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  • 1. Arruda, Lynnyngs Kelly
    et al.
    Lenells, Jonatan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Long-time asymptotics for the derivative nonlinear Schrodinger equation on the half-line2017In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 30, no 11, p. 4141-4172Article in journal (Refereed)
    Abstract [en]

    We derive asymptotic formulas for the solution of the derivative nonlinear Schrodinger equation on the half-line under the assumption that the initial and boundary values lie in the Schwartz class. The formulas clearly show the effect of the boundary on the solution. The approach is based on a nonlinear steepest descent analysis of an associated Riemann-Hilbert problem.

  • 2. Atkin, Max R.
    et al.
    Charlier, Christophe
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Zohren, Stefan
    On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble2018In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 31, no 4, p. 1155-1196Article in journal (Refereed)
    Abstract [en]

    We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the n x n Laguerre unitary ensemble. The probability that this ratio is greater than r > 1 is expressed in terms of an n x n Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as n -> infinity is found as an integral over (0, infinity) containing two functions q(1)(x) and q(2)(x). These functions satisfy a system of two coupled Painleve V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. We compute asymptotic behaviours of these functions as rx -> 0(+) and (r - 1)x -> infinity, as well as large n asymptotics for the associated Hankel determinants in several regimes of r and x.

  • 3. Beardmore, R. E.
    et al.
    Laister, R.
    Peplow, Andrew
    KTH, Superseded Departments, Vehicle Engineering.
    Trajectories of a DAE near a pseudo-equilibrium2004In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 17, no 1, p. 253-279Article in journal (Refereed)
    Abstract [en]

    We consider a class of differential-algebraic equations (DAEs) defined by analytic nonlinearities and study its singular solutions. The main assumption used is that the linearization of the DAE represents a Kronecker index-2 matrix pencil and that the constraint manifold has a quadratic fold along its singularity. From these assumptions we obtain a normal form for the DAE where the presence of the singularity and its effects on the dynamics of the problem are made explicit in the form of a quasi-linear differential equation. Subsequently, two distinct types of singular points are identified through which there pass exactly two analytic solutions: pseudo-nodes and pseudo-saddles. We also demonstrate that a singular point called a pseudo-node supports an uncountable infinity of solutions which are not analytic in general. Moreover, akin to known results in the literature for DAEs with singular equilibria, a degenerate singularity is found through which there passes one analytic solution such that the singular point in question is contained within a quasi-invariant manifold of solutions. We call this type of singularity a pseudo-centre and it provides not only a manifold of solutions which intersects the singularity, but also a local flow on that manifold which solves the DAE.

  • 4.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Attractors in the quasi-periodically perturbed quadratic family2012In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 25, no 5, p. 1537-1545Article in journal (Refereed)
    Abstract [en]

    We give a geometric description of an attractor arising in quasi-periodically perturbed maps T x [0, 1] (sic) (theta, x) bar right arrow (theta + omega, c(theta)x(1 - x)) is an element of T x [0, 1] for certain choices of smooth c : T -> [1.5, 4] and Diophantine omega. The existence of the 'strange' attractor was established in Bjerklov 2009 Commun. Math. Phys. 286 137.

  • 5.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Quasi-periodic perturbation of unimodal maps exhibiting an attracting 3-cycle2012In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 25, no 3, p. 683-741Article in journal (Refereed)
    Abstract [en]

