E. Benoit, J. L. Callot, F. Diener, M. Diener, Chasse au canard, Collect. Math., 31--32 (1--3), 37--119 (1981).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....tools of invariant manifold theory can be used. With that, asymptotic expressions for the ow near the singularity can by obtained. Generally the singularities considered are the fold (whichis the saddle node with a slow drift) the transcritical and pitchfork with a drift, and the canard [4], see also [2] 3] 7] 11] and [12] which involves a drift along an unstable critical manifold. An approach using invariant manifolds and hyperbolicity has also been developed by Liu [33] 34] 35] Our studies here concentrate on the simplest nontrivial singularity, the generic fold in ....

 E. Beno^t, J.L. Callot, F. Diener, and M. Diener, Chasse au canards, Collect. Math. ## (1981), pp. 37-119.

....that the mechanism of localization is based on the canard phenomenon that occurs in single relaxation oscillators. The canard phenomenon is a very rapid change in the amplitude of the limit cycle of a relaxation oscillator as the inhibitor nullcline moves with respect to the activator nullcline [11, 12, 13, 14, 15, 16]. It arises in the context of experiments and simulations of nonlinear chemical dynamics [17, 18, 19] and in a two pool model describing the mechanism of calcium induced calcium release [20, 21] The mechanism of localization in oscillatory systems, not of the relaxation type, has been studied in ....

.... the limit cycle, given by the maximum and minimum values of v and w (v min , v max , w min and w max ) as a function of for the sigmoid version of the FHN equations (5) This rapid change from a small amplitude limit cycle to a large amplitude limit cycle is known as the canard phenomenon [11, 12, 14, 15, 16, 27]. In this case the canard phenomenon has been induced by changes in . Here we concentrate on the canard phenomenon near v = v m . The canard phenomenon was discovered by Benoit et al. 14] for the VDP oscillator. In their work [14] they show that there exists a critical value c ( of such ....

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E. Benoit, J. L. Callot, F Dienner, and Dienner M. Chasse au Canard. IRMA, Strasbourg, 1980.

....we study a related Riccati equation. More precisely, the transformation y y 0 = Gamma 0 0 u yields an equation of the form u 0 = f(x)u P (x; u; 4) where f = Gamma 0 and P is analytic ( cf. 26) Equation (4) is of slow fast type and hence the theory of canard solutions [4] applies. The precise relation between canard solutions of (4) and resonant solutions of (1) is presented in section 4. This connection had already been established by J.L. Callot in his thesis [7] In order to simplify matters for readers unfamiliar with nonstandard analysis, we present ....

....0 ff exp i x 2 Gamma 2 2 j d : 8) If the asymptotic expansion of ff = ff( as tends to 0 is the zero series, then equation (7) has a formal solution without poles: The zero series. On the other hand, if log ff( tends to a limit l then the equation has a boundary ( but ee [4]) in the points x = Sigma p 2l. For example, if ff = exp i Gamma 1 p j then there are no canard solutions, if ff = exp i Gamma 1 j then there are local canard solutions but no global canard solutions whenever a Gamma p 2 and p 2 b. The above theorem 3 states that ....

E. Beno it, J.L. Callot, F. Diener, M. Diener, Chasse au canard, Collect. Math., 31, 1-3 (1981) 37-119.

....l equation de Riccati associ ee. Le changement d inconnue y 0 y = Gamma 0 0 u aboutit a une equation de la forme u 0 = f(x)u P (x; u; 4) avec f = Gamma 0 et P r eguli ere (cf. formule (24) Cette equation (4) est du type lentrapide, sur laquelle la th eorie des canards [4] s applique. Le lien pr ecis entre les canards de l equation (4) et les solutions r esonnantes de l equation (1) est etabli dans la partie 4. Ce lien avait d ej a et e etabli par J.L. Callot dans sa th ese [7] Cependant, pour faciliter la tache au lecteur classique , nous pr esentons ici des ....

