N. Fenichel, Geometric singular perturbation theory for ordinary di#erential equations, J. Di#. Eq., 31, 53--98 (1979).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....equilibria are saddles. In the whole (2n 2) dimensional phase space of the system (4) we get the following picture: at = 0 the system has an invariant manifold (namely, the slow manifold) lled with equilibria, and this manifold is normally hyperbolic in the usual sense, see, for example, [HPSh77, Fen79]. The normal hyperbolicity implies that this submanifold persists for 0 small enough being a symplectic (with respect to the restriction to it of 2 form dy dx 1 P n i=1 dv i du i ) invariant normally hyperbolic submanifold, and the restriction of the system (4) to this submanifold is an ....

N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J. Di. Equa., 31 (1979), pp.53-98.

.... u 3 persists and is in fact the stable manifold of (u 3 ; v 3 ) A heteroclinic wave exists i the two manifolds intersect and (since the are both one dimensional) coincide near u 2 . Since at the point (u 2 ; v 2 ) normal hyperbolicity of the slow manifold fails, the analysis along the lines of [Fen79] is not possible but using the method of rotated vector elds [Duf53, Per93] it is possible to show that for small, there is a unique value s( such that a heteroclinic orbit from u 1 to u 3 exists. A basic phase plane analysis or a version of the blow up method of Krupa Szmolyan [KS99] shows ....

N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J.Di.Eq., 31:53-98, 1979.

....= f(x; y; y = g(x; y; 1:1) upon rescaling time by #= 0. Here (x; y) # IR and # IR is a parameter close to zero. Systems such as (1. 1) arise in a tremendous number of problems, and a great many techniques and results have been developed to study them (see, for example, 13] [14], 18] 26] 31] 32] 37] 40] and [45] In this paper we seek to understand when two such systems are equivalentinthesense of having the same (homeomorphic or di eomorphic) phase portraits, much as in the spirit of the classical Hartman Grobman theorem [16] 19] Our approach is also ....

....generally giverisetosuch diculties. Problems (1. 1) in which the slow manifold g(x; y; 0) 0 is normally hyperbolic, that is Re #=0for all eigenvalues of D y g(x; y; 0) whenever g(x; y; 0) 0,are quite readily treated using the invariant manifold and invariant foliation theory of Fenichel [14]. Indeed, suppose one has g(x; # (x) 0) 0 identically along some smooth manifold y = # (x) which is normally hyperbolic for x in some open subset U of IR . Then there exists for any open subset V with compact closure V # U , and for near zero, a smooth invariantcenter manifold y = x; ....

N. Fenichel, Geometric singular perturbation theory for ordinary dierential equations, J. Di. Eq. ## (1979), pp. 53-98.

....y(t) 0; y) 2.5) Similarly, but going backwards in time instead, we obtain the foliation of the centerunstable manifold W by strong unstable manifolds W (y) See for example [HPS77] y x 1 x 2 Figure 2.1: A normally hyperbolic line of equilibria with ow invariant foliation. Fen77, Fen79] Aul84] for additional background and technical details. Tangent spaces to W (y) and W (y) at (0; y) for example, are given by the eigenspaces of A(y) corresponding to the spectrum strictly in the left and right complex half plane, respectively. By (2.4) these eigenvalues are precisely ....

N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J. Di. Eq., 31:53-89, 1979.

....= 0 u = u0 vc = 0 hence v = v0 c wc = w sin u c wc = w sin u0 ( vfi )o (vfi)o Thus, the critical points (slow manifold) of the system are w0 = sinu0, which are attracting if tic 0. This manifold is also clearly normally hyperbolic if tic O. So for e small and tic 0, Fenichel s Theorem [7] gives the existence of an invariant (center) manifold w = h(u, v, sin u ehl (u, v, E) where h (u, v, e) satisfies h(u,v,e) 7 v Hence in lowest order, The dynamics on the slow manifold is given by v cosu (V ] s u u v (sinu zh) vv] v (2.6) v = sinu eh(u,v,O) O(e 2) This ....

N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eq., 31:53-98, 1979.

....method has been developed further and applied, in particular, to several problems of enzyme kinetics and metabolism in Refs. 44, 51, 52, 16, 54, 55, 56, 57] A natural framework for the analysis of these and similar reduction meth3 ods is provided by geometric singular perturbation theory (GSPT) [14, 58, 22, 25]. The presence of a fast and a slow time scale leads naturally to the introduction of a small positive parameter measuring the ratio of the characteristic times. If, in the limit as # 0 (infinite separation of time scales) the system of kinetics equations has a slow manifold, M 0 , in phase ....