    We study a class of smooth maps Phi : T x [0, 1]. T x [0, 1] of the form theta bar right arrow theta + omega x bar right arrow c(theta)h(x) where h : [0, 1] --> [0, 1] is a unimodal map exhibiting an attracting periodic point of prime period 3, and omega is irrational (T = R/Z). We show that the following phenomenon can occur for certain h and c : T --> R: There exists a single measurable function psi : T --> [0, 1] whose graph attracts (exponentially fast) a.e. (theta, x) is an element of T x [0, 1] under forward iterations of the map Phi. Moreover, the graph of psi is dense in a cylinder M subset of T x [0, 1]. Furthermore, for every integer n >= 1 there exists n distinct repelling continuous curves Gamma(k) : (theta, phi(k)(theta))(theta is an element of T), all lying in M, such that Phi(Gamma(k)) = Gamma(k+1) (k < n) and Phi(Gamma(n)) = Gamma(1). We give concrete examples where both c(theta) and h(x) are real-analytic, but in the analysis we only need that they are C-1. In our setting the function c(theta) will be very close to 1 for all theta outside a tiny interval; on the interval c(theta) > 1 makes a small bump. Thus we cause the perturbation of h by rare quasi-periodic kicking.

  • 6. Bjerklöv, Kristian
    et al.
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Universal asymptotics in hyperbolicity breakdown2008In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 21, no 3, p. 557-586Article in journal (Refereed)
    Abstract [en]

    We study a scenario for the disappearance of hyperbolicity of invariant tori in a class of quasi-periodic systems. In this scenario, the system loses hyperbolicity because two invariant directions come close to each other, losing their regularity. In a recent paper, based on numerical results, Haro and de la Llave (2006 Chaos 16 013120) discovered a quantitative universality in this scenario, namely, that the minimal angle between the two invariant directions has a power law dependence on the parameters and the exponents of the power law are universal. We present an analytic proof of this result.

  • 7. Bruell, Gabriele
    et al.
    Ehrnstrom, Mats
    Geyer, Anna
    Pei, Long
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Symmetric solutions of evolutionary partial differential equations2017In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 30, no 10, p. 3932-3950Article in journal (Refereed)
    Abstract [en]

    We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the equations. The first principle covers equations that allow for steady solutions and shows that any spatially symmetric solution is in fact steady with a speed determined by the motion of the axis of symmetry at the initial time. The second principle includes equations that admit breathers and steady waves, and therefore is less strong: it holds that the axes of symmetry are constant in time. The last principle is a mixed case, when the equation contains terms of the kind from both earlier principles, and there may be different outcomes; for a class of such equations one obtains that a spatially symmetric solution must be constant in both time and space. We list and give examples of more than 30 well-known equations and systems in one and several dimensions satisfying these principles; corresponding results for weak formulations of these equations may be attained using the same techniques. Our investigation is a generalisation of a local and one-dimensional version of the first principle from EhrnstrOm et al (2009 Int. Math. Res. Not. 2009 4578-96) to nonlocal equations, systems and higher dimensions, as well as a study of the standing and mixed cases.

  • 8. Constantin, A.
    et al.
    Ivanov, R. I.
    Lenells, Jonatan
    Institut für Angewandte Mathematik, Leibniz Universität Hannover, Hannover, Germany .
    Inverse scattering transform for the Degasperis-Procesi equation2010In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 23, no 10, p. 2559-2575Article in journal (Refereed)
    Abstract [en]

    We develop the inverse scattering transform (IST) method for the Degasperis- Procesi equation. The spectral problem is an sl(3) Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST, such as the construction of fundamental analytic solutions, the formulation of a Riemann-Hilbert problem, and the implementation of the dressing method, are presented.

  • 9.
    Duits, Maurice
    et al.
    CALTECH, Pasadena.
    Geudens, Dries
    Kuijlaars, Arno B. J.
    A vector equilibrium problem for the two-matrix model in the quartic/quadratic case2011In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 24, no 3, p. 951-993Article in journal (Refereed)
    Abstract [en]

    We consider the two sequences of biorthogonal polynomials (p(k,n))(k=0)(infinity) and (q(k,n))(k=0)(infinity) related to the Hermitian two-matrix model with potentials V (x) = x(2)/2 and W(y) = y(4)/4 + ty(2). From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p(n,n) as n -> infinity. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t = 0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behaviour for a certain negative value of t.