....2 2 j d : 8) Si ff = ff( admet la s erie nulle pour d eveloppement asymptotique lorsque tend vers 0, alors l equation (6) a pour solution formelle la s erie nulle. En revanche, si ff est telle que la fonction log jff( j a une limite l quand 0, alors l equation pr esente une but ee [4] aux points Sigma p 2l. Par exemple, si ff = exp i Gamma 1 p j alors il n y a pas de canards, et si ff = exp i Gamma 1 j , alors il y a des canards locaux mais pas globaux d es que a Gamma p 2 et p 2 b. Le th eor eme 3 pr ec edent affirme qu il n y a pas de but ee pour ....

E. Beno it, J.L. Callot, F. Diener, M. Diener, Chasse au canard, Collect. Math., 31, 1-3 (1981) 37-119.

....= f(Z(t) C(t) t) Gamma f(C(t) t) Deltat=ffl. Remark 3.5 The results of this section all have analogues in the literature on differential equations, which is surveyed in [28] 9] gives an informal introduction to the theory. A proof of the differential analogue of our theorem is given in [3], and can also be found in [28] The proofs there are given for a slightly different (and more complex) setting; a generalisation of the van der Pol equation. The basic method is to use an exponential microscope to determine the behaviour of paths near the canard, and can be easily applied to ....

E. Benoit, J.L. Callot, F. Diener, and M. Diener, Chasse au canard, Collectanea Mathematica 31 (1980), 37--119.

....and also in the proposition we have suppressed the smooth dependence of fl and c ( on the parameter . In singularly perturbed systems an orbit like fl is called a (maximal) canard. More generally orbits which follow a repelling slow manifold for an O(1) amount of time are called canards [4, 6]. Canards were discovered and first studied by means of nonstandard analysis in the Van der Poloscillator [4] Canards arise in one parameter families of planar singularly perturbed systems if an equilibrium passes through the fold point as the parameter varies. For a treatment of this situation ....

....singularly perturbed systems an orbit like fl is called a (maximal) canard. More generally orbits which follow a repelling slow manifold for an O(1) amount of time are called canards [4, 6] Canards were discovered and first studied by means of nonstandard analysis in the Van der Poloscillator [4]. Canards arise in one parameter families of planar singularly perturbed systems if an equilibrium passes through the fold point as the parameter varies. For a treatment of this situation which also leads to the so called singular Hopf bifurcation by matched asymptotic expansions we refer to [8, ....

E. Benoit, J.-L. Callot, F. Diener, and M. Diener, Chasse au canard, Collect. Math., 31 (1981), pp. 37--119.

....of some point x 0 . More precisely, we want to study overstable solutions that tend, as 0, to some given slow curve of (1.1) say OE 0 (x) satisfying F (x; OE 0 (x) a 0 ; 0) 0. Questions of this nature appear in the study of phenomena like bifurcation delay [12] canard solutions [4] and adiabatic invariants [24] We propose a general framework for the local study of overstable solutions of (1.1) proving their existence and asymptotic properties under weak conditions. If the Jacobian A 0 (x) F y (x; OE 0 (x) a 0 ; 0) is invertible at x = x 0 , the existence of such ....

E. Benoit, J.L. Callot, F. Diener and M. Diener, Chasse au canard, Collect. Math. 31, 13 (1981) 37119.

....x 0 a un caract ere exceptionnel. Ce sont les solutions surstables, qui g en eralisent les canards. En France, l etude des equations diff erentielles lentes rapides a d emarr e sous l impulsion de Georges Reeb. Les canards ont et e d ecouverts pour la premi ere fois sur l equation de van der Pol [3]. Il a et e montr e dans un cadre r eel assez g en eral pour des familles a un param etre et sous certaines conditions de transversalit e les r esultats suivants. 1. L existence de solutions canards pour certaines valeurs du param etre est in eluctable. 2. Les valeurs a canard et les solutions ....

