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Document Server@UHasselt (58.026 recursos)

Repository of the University of Hasselt containing publications in the fields of statistics, computer science, information strategies and material from the Institute for behavioural sciences.

Dynamical Systems

Mostrando recursos 1 - 20 de 28

  1. Cyclicity of a fake saddle inside the quadratic vector fields

    DE MAESSCHALCK, Peter; Rebollo-Perdomo, S.; Torregrosa, J.
    This paper concerns the study of small-amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity. This degenerate singularity is also known as an impassable grain. The canonical form of the unperturbed vector field is like a degenerate flow box. Near the singularity, the phase portrait consists of parallel fibers, all but one of which have no singular points, and at the singular fiber, there is one node. We demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic. (C) 2014 Elsevier...

  2. Slow divergence integrals in generalized Li??nard equations near centers

    Huzak, Renato; De Maesschalck, Peter
    Using techniques from singular perturbations we show that for any n???6 and m???2 there are Li??nard equations {x??=y???F(x), y??=G(x)}, with F a polynomial of degree n and G a polynomial of degree m, having at least 2[n???2/2]+[m/2] hyperbolic limit cycles, where [???] denotes "the greatest integer equal or below".

  3. Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations

    HUZAK, Renato; DE MAESSCHALCK, Peter; DUMORTIER, Freddy
    In this paper we continue the study of ``large" small-amplitude limit cycles in slow-fast codimension 3 elliptic bifurcations which is initiated in [8]. Our treatment is based on blow-up and good normal forms.

  4. Slow Divergence Integrals in Classical Li??nard Equations Near Centers

    DE MAESSCHALCK, Peter; HUZAK, Renato
    We significantly improve lower bounds for the number of limit cycles for polynomial classical Li??nard equations, aiming at stating lower bounds that are reasonable enough to be optimal. The techniques used are the notion of slow divergence integral from the geometric theory of planar slow-fast systems.

  5. Canard cycle transition at a slow-fast passage through a jump point

    DE MAESSCHALCK, Peter; DUMORTIER, Freddy; Roussarie, Robert
    We introduce transitory canard cycles for slow???fast vector fields in the plane. Such cycles separate ???canards without head??? and ???canardswithhead???,like for example in the Van der Pol equation. We obtain optimal upper bounds on the number of periodic orbits that can appear near the cycle underwhatever condition on the related slow divergence integralI,including the challenging caseI=0.

  6. Gevrey asymptotics of series in Mourtada-type compensators used for linearization of an analytic 1:-1 resonant saddle

    Bonckaert, Patrick
    Given a 1:-1 resonant saddle singularity of a planar analytic vector field, we provide a linearization procedure using a series expansion in compensators of Mourtada-type, and show that this series has Gevrey-1 asymptotics. In case of an analytic Poincar\'e-Dulac normal form we show that this transformation is analytic as a function of the compensators

  7. Three Time-Scales In An Extended Bonhoeffer???Van Der Pol Oscillator

    DE MAESSCHALCK, Peter; Popovic, Nikola; KUTAFINA, Ekaterina
    We consider an extended three-dimensional Bonhoeffer-van der Pol oscillator which generalises the planar FitzHugh-Nagumo model from mathematical neuroscience, and which was recently studied by Sekikawa et al. and by Freire and Gallas. Focussing on a parameter regime which has hitherto been neglected, and in which the governing equations evolve on three distinct time-scales, we propose a reduction to a model problem that was formulated by Krupa et al. as a canonical form for such systems. Based on previously obtained results in, we characterise completely the mixed-mode dynamics of the resulting three time-scale extended Bonhoeffer-van der Pol oscillator from the point of...

  8. Mixed mode oscillations in the Bonhoeffer-van der Pol oscillator with weak periodic perturbation

    KUTAFINA, Ekaterina
    Following the paper of Shimizu et al. (Phys Lett A 375:1566, 2011), we consider the Bonhoeffer-van der Pol oscillator with non-autonomous periodic perturbation. We show that the presence of mixed mode oscillations reported in that paper can be explained using the geometric singular perturbation theory. The considered model can be re-written as a four-dimensional (locally three-dimensional) autonomous system, which under certain conditions has a folded saddle-node singularity and additionally can be treated as a three time scale one.

  9. Configurations of limit cycles in Lienard equations

    Coll, B.; DUMORTIER, Freddy; Prohens, R.
    We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cy-cles of a polynomial Liénard equation.The related vector field X is Morse–Smale. Moreover it has the minimum numberof singulari-ties required fo rrealizing the configuration in a Liénardequation. We provide an explicit upper bound on the degree of X, which is lower than the results obtained before, obtained in the context of general polynomial vector fields. ©2013ElsevierInc. All rights reserved.

  10. ABOUT THE UNFOLDING OF A HOPF-ZERO SINGULARITY

    Dumortier, Freddy; Ibanez, Santiago; Kokubu, Hiroshi; Simo, Carles
    We study arbitrary generic unfoldings of a Hopf-zero singularity of codimension two. They can be written in the following normal form: {x' = -y + mu x - axz + A(x,y,z,lambda,mu) y' = x + mu y - ayz + B(x,y,z,lambda,mu) z' = z(2) vertical bar lambda vertical bar b(x(2) vertical bar y(2)) vertical bar C(x,y,z,lambda,mu), with a > 0, b > 0 and where A, B, C are C-infinity or C-omega functions of order O(vertical bar vertical bar(x,y,z,lambda,mu)vertical bar vertical bar(3)). Despite that the existence of Shilnikov homoclinic orbits in unfoldings of Hopf-zero singularities has been discussed previously in...