  • 10. Fokas, A. S.
    et al.
    Lenells, Jonatan
    Explicit soliton asymptotics for the Korteweg-de Vries equation on the half-line2010In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 23, no 4, p. 937-976Article in journal (Refereed)
    Abstract [en]

    There exists a distinctive class of physically significant evolution PDEs in one spatial dimension which can be treated analytically. A prototypical example of this class (which is referred to as integrable) is the Korteweg-de Vries equation. Integrable PDEs on the line can be analysed by the so-called inverse scattering transform (IST) method. A particularly powerful aspect of the IST is its ability to predict the large t behaviour of the solution. Namely, starting with initial data u(x, 0), IST implies that the solution u(x, t) asymptotes to a collection of solitons as t → ∞, x/t = O(1); moreover, the shapes and speeds of these solitons can be computed from u(x, 0) using only linear operations. One of the most important developments in this area has been the generalization of the IST from initial to initial-boundary value (IBV) problems formulated on the half-line. It can be shown that again u(x, t) asymptotes into a collection of solitons, where now the shapes and the speeds of these solitons depend both on u(x, 0) and on the given boundary conditions at x = 0. A major complication of IBV problems is that the computation of the shapes and speeds of the solitons involves the solution of a nonlinear Volterra integral equation. However, for a certain class of boundary conditions, called linearizable, this complication can be bypassed and the relevant computation is as effective as in the case of the problem on the line. Here, after reviewing the general theory for KdV, we analyse three different types of linearizable boundary conditions. For these cases, the initial conditions are (a) restrictions of one- and two-soliton solutions at t = 0; (b) profiles of certain exponential type and (c) box-shaped profiles. For each of these cases, by computing explicitly the shapes and the speeds of the asymptotic solitons, we elucidate the influence of the boundary.

  • 11. Froehlich, J.
    et al.
    Jonsson, B. Lars G.
    KTH, School of Electrical Engineering (EES), Electromagnetic Engineering.
    Lenzmann, E.
    Effective dynamics for boson stars2007In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 20, no 5, p. 1031-1075Article in journal (Refereed)
    Abstract [en]

    We study solutions close to solitary waves of the pseudo-relativistic Hartree equation describing boson stars under the influence of an external gravitational field. In particular, we analyse the long-time effective dynamics of such solutions. In essence, we establish a ( long-time) stability result for solutions describing boson stars that move under the influence of an external gravitational field. The proof of our main result tackles difficulties that are absent when deriving similar results on effective solitary wave motions for nonlinear Schrodinger equations or nonlinear wave equations. This is due to the fact that the pseudo-relativisitic Hartree equation does not exhibit Galilean or Lorentz covariance.

  • 12.
    Gaidash, Denis G.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Cylinder renormalization for Siegel discs and a constructive measurable Riemann mapping theorem2007In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 20, no 3, p. 713-741Article in journal (Refereed)
    Abstract [en]

    The boundary of the Siegel disc of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this explanation, the hyperbolicity of renormalization, has not yet been proved. The present work considers a cylinder renormalization-a novel type of renormalization for holomorphic maps with a Siegel disc which is better suited for a hyperbolicity proof. A key element of a cylinder renormalization of a holomorphic map is a conformal isomorphism of a dynamical quotient of a subset of C to a bi-infinite cylinder C/Z. A construction of this conformal isomorphism is an implicit procedure which can be performed using the measurable Riemann mapping theorem. We present a constructive proof of the measurable Riemann mapping theorem and obtain rigorous bounds on a numerical approximation of the desired conformal isomorphism. Such control of the uniformizing conformal coordinate is of key importance for a rigorous computer-assisted study of cylinder renormalization.