E. BENOiT, J.L. CALLOT, F. DIENER, M. DIENER, Chasse au canard, Collect. Math., 31, 1-3 (1981) 37-119.

....burning reaction led to an unexpected and striking fact: duck trajectory appeared as a model of a critical regime. The term canard or French duck is comparatively recent in scienti c literature. It has been introduced by French mathematicians investigating van der Pol s equation [3, 5]. A trajectory of a singularly perturbed system of dioeerential equations is called a duck trajectory, if it follows at rst a stable integral manifold, and then an unstable one. In both cases the passed distances are not in nitesimally small. Bibliographies on this theme can be found in [1, 6, ....

....at rst a stable integral manifold, and then an unstable one. In both cases the passed distances are not in nitesimally small. Bibliographies on this theme can be found in [1, 6, 13, 17] In most papers devoted to duck trajectories the non standard analysis is the main tool of investigations [2, 3, 17]. Therefore, the opinion is widely used that the duck trajectories are exotic objects and are of interest for the theory of dioeerential equations only. Applying duck trajectories for modelling critical regimes promoted us to solve some interesting and important problems of combustion theory [7, ....

E. Benoit, J.L. Callot, F. Diener and M. Diener. Chasse au canard. Collect. Math., 3132(13) (1981), pp. 37119.

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E. Beno^ t, J.L. Callot, F. Diener, M. Diener, Chasse au canard, Collect. Math., 31, 1-3 (1981) 37-119.

....of # is reached in the interior of some standard given, not too small, interval. Such functions are called visibles and they match the canards (with or without poles) of equation (2) see [4] In the case of a simple well as potential, the work of J.L. Callot shows with non standard methods(see[2,10], theexistenceoftheenergylevels,theirexponentiallysmall thickness, and he computes the first term of their asymptotic expansion (see [4, 14] In this paper, I will give an e#ective algorithm (and his program in Maple language) to compute the asymptotic expansion of the n th energy level, for ....

....L( # f(x, t, e) 0) has a singularity. In the generic case, this singularity is the transverse intersection of two branches, as in figure 1. # A canard is a solution of a slow fast vector field which first go along an attractive branch of the slow curve, next go along a repulsive branch (see [2, 10]) In figures 1 to 4, I took one specific equation which is interesting only for the pictures. The reader has to look for the qualitative behaviour of the curves and for the order of magnitude of the distances. The horizontal axis is the t axis, and the vertical one is the x axis. Figure 1: A ....

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E. Benot, J.L. Callot, F. Diener, and M. Diener. Chasse au canard. Collectanea Mathematica, 31--32(1--3):37--119, 1981.

....of 1. 1 A discussion of some notations of Nonstandard Analysis can be found at the end of the article The sudden change of size of the cycle near some ff i close to 1 lead E. Benoit, J.L. Callot, F. and M. Diener to the discovery of surprizing trajectories. The so called canards [2] are cycles that follow the cubic on a part containing appreciably the point A. These authors proved the following results. 1. There exists some ff 1 (not unique) called a long canard value such that the limit cycle follows the cubic until the point ( Gamma1; 2 3 ) is reached. 2. For ff 1 ....

....and the corollary of this section prove the statements of theorem 1. Notation. The article is written in the framework of Nonstandard Analysis in its axiomatic approach IST given by E.Nelson [15] Nevertheless this is not essential for understanding. Contrary to previous works on canards [2] the proofs presented here involve very classical tools. For the reader who is not acquainted with Nonstandard Analysis, here is a sort of dictionary which should not be used for other articles using nonstandard methods. Nonstandard approach Classical approach fixed i small positive number ....

E. Benoit, J.L. Callot, F. Diener and M. Diener, Chasse au canard, Collect. Math. 31, 1--3 (1981) 37--119.

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E. Benoit, J. L. Callot, F. Diener, M. Diener, Chasse au canard, Collect. Math., 31--32 (1--3), 37--119 (1981).

No context found.

E. BENO IT, J.L. CALLOT, F. DIENER, M. DIENER, Chasse au canard, Collect. Math., 31, 1-3 (1981) 37-119.

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