The space of initial conditions for linearisable mappings

Topological methods in nonlinear analysis

We consider the operator $F(u) = u' + f(t,u(t))$ acting on periodic real valued functions. Generically, critical points of $F$ are infinite dimensional Morin-like singularities and we provide operational characterizations of the singularities of different orders. A global Lyapunov-Schmidt decomposition of $F$ converts $F$ into adapted coordinates, $\Fbd(\tilde v, \overline u) = (\tilde v, \overline v)$, where $\tilde v$ is a function of average zero and both $\overline u$ and $\overline v$ are numbers. Thus, global geometric aspects of $F$ reduce to the study of a family of one-dimensional maps: we use this approach to obtain normal forms for several nonlinearities $f$. For example, we characterize autonomous nonlinearities giving rise to global folds and, in general, we show that $F$ is a global fold if all critical points are folds. Also, $f(t,x) = x^3 - x$, or, more generally, the Cafagna-Donati nonlinearity, yield global cusps; for $F$ interpreted as a map between appropriat...

Journal of Physics A: Mathematical and Theoretical, 2018

We examine the notion of anticonfinement and the role it has to play in the singularity analysis of discrete systems. A singularity is said to be anticonfined if singular values continue to arise indefinitely for the forward and backward iterations of a mapping, with only a finite number of iterates taking regular values in between. We show through several concrete examples that the behaviour of some anticonfined singularities is strongly related to the integrability properties of the discrete mappings in which they arise, and we explain how to use this information to decide on the integrability or non-integrability of the mapping.

2018

The work is devoted to the results of a fundamental study on the arithmetical plane of a broad special family of differential dynamic systems having polynomial right parts. Let those polynomials be a cubic and a square reciprocal forms. A task of a whole investigation was to find out all topologically different phase portraits in a Poincare circle and indicate close to coefficient criteria of them. To achieve this goal a Poincare method of the central and the orthogonal consecutive displays (or mappings) has been used. As a rezult more than 250 topologically different phase portraits in a total have been constructed. Every portrait we depict with a special table called a descriptive phase portrait. Each line of such a special table corresponds to one invariant cell of the phase portrait and describes its boundary, a source of its phase flow and a sink of it. All finite and infinitely remote singularities of dynamic systems under consideration were fully investigated. Namely infinite...

American Journal of Mathematics, 2010

It is shown how a selection of prominent results in singularity theory and differential geometry can be deduced from one theorem, the Rank Theorem for maps between spaces of power series.

Journal of Differential Equations, 2019

Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate timescale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates.

Advances in Mathematics and Applications, 2018

The aim of this paper is to provide a discussion on current directions of research involving typical singularities of 3D nonsmooth vector fields. A brief survey of known results is presented. The main purpose of this work is to describe the dynamical features of a fold-fold singularity in its most basic form and to give a complete and detailed proof of its local structural stability (or instability). In addition, classes of all topological types of a fold-fold singularity are intrinsically characterized. Such proof essentially follows firstly from some lines laid out by Colombo, García, Jeffrey, Teixeira and others and secondly offers a rigorous mathematical treatment under clear and crisp assumptions and solid arguments. One should to highlight that the geometric-topological methods employed lead us to the completely mathematical understanding of the dynamics around a T-singularity. This approach lends itself to applications in generic bifurcation theory. It is worth to say that such subject is still poorly understood in higher dimension.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 1996

In this paper we are concerned with the dynamics of noninvertible transformations of the plane. Three examples are explored and possibly a new bifurcation, or ''eruption,'' is described. A fundamental role is played by the interactions of fixed points and singular curves. Other critical elements in the phase space include periodic points and an invariant line. The dynamics along the invariant line, in two of the examples, reduces to the one-dimensional Newton's method which is conjugate to a degree two rational map. We also determine, computationally, the characteristic exponents for all of the systems. An unexpected coincidence is that the parameter range where the invariant line becomes neutrally stable, as measured by a zero Lyapunov exponent, coincides with the merging of a periodic point with a point on a singular curve.

Physics Letters A, 2005

Experiments observing the liquid surface in a vertically oscillating container have indicated that modeling the dynamics of such systems require maps that admit states at infinity. In this paper we investigate the bifurcations in such a map. We show that though such maps in general fall in the category of piecewise smooth maps, the mechanisms of bifurcations are quite different from those in other piecewise smooth maps. We obtain the conditions of occurrence of infinite states, and show that periodic orbits containing such states are superstable. We observe period-adding cascade in this system, and obtain the scaling law of the successive periodic windows.

Differential Equations, 2010

We study the topological structure of singular (in the sense of the Feigenbaum-Sharkovskii-Magnitskii theory) attractors of nonlinear dissipative systems of differential equations. We show that any such attractor is a stable nonperiodic trajectory lying on a twodimensional infinitely folded heteroclinic separatrix manifold generated by the unstable two-dimensional invariant manifold of the original singular cycle as the bifurcation parameter of the system varies. The results obtained for two-dimensional nonautonomous and threedimensional autonomous dissipative systems are generalized to autonomous multi-and infinitedimensional dissipative systems as well as to conservative (in particular, Hamiltonian) systems.

2014

In analogy to what happens in finite dimensions we state the Normal Form Theorem for ksingularities, introduced in the previous paper of the series. By means of that we study the local behaviour near a singularity i.e. we deduce local results of existence and multiplicity of solutions for the equation F (x) = y where F is a 0-Fredholm map and x belongs to a suitable neighbourhood of a singular point x o , once x o is identified as one of the three kinds of singularities defined in the first paper. To this end we also start to seek alternative strategies for the determination of the type of a given singularity according to our classification. Here we give a pointwise approach for lowerorder singularities which is coherent with the Thom-Boardman classification in finite dimensions. We conclude by applying the pointwise condition to a differential problem where, under suitable hypotheses, we determine a swallow's tail singularity (3-singularity).