A Brief Survey on Normal Forms of Singularities

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 results from Dumortier et. al.. Once the equivalence relation is determined, certain representatives (determined by some rules) from each equivalent class is called a normal 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 nonlinear 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 Poincaréin 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 assumed. 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 singularities; [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 coordinates 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 parametric 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 parametric 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 problems 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 associated 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. A main tool for finding a meaningful decomposition is a sl(2)-Lie algebra representation; see [3, 33]. Indeed, this can contribute to computing possible first integrals, Hamiltonian, Eulerian, or Lagrangian of the motion. For instance we in [7–9] decomposed a Hopf-zero normal form singularity into a conservative and a non-conservative part. Therefore, the normal form computation of a given Hopf-zero singularity develops an approach for finding the set of all incompressible normal form vector fields. An important new result (from our in-progress project) on parametric normal forms provides a method for computing any possible local first integrals; recall that detecting and computing first integrals of the motion is a fundamentally important achievement for dynamics analysis. 5

3. Individual applications There are many individual cases in different areas of Mathematics that have used normal forms. In this section we provide a few references as examples in which demonstrates the theory’s strength. Traction and approximation of possible invariant manifolds [22], dynamically meaningful decomposition of vector fields [3,7], (existence results for) solutions of certain PDEs [34] and/or finding their associated life-span [20], development of efficient numerical ODE solvers [30], state-dimension reduction [29], deriving examples and counter-examples in dynamical systems [24], etc. Golubitsky et. al. [18] describe how normal form classification and their associated analysis help to find organizing centers and modeling of real life problems. As for application of this theory in partial differential equations, normal forms for Hamiltonian partial differential equations have been extensively worked out; see [13] for a recent survey. Hunter and Infrim [20] recently used near identity transformations to put a Burgers–Hilbert equation into a normal form and finding an enhanced (compared with directly working with the original system) life spans for smooth solutions. Similarly, Shatah [34] applied a near identity transformations of the dependent variable for a nonlinear Klein-Gordon equation with nonzero quadratic terms. He removed the quadratic terms, at the expense of a cubic nonlinearity, and was able to solve the system. This approach proved that his original problem had the same as- ymptotic behavior as the linear Klein-Gordon equation. Liapunov-Schmidt reduction is an approach that may reduce the steady-state analysis of certain PDEs (and many other operators from different origins) into the roots’ study of smooth germs. This study falls within the scope of singularity theory that uses contact equivalence; unfolding and normal forms play the key and essential roles; see [18]. We have recently paid attention to this problem (in progress results). We are developing some tools into singularity theory using some notions similar to the parametric normal forms of vector fields. However, the required tools for our project is from an essentially different discipline, that is, algebraic geometry. Indeed, we must develop concepts such as comprehensive standard basis and standard basis system (that have intimate relations with comprehensive Gröbnerbasis and Gröbnerbasis system) for tangent spaces associated with germs of singularities in our project.

References 1. S.N. Chow, C.Z. Li, D. Wang, “Normal forms and bifurcation of planar vector fields,” Cambridge University Press, Cambridge, 1994. 2. R. C. Churchill, M. Kummer, A unified approach to linear and nonlinear normal forms for Hamiltonian systems, J. Symbolic Comput. 27 (1999) 49–131. 3. R. Cushman, J.A. Sanders, Nilpotent normal forms and representation theory of sl(2, R), Contemp. Math. Amer. Math. Soc., Providence, RI, 56 (1986) 31–51. 4. B. Dionne, M. Golubitsky, I. Stewart, Coupled cells with internal symmetry Part I: wreath products, Nonlinearity 9 (1996) 559–574. 5. F. Dumortier, S. Ibańez,˜ H. Kokubu, C. Simo´, About the unfolding of a Hopf-zero singularity, Discrete and Continuous Dynamical Systems - Series A 33 (2013) 4435–4471. 6