....each y 2 K) and GSPT can be applied to each local portion. In the absence of singularities, these local functions can be pieced together to form a smooth global function over the entire domain under consideration. Under the above conditions, standard asymptotic theory (see, for example, Refs. [69, 30, 14, 5, 45, 41, 22]) guarantees that, when is positive but arbitrarily small, there exists a slow manifold M that is invariant under the dynamics of the system of Eqs. 2.1) 2.2) has the same dimension as M 0 , and lies near M 0 . All nearby solutions relax exponentially fast to M , and their long term ....

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N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq. 31 (1979) 53--98

....relevant pulse solution satis es lim x 1 jA(x; t)j = lim x 1 jB(x; t)j = 0. Therefore, we look for homoclinic solutions h ( to (1.5) that satisfy lim 1 h ( 0; 0; 0; 0) where S = 0; 0; 0; 0) is a xed point of (1. 5) By the methods of geometric singular perturbation theory [12, 18] the existence of such homoclinic solutions, and thus the persistence of the limit pulse, is established. We rst consider the system for = 0 and show for all 0 and 6= 0 that there is a unique homoclinic orbit h ( that merges with the scalar pulse in the limit 0. The associated ....

....pulses of the uncoupled Ginzburg Landau equation may not persist, or persist in two di erent forms when 6= 0. In section 2.5 we will give a more precise description of this situation. Sections 2.1 to 2. 4 are devoted to the proof of both Theorems using a geometrical singular perturbation approach [12, 18]. 2.1 The reduced fast and slow systems The fast reduced system (2.2) is in essence a two parameter (b 0 and d 0 ) family of planar, integrable systems (Figure 1) Since we are interested in homoclinic orbits, we assume throughout this paper that 1 b 0 0: 2.5) In that case, 2:2) possesses ....

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N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J. Dierential Equations, 31(1):53-98, 1979.

....= Av(v ) 1 v) for which v relaxes rapidly onto the attracting parts of the curve w = f(v) w0 , followed by slow evolution along this curve until near a turning point. For such problems one can do much better than just approximations. The theory of normally hyperbolic invariant manifolds applies [Fen], which proves that there is a locally invariant slow manifold nearby, together with many other useful results (smoothness, persistence, forwards and backwards contracting foliations, local linearisability in the normal direction, and local maximality and uniqueness in case of full invariance; ....

Fenichel N, Geometric singular perturbation theory for ordinary dierential equations, J Di Eq 31 (1979) 53-98.

....part of the level set k(x; y) 0. For simplicity the functions f and g, which depend upon a parameter 2 R , are chosen to preserve these lines, but since Gamma 0 is normally hyperbolic, a nearby slow manifold Gamma = Gamma 0 O( is preserved for any C small perturbations f; g [4]. Our main goal is to study the stable and unstable manifolds W ( Gamma ) W ( Gamma ) of Gamma and their intersections: the homoclinic and heteroclinic orbits referred to above. In [3] we took the quadratic functions f = y(b cz) g = aex z , for which a saddle node bifurcation ....

.... Gamma = Gamma = fx = y = 0g. For = 0 there exists a homoclinic manifold H, corresponding to k = 0 (1.2) which connects Gamma to itself. For 0, H splits into stable (resp. unstable) manifolds W ( Gamma) compact pieces of which contain orbits O( close to H for t 0 (resp. t 0) [4]. We study W ( Gamma) via their intersections with the Poincar e cross section V = fy = 0; 1 x O( g. We denote the first intersections of W ( Gamma) with fy = 0g, lying O( close to the component fy = 0; x = 2 g of H fy = 0g, by P ( Gamma) respectively P ( Gamma) P is ....

N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eq., 31:53--98, 1979.

.... unstable manifolds, W ( Gamma) of a certain slow manifold Gamma can be considered as the backbone of the geometrical approach (section 3 and 4) This slow manifold must exist since we study the ODE reductions near an integrable limit which possesses a family of homoclinic orbits (section 3, [13], 33] This interpretation allows us to prove the existence of large families of multi circuit heteroclinic orbits which so far has not been noted in the literature. The existence of these orbits follows from a theorem which was already proved in [8] for a fairly simple model problem. In ....

....) g: 2.19) The curve C sn is tangent to C d as X . In section 5 we will study these, and other, curves in more detail. Note that all perturbed saddle points are on the slow manifold Gamma which exists O( near the curve of saddle points to the unperturbed system (see subsection 3. 2, [13], 33] The direction of the flow on Gamma is determined by z and thus by the weak (O( eigenvalues of the perturbed saddles. The slow manifold Gamma is not influenced by the degeneration at z = 0. 3 Perturbation analysis 3.1 The approximation of a Poincar e map Since we are only ....