  11. Canard-cycle transition at a fast-fast passage through a jump point

    DE MAESSCHALCK, Peter; DUMORTIER, Freddy; Roussarie, Robert
    We consider transitory canard cycles that consist of a generic breaking mechanism, i.e. a Hopf or a jump breaking mechanism, in combination with a fast-fast passage through a jump point. Such cycle separates two types of canard cycles with a different shape. We obtain upper bounds on the number of periodic orbits that can appear near the canard cycle, and this under very general conditions.

  12. Normal forms with exponentially small remainder and gevrey normalization for vector fields with a nilpotent linear part

    BONCKAERT, Patrick; VERSTRINGE, Freek
    We explore the convergence/divergence of the normal form for a singularity of a vector field on C-n with nilpotent linear part. We show that a Gevrey-alpha vector field X with a nilpotent linear part can be reduced to a normal form of Gevrey-1 + alpha type with the use of a Gevrey-1 + alpha transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

  13. Canard cycle transition at a slow-fast passage through a jump point

    De Maesschalck, Peter; Dumortier, Freddy; Roussarie, Robert
    We introduce transitory canard cycles for slow-fast vector elds in the plane. Such cycles separate "canards without head" and "canards with head", like for example in the Van der Pol equation. We obtain optimal upper bounds on the number of periodic orbits that can appear near the cycle under whatever condition on the related slow divergence integral I, including the challenging case I = 0.

  14. Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations

    De Maesschalck, Peter; Huzak, Renato; Dumortier, Freddy
    In this paper we study singular perturbation problems occuring in planar slow-fast systems. We distinguish between slow-fast elliptic and slow-fast saddle bifurcations of codimension 3. These bifurcations are singular limits of well-known codimension 3 bifurcations. In this paper we focus on the appearance of small-amplitude limit cycles, using tools from geometric singular perturbation theory, including blow-up.

  15. Gevrey normal forms for nilpotent contact points of order two

    DE MAESSCHALCK, Peter
    This paper deals with normal forms about contact points (turning points) of nilpotent type that one frequently encounters in the study of planar slow-fast systems. In case the contact point of an analytic slow-fast vector field is of order two, we prove that the slow-fast vector field can locally be written as a slow-fast Li??nard equation up to exponentially small error. The proof is based on the use of Gevrey asymptotics. Furthermore, for slow-fast jump points, we eliminate the exponentially small remainder.

  16. Detecting hypermotor seizures using extreme value statistics

    Luca, Stijn; Karsmakers, Peter; Cuppens, Kris; Croonenborghs, Tom; Van de Vel, Anouk; Ceulemans, Berten; Lagae, Lieven; Van Huffel, Sabine; Vanrumste, Bart
    Nocturnal home monitoring of epileptic children is often not feasible due to the cumbersome manner of seizure detection with the standard method of video/EEG-monitoring. We propose a method for hypermotor seizure detection based on accelerators attached to the extremities. Hypermotor seizures often involve violent movements with the arms or legs, which increases the need for an alarm system as the patient can injure himself during the seizure. In the literature, classification models are commonly estimated in a supervised manner. Such models are estimated using annotated examples. This annotation of data requires expert (human) interaction and results therefore in a substantial...

  17. Alien limit cycles in Lienard equations

    Coll, B.; Dumortier, F.; Prohens, R.
    This paper aims at providing an example of a family of polynomial Lienard equations exhibiting an alien limit cycle. This limit cycle is perturbed from a 2-saddle cycle in the boundary of an annulus of periodic orbits given by a Hamiltonian vector field. The Hamiltonian represents a truncated pendulum of degree 4. In comparison to a former polynomial example, not only the equations are simpler but a lot of tedious calculations can be avoided, making the example also interesting with respect to simplicity in treatment. (C) 2012 Elsevier Inc. All rights reserved.

  18. Sharp upperbounds for the number of large amplitude limit cycles in polynomial lienard sytems

    DUMORTIER, Freddy
    In [1] and [2] upperbounds have been given for the number of large amplitude limit cycles in polynomial Lienard systems of type (m, n) with m < 2n + 1, m and n odd. In the current paper we improve the upperbounds from [1] and [2] by one unity, obtaining sharp results. We therefore introduce the "method of cloning variables" that might be useful in other cyclicity problems.

  19. Numerical continuation techniques for planar slow-fast systems

    De Maesschalck, P.; Desroches, M.
    Continuation techniques have been known to successfully describe bifurcation diagrams appearing in slow-fast systems with more than one slow variable. In this paper we investigate the usefulness of numerical continuation techniques dealing with some solved and some open problems in the study of planar singular perturbations. More precisely, we first verify known theoretical results (thereby showing the reliability of the numerical tools) on the appearance of multiple limit cycles of relaxation-oscillation type and on the existence of multiple critical periods in well-chosen annuli of slow-fast periodic orbits in the plane. We then apply the technique to study a notion of...

  20. On the non-local hydrodynamic-type system and its soliton-like solution

    Kutafina, Ekaterina; Vladimirov, V.A.; Zorychta, B.
    We consider a hydrodynamic system of balance equations for mass and momentum. This system is closed by the dynamic equation of state, taking into account the effects of spatio-temporal non-localities. Using group theory reduction, we obtain a system of ODEs, describing a set of approximate traveling wave solutions to the source system. The factorized system, containing a small parameter, proves to be Hamiltonian when the parameter is zero. Using Melnikov's method, we show that the factorized system possesses, in general, a one-parameter family of homoclinic loops, corresponding to the approximate soliton-like solutions of the source system.

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