. Of course, combination of the infinite changes of coordinates is possible given the well-known filtration topology. One may distinguish three different types of normal forms for autonomous ODEs, namely normal forms [32], orbital normal forms and parametric normal forms. In this context, normal forms refers to when only changes of state variables are used, and orbital normal forms further utilizes time rescaling in addition to new state variables. Parametric normal forms establishes a broader context that include both unfolding theory as well as finding normal forms for para- metric systems. Parametric normal forms applies reparametrization along with time rescaling and changes of state variables; see [6,8{12]. Unfolding theory would also fall within the scope of normal forms. Any small perturbation of an object (a system) may no longer stay equivalent to the normal form of the unperturbed system. Since small perturbations or errors are unavoidable in any real life problems, one has to look for ways to investigate not only the given problems but also any of its small deviations. By adding small perturbations to a given system, one obtains a parametric system (parameters are called unfolding parameters) which is called an unfolding for the original system. A versal unfolding of a system is an unfolding of the system such that any small perturbation of the system is equivalent to the versal unfolding system for some specified values of unfolding parameters. Of course, one is interested in a versal unfolding with minimum number of parameters (the number refers to the system's codimension). Such a versal unfolding is named (universal) mini-versal unfolding; e.g., see [1,18,25]. The universal unfolding of ODEs can be computed via parametric normal forms; see [6,8{12]. Parametric normal forms in the context of autonomous ODEs [10] is designed to obtain a universal unfolding for different singularities and also finding the simplest 4 normal forms for parametric systems. The traditional approach to deal with a para- metric system is described as follows. One folds the parametric system by setting the parameters to zero (assuming that parameters are just small perturbations) and then, one obtains the (classical) normal form of the folded system. Next, one tries to unfold (add small parametric linear terms to) the system using unfolding theory of matrices (linear unfolding). This however is not useful for practical engineering prob- lems due to having no explicit relations between the original parametric system and the unfolded normal form. This mainly signifies the necessity of parametric normal forms. Further for cases with additional nonlinear degeneracies, linear unfolding is not sufficient to provide unfolding for the normal form of the folded system. 2. Symmetries and normal forms One of the prominent applications of normal forms is to reveal any possible hidden symmetries and/or detect the existing symmetries of the system. Once a symmetry is detected, any further normalization must preserve the symmetry. Symmetric systems may be distinguished by two categories, namely equivariant and invariant systems. Most real life symmetry problems come from symmetries that have a compact Lie group representation. In this context, the equivalence relation must be defined in a way that the symmetry would be preserved by the relation. Since the relation are usually determined via a group of transformations, the transformation group must preserve the symmetry group in order that the normal form would work well. Certain other dynamical structures may also be thought as a symmetric structure and be treated the same way as the equivariant systems are dealt with. These include Hamiltonian [2], reversible, incompressible and/or conservative systems (see [8, 23, 29]), coupled cell networks [4,17,31], invariant and equivariant systems with different Lie group symmetries [1,18]. Introduction and classification of important families of vector fields, and their as- sociated differential systems, is one of the main and important features of our recent contributions. Dynamically meaningful decomposition of normal forms contributes to finding certain symmetries and dynamical structures.