N. Fenichel (1979) Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq. 31 53-98.

....0. Both, most stable (see [20] Stokes wave solutions, A = const. B j 0) and (B = const. A j 0) satisfy 2 = 0 and are thus described by (1. 8) This 4 dimensional system can be analysed (for instance) by the geometric theory for singularly perturbed systems, originally developed by Fenichel [11], see also the contribution of Jones to [1] Due to the results of Fenichel we establish the existence of two so called slow, invariant, manifolds Gamma l and Gamma r . We find a very rich structure of heteroclinic and homoclinic orbits which jump up and down between Gamma l and Gamma r . ....

....of distinct slow and fast parts. A solution evolves slowly if it is close to a so called slow manifold of the system. A slow manifold is an invariant 19 manifold on which the flow is O(ffi) slow. The existence of these manifolds follows from the theory originally developed by Fenichel (see [11] and references there, or [1] There it is shown that a manifold of critical points of the unperturbed limit ffi 0 with a normally hyperbolic structure persists under the perturbation as a slow manifold. These slow manifolds play an important role in the organisation of the total flow induced ....

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N. Fenichel (1979) Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq. 31 53-98.

....are mostly of a geometrical nature, the problems when taking the next step will have a computational, but not a conceptual, character. In order to describe the homoclinic saddle node bifurcation to be studied in this paper, we first need to appeal to some fundamental results due to Fenichel ([8], 12] Since Gamma (1.2) is normally hyperbolic in the limit 0, both Gamma and its (two dimensional) stable and unstable manifolds, W ( Gamma) persist under the perturbation (see section 2 for more details: we do not need to distinguish between Gamma 0 (i.e. 0) and Gamma ....

....the fast flow counteracts the slow flow. The approach of the analysis in this paper is mainly geometrical and does not rely on a specific form of the model system. This includes the quantitative results (Theorems 4.2, 5. 1) these are also based on a geometrical singular perturbation theory ([8], 15] 13] 12] This approach enabled us to unravel much of the complex structure of W ( Gamma) in (1.1) Due to the nature of the analysis in this paper, it is likely that a similar complexity can be observed in a more general class of singularly perturbed systems. Remark 1.1 The ....

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N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eq., 31:53--98, 1979.

....cycles, and of the way their properties scale with . It relies both on existing and new results, including in particular the following: ffl If x ( is a hyperbolic equilibrium branch of (1) one contructs a particular solution of (3) tracking this branch at a distance of order . See [14, 15] for related results. ffl The motion near this particular solution is analysed by local methods such as dynamic diagonalization of the equation, and the construction of invariant manifolds (see Section 3) Results in this direction have been obtained in [9] and [14] In [5] we extend them to a ....

....a distance of order . See [14, 15] for related results. ffl The motion near this particular solution is analysed by local methods such as dynamic diagonalization of the equation, and the construction of invariant manifolds (see Section 3) Results in this direction have been obtained in [9] and [14]. In [5] we extend them to a more contructive method, allowing to determine solutions up to exponentially small order in (see [6] for a summary) ffl The motion near bifurcation points, which is responsible for hysteresis and nontrivial scaling laws, is rst simplied by a center manifold ....

N. Fenichel, Geometric singular perturbation theory for ordinary dioeerential equations, J. Dioe. Eq. 31:5398 (1979).

....these equilibria is a saddle. In the whole (2n 2) dimensional phase space of the system (4) we get the following picture: at = 0 the system has an invariant manifold (namely, the slow manifold) filled with equilibria, and this manifold is normally hyperbolic in the usual sense, see, for example, [HPSh77, Fen79]. The normal hyperbolicity implies that this submanifold persists for 0 small enough being a symplectic (with respect to the 2 form dy dx Gamma1 P n i=1 dv i du i ) invariant normally hyperbolic submanifold, and the restriction of the system (4) to this submanifold is an n ....

N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Diff. Equa., 31 (1979), pp.53-98.

....Newtonian equations. By the de nition of as the reciprocal of a power of the speed of light, it is reasonable to assume that is a small parameter, at least in the low velocity regime. To recover the correct Newtonian model, we can apply Fenichel s geometric singular perturbation theory (see [16, 17]) 9 For the electrodynamic oscillator, we have the equivalent (singularly perturbed) rst order system given by x = y; y = z; z = z x 1: 4) For the gravitodynamic oscillator, the appropriate rst order system is given by z = u; u = v; v = w; w = x; x = 1 z v ....

Fenichel N (1979) Geometric singular perturbation theory for ordinary dierential equation J. Di. Eqs. 31 53-98.