  • 13.
    Gaidashev, Denis
    et al.
    Uppsala univ., Uppsala, Sweden .
    Winckler, Björn
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Existence of a Lorenz renormalization fixed point of an arbitrary critical order2012In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 25, no 6, p. 1819-1841Article in journal (Refereed)
    Abstract [en]

    We present a proof of the existence of a renormalization fixed point for Lorenz maps of the simplest non-unimodal combinatorial type ({0, 1}, {1, 0, 0}) and with a critical point of arbitrary order rho > 1.

  • 14.
    Glendinning, P.
    et al.
    Univ Manchester, Manchester, Lancs, England.
    Kowalczyk, P.
    Manchester Metropolitan Univ, Manchester, Lancs, England.
    Nordmark, Arne B.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Attractors near grazing-sliding bifurcations2012In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 25, no 6, p. 1867-1885Article in journal (Refereed)
    Abstract [en]

    In this paper we prove, for the first time, that multistability can occur in three-dimensional Fillipov type flows due to grazing-sliding bifurcations. We do this by reducing the study of the dynamics of Filippov type flows around a grazing-sliding bifurcation to the study of appropriately defined one-dimensional maps. In particular, we prove the presence of three qualitatively different types of multiple attractors born in grazing-sliding bifurcations. Namely, a period-two orbit with a sliding segment may coexist with a chaotic attractor, two stable, period-two and period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three orbit with two sliding segments may coexist.

  • 15.
    Kurlberg, Pär
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Rosenzweig, Lior
    Rudnick, Zeev
    Matrix elements for the quantum cat map: fluctuations in short windows2007In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 20, no 10, p. 2289-2304Article in journal (Refereed)
    Abstract [en]

    We study fluctuations of the matrix coefficients for the quantized cat map. We consider the sum of matrix coefficients corresponding to eigenstates whose eigenphases lie in a randomly chosen window, assuming that the length of the window shrinks with Planck's constant. We show that if the length of the window is smaller than the square root of Planck's constant, but larger than the separation between distinct eigenphases, then the variance of this sum is proportional to the length of the window, with a proportionality constant which coincides with the variance of the individual matrix elements corresponding to Hecke eigenfunctions.

  • 16.
    Lenells, Jonatan
    et al.
    Baylor University, United States .
    Fokas, A. S.
    Boundary-value problems for the stationary axisymmetric Einstein equations: A rotating disc2011In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 24, no 1, p. 177-206Article in journal (Refereed)
    Abstract [en]

    The stationary, axisymmetric reduction of the vacuum Einstein equations, the so-called Ernst equation, is an integrable nonlinear PDE in two dimensions. There now exists a general method for analysing boundary-value problems (BVPs) for integrable PDEs, and this method consists of two steps: (a) Construct an integral representation of the solution characterized via a matrix RiemannHilbert (RH) problem formulated in the complex k-plane, where k denotes the spectral parameter of the associated Lax pair. This representation involves, in general, some unknown boundary values, thus the solution formula is not yet effective. (b) Characterize the unknown boundary values by analysing a certain equation called the global relation. This analysis involves, in general, the solution of a nonlinear problem; however, for certain BVPs called linearizable, it is possible to determine the unknown boundary values using only linear operations. Here, we employ the above methodology for the investigation of certain BVPs for the elliptic version of the Ernst equation. For this problem, the main novelty is the occurrence of the spectral parameter in the form of a square root and this necessitates the introduction of a two-sheeted Riemann surface for the formulation of the relevant RH problem. As a concrete application of the general formalism, it is shown that the particular BVP corresponding to the physically significant case of a rotating disc is a linearizable BVP. In this way the remarkable results of Neugebauer and Meinel are recovered.

  • 17.
    Lenells, Jonatan
    et al.
    University of Cambridge, United Kingdom .
    Fokas, A. S.
    On a novel integrable generalization of the nonlinear Schrödinger equation2009In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 22, no 1, p. 11-27Article in journal (Refereed)
    Abstract [en]

    We consider an integrable generalization of the nonlinear Schrödinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way as the Camassa-Holm equation is related to the KdV equation. In this paper we (a) use the bi-Hamiltonian structure to write down the first few conservation laws, (b) derive a Lax pair, (c) use the Lax pair to solve the initial value problem and (d) analyse solitons.