6. M. Gazor, M. Moazeni, Parametric normal forms for Bogdanov–Takens singularity, arXiv preprint arXiv:1304.7329 (2013). 7. M. Gazor, F. Mokhtari, Normal Forms of Hopf-zero singularity, arXiv preprint arXiv:1210.4467 (2012). 8. M. Gazor, F. Mokhtari, Volume-preserving normal forms of Hopf–zero singularity, Nonlin- earity 26 (2013) 2809–2832. 9. M. Gazor, F. Mokhtari, J.A. Sanders, Normal forms for Hopf-zero singularities with nonconservative nonlinear part, J. Differential Equations 254 (2013) 1571–1581. 10. M. Gazor, P. Yu, Spectral sequences and parametric normal forms, J. Differential Equations 252 (2012) 1003–1031. 11. M. Gazor, P. Yu, Formal decomposition method and parametric normal forms, Internat. J. Bifur. Chaos 11 (2010) 3487–3515. 12. M. Gazor, P. Yu, Infinite order parametric normal form of Hopf singularity, Internat. J. Bifur. Chaos 11 (2008) 3393–3408. 13. B. Grebert´ , Birkhoff normal form and Hamiltonian PDEs, arXiv preprint arXiv:math/0604132 (2006). 14. M. Golubitsky, An introduction to catastrophe theory and its applications, SIAM Rev. 20 (1978) 352–387. 15. M. Golubitsky, W.F. Langford, Classification and unfoldings of degenerate Hopf bifurcation, J. Differential Equations 41 (1981) 375–415. 16. M. Golubitsky and M. Krupa, Stability computations for nilpotent Hopf bifurcations in coupled cell systems, Internat. J. Bifur. Chaos 17 (2007) 2595-2603. 17. M. Golubitsky and C. Postlethwaite, Feed-forward networks, center manifolds, and forcing, Discrete and Continuous Dynamical Systems - Series A 32 (2012) 2913–2935. 18. M. Golubitsky, I. Stewart, D.G. Schaeffer, “Singularities and groups in bifurcation theory,” Vol. I and II. Applied Mathematical Sciences, 69. Springer-Verlag, New York, 1986 and 1988. 19. M. Guardia, T.M. Seara, M.A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, J. Differential Equations 250 (2011) 1967–2023. 20. J.K. Hunter, M. Ifrim, Enhanced life span of smooth solutions of a Burgers-Hilbert equation, SIAM J. MATH. ANAL. 44 (2012) 2039–2052. 21. G. Iooss, E. Lombardi, Polynomial normal forms with exponentially small remainder for analytic vector fields, J. Differential Equations 212 (2005) 1–61. 22. G. Iooss, E. Lombardi, Approximate invariant manifolds up to exponentially small terms, J. Differential Equations 248 (2010) 1410–1431. 23. J. Llibre, C. Pantazi, S. Walcher, First integrals of local analytic differential systems, Bull. Sci. Math. 136 (2012) 342–359. 24. K.R. Meyer, Counter-examples in dynamical systems via normal form theory, SIAM Rev. 28 (1986) 41–51. 25. J. Murdock, “Normal Forms and Unfoldings for Local Dynamical Systems,” Springer-Verlag, New York, 2003. 26. J. Murdock, Hypernormal form theory: foundations and algorithms, J. Differential Equations 205 (2004) 424–465. 27. J. Murdock, D. Malonza, An improved theory of asymtotic unfoldings, J. Differential Equations 247 (2009) 685–709. 28. N.S. Namachchivaya, Y.K. Lin, Method of stochastic normal forms, Internat. J. Non-Linear Mech. 26 (1991) 931–943. 29. J. Palacian,´ P. Yanguas, Reduction of polynomial planar Hamiltonians with quadratic unperturbed part, SIAM Rev. 42 (2000) 671–691. 7

30. J. Palacian,´ P. Yanguas, Reduction of polynomial Hamiltonians by the construction of formal integrals, Nonlinearity 13 (2000) 1021–1054. 31. B. Rink, J. Sanders, Coupled cell networks: Semigroups, Lie algebras and normal forms, Trans. Amer. Math. Soc. (2013) to appear. 32. J.A. Sanders, Normal form theory and spectral sequences, J. Differential Equations 192 (2003) 536–552. 33. J.A. Sanders, F. Verhulst, J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Second edition. Applied Mathematical Sciences, 59 Springer-Verlag, New York, 2007. 34. J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985) 685–696. 35. S. Siegmund, Normal forms for nonautonomous differential equations, J. Differential Equations 178 (2002) 541–573. 36. S. Wiggins, “Introduction to applied nonlinear dynamical systems and chaos,” Texts in Applied Mathematics, Springer-Verlag, New York, 1990.