  • 18.
    Nordmark, Arne B.
    KTH, Superseded Departments, Mechanics.
    Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators2001In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 14, no 6, p. 1517-1542Article in journal (Refereed)
    Abstract [en]

    Grazing bifurcations are local bifurcations that can occur in dynamical models of impacting mechanical systems. The motion resulting from a grazing bifurcation can be complex. In this paper we discuss the creation of periodic orbits associated with grazing bifurcations, and we give sufficient conditions for the existence of a such a family of orbits. We also give a numerical example for an impacting system with one degree of freedom.

  • 19.
    Nordmark, Arne B.
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Biomechanics.
    Kowalczyk, P.
    A codimension-two scenario of sliding solutions in grazing-sliding bifurcations2006In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 19, no 1, p. 1-26Article in journal (Refereed)
    Abstract [en]

    This paper investigates codimension-two bifurcations that involve grazing-sliding and fold scenarios. An analytical unfolding of this novel codimension-two bifurcation is presented. Using the discontinuity mapping techniques it is shown that the fold curve is one-sided and cubically tangent to the grazing curve locally to the codimension-two point. This theory is then applied to explain the dynamics of a dry-friction oscillator where such a codimension-two point has been found. In particular, the presence and the character of essential bifurcation curves that merge at the codimension-two point are confirmed. This allows us to study the dynamics away from the codimension-two point using a piecewise affine approximation of the normal form for grazing-sliding bifurcations and explain the dynamics observed in the friction system.

  • 20.
    Rodrigues, Ana
    et al.
    CMUP and Departamento de Matemática Pura, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal .
    Dias, Ana Paula S.
    CMUP and Departamento de Matemática Pura, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal .
    Hopf bifurcation with SN-symmetry2009In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 22, no 3, p. 627-666Article in journal (Refereed)
    Abstract [en]

    We study Hopf bifurcation with SN-symmetry for the standard absolutely irreducible action of SN obtained from the action of SN by permutation of N coordinates. Stewart (1996 Symmetry methods in collisionless many-body problems, J. Nonlinear Sci. 6 543–63) obtains a classification theorem for the C-axial subgroups of SN × S1. We use this classification to prove the existence of branches of periodic solutions with C-axial symmetry in systems of ordinary differential equations with SN-symmetry posed on a direct sum of two such SN-absolutely irreducible representations, as a result of a Hopf bifurcation occurring as a real parameter is varied. We determine the (generic) conditions on the coefficients of the fifth order SN × S1-equivariant vector field that describe the stability and criticality of those solution branches. We finish this paper with an application to the cases N = 4 and N = 5.

  • 21. Runborg, Olof
    et al.
    Theodoropoulos, C.
    Kevrekidis, I. G.
    Effective bifurcation analysis: a time-stepper-based approach2002In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 15, no 2, p. 491-511Article in journal (Refereed)
    Abstract [en]

    We introduce a numerical approach to perform the effective (coarse-scale) bifurcation analysis of solutions of dissipative evolution equations with spatially varying coefficients. The advantage of this approach is that the 'coarse model' (the averaged, effective equation) need not be explicitly constructed. The method only uses a time-integrator code for the detailed problem and judicious choices of initial data and integration times; the bifurcation computations are based on the so-called recursive projection method.

  • 22.
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Domain of analyticity of normalizing transformations2006In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 19, no 7, p. 1581-1599Article in journal (Refereed)
    Abstract [en]

    We investigate questions of divergence or local convergence of (formal) normalizing transformations associated with the Birkhoff normal form (BNF) at the origin of a holomorphic Hamiltonian system. These questions are addressed for systems for which the BNF is a quadratic function H-Lambda = Sigma(d)(j=1) lambda(j) x(j) y(j), Lambda := (lambda(1),..., lambda(d)) being a non-resonant, either real or purely imaginary, vector. We prove that for a generic Lambda is an element of R-d or i Lambda is an element of R-d one can define Hamiltonians H = H-Lambda + (H) over cap satisfying the following properties: (i) H is real-analytic, holomorphic in the unit polydisc D(1), and H is defined arbitrarily close to H-Lambda, (ii) the BNF of H equals H-Lambda and (iii) any symplectic normalizing transformation diverges, or given any 0 < rho < 1 any normalizing transformation diverges outside the polydisc of radius rho, and there is a real-analytic normalizing transformation (converging in a smaller domain).

  • 23.
    Schnellmann, Daniel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Positive Lyapunov exponents for quadratic skew-products over  a Misiurewicz-Thurston map2009In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 22, p. 2681-2695Article in journal (Refereed)
    Abstract [en]

    We study a class of skew-products of quadratic maps-also called Viana maps-where the base dynamics is given by a high enough iteration of a Misiurewicz-Thurston quadratic map. We show that these systems admit two positive Lyapunov exponents.

  • 24. Thunberg, Hans
    Unfolding of chaotic unimodal maps and the parameter dependence of natural measures2001In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 14, no 2, p. 323-337Article in journal (Refereed)
    Abstract [en]

    We consider one-parameter families f(a) of interval maps, and discuss the structure in parameter space and the (dis)continuity properties of the natural measure as a function of the parameter near certain strongly chaotic maps (post-critically finite Misiurewicz maps and Benedicks-Carleson maps). In particular, it is shown that the mapping a bar right arrow mu (a) (the natural measure of f(a)) is severely discontinuous at these strongly cli8dtii: maps and is not continuous on any full measure set of parameters in full, generic families. Going in the other direction, it is also shown that if such a chaotic map has a measure for which the critical point is generic, then this measure can be,approximated with measures supported on periodic attractors of nearby maps. The main idea is to construct cascades of post-critically finite Misiurewicz map and cascades of maps with periodic attractors, whose critical oibits reproduce various invariant sets of the unperturbed map. In the special case of the quadratic family, generalizations can be made to any non-renormalizable maps.

  • 25.
    Timoudas, Thomas Ohlson
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Power law asymptotics in the creation of strange attractors in the quasi-periodically forced quadratic family2017In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 30, no 12, p. 4483-4522Article in journal (Refereed)
    Abstract [en]

    Let Phi be a quasi-periodically forced quadratic map, where the rotation constant omega is a Diophantine irrational. A strange non-chaotic attractor (SNA) is an invariant (under Phi) attracting graph of a nowhere continuous measurable function psi from the circle T to [0, 1]. This paper investigates how a smooth attractor degenerates into a strange one, as a parameter beta approaches a critical value beta(0), and the asymptotics behind the bifurcation of the attractor from smooth to strange. In our model, the cause of the strange attractor is a so-called torus collision, whereby an attractor collides with a repeller. Our results show that the asymptotic minimum distance between the two colliding invariant curves decreases linearly in the parameter beta, as beta approaches the critical parameter value beta(0) from below. Furthermore, we have been able to show that the asymptotic growth of the supremum of the derivative of the attracting graph is asymptotically bounded from both sides by a constant times the reciprocal of the square root of the minimum distance above.

  • 26.
    Winckler, Björn
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A renormalization fixed point for Lorenz maps2010In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 23, no 6, p. 1291-1302Article in journal (Refereed)
    Abstract [en]

    A Lorenz map is a Poincare map for a three-dimensional Lorenz flow. We describe the theory of renormalization for Lorenz maps with a critical point and prove that a restriction of the renormalization operator acting on such maps has a hyperbolic fixed point. The proof is computer assisted and we include a detailed exposition on how to make rigorous estimates using a computer as well as the implementation of the estimates.

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