....system which is linearly perturbed. The dynamics consists of slow fast dynamics. The fast dynamics corresponds to the motion on two spheres described in the above paragraph. The slow dynamics is the motion from one sphere to another along the direction of the curves of critical points. In [7], Fenichel proved the existence of an invariant manifold where the slow dynamics takes place. This slow manifold is actually a perturbation of the manifold of equilibria which exists for the unperturbed case. The conditions that have to be satisfied are the unperturbed manifold should be normally ....

Fenichel, N., Geometric Singular Perturbation Theory for Ordinary Di#erential Equations, Journal of Di#erential Equations 31, pp. 53-98, 1979.

....that the w components continue to be O(# ) apart through this part of the trajectory. Lemma 3.2. If the conclusion of Lemma 3.1 is satisfied, then w 1 (T 1 ) and w 2 (T 2 ) are O(# ) apart (see Fig. 4) Proof. We need to apply results from Fenichel s geometric singular perturbation theory [6]. Before applying this theory, we briefly summarize the results which are needed. For small # , there is a center manifold R # for (2.6) that is O(# ) close to the right branch R 0 of the cubic C 1 .Ina neighborhood of R # , there is a Fenichel fibration. This associates to each p 0 R # a ....

....i 1, 2, 3. Recall that these are the points on R # which have the same asymptotic phase as cell i. Because of our choice of # com , cell 1 now has the largest w value. The other two do not reverse orientation, so w 2 (see Fig. 10) We now need results concerning the Fenichel fibration [6]. These results state that the fibers depend smoothly on both # and their position along R # . Since the fibers are horizontal when # 0, it follows that there exists M 6 such that, for # sufficiently small, #(1 M 2 ) Let ( w i (t) be the solution of (2.6) which passes through p ....

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1) (1979) 53--98.

....motion is sometimes called the boundary layer behaviour. For larger times, solutions of (1.1) remain in an neighbourhood of the slow manifold, and are thus well approximated by solutions of the reduced equation (1. 3) This result was rst proved by Grad ste n [15] and Tihonov [26] Fenichel [11] has given results allowing for a geometrical description of these phenomena in terms of invariant manifolds. He showed, in particular, the existence of an invariant manifold x = x(y; with x(y; x (y) O( 1.6) for suciently small , whenever x (y) is a family of hyperbolic ....

....which remain in a neighbourhood of order of the slow manifold. If, moreover, the slow manifold is asymptotically stable, then the solutions starting in a neighbourhood of order 1 of the slow manifold converge exponentially fast in t= to an neighbourhood of the slow manifold. Fenichel [11] has given extensions of this result based on a geometrical approach. If (2.3) admits a hyperbolic slow manifold, then there exists, for suciently small , an invariant manifold y = x(y; x (y) O( y 2 D 0 : 2.4) Here invariant means that if y 0 2 D 0 and x 0 = x(y 0 ; then x t ....

N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J. Dierential Equations, 31(1):53-98, 1979.

....implemented with methods which are natural for pde s. Remark 2: Fiber representions are constructions from geometric singular perturbation theory which permit one to follow the long time fate of the full motion with an orbit totally restricted to a slow manifold. These were developed by Fenichel [65, 66, 67, 68] to provide a geometric understanding of singular perturbation methods, such as those of Howard and Kopell [110, 92] Recent descriptions of these fibrations, with explicit examples, may be found in [97, 145, 147] Remark 3: The argument is a shooting method , with the final target the stable ....

N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J Diff Eqns, 31:53--98, 1979.

....Wilde and Kokotovic (1972) applied dichotomic transformations to linear optimal control problems. Chow (1979) used dichotomic transformations for nonlinear two time scale HBVPs in standard form with linear boundary layer dynamics. We use the geometric characterization of two time scale systems by Fenichel (1979) and the stable and unstable sub bundles of Sacker and Sell (1980) to extend the use of dichotomic transformations to nonlinear HBVPs. Although we only present a method for completely hyper sensitive problems here, our approach has the potential for extension to the general two time scale case as ....

....eigenspace and an n dimensional unstable eigenspace. By allowing the base space Q to be an overflowing invariant manifold with a boundary, a segment of the stable manifold (with time reversed) or the unstable manifold, that includes the equilibrium point, can be taken as the base space (see Fenichel, 1979). We go a step further and modify the dichotomy definition such that the base space can be a non invariant, bounded, open set. In particular, let #LP be a neighborhood of the optimal phase trajectory, composed of extremals of duration t # or longer. Definition 3. The linear Hamiltonian flow on ....

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Fenichel, N. (1979). Geometric singular perturbation theory for ordinary di#erential equations. Journal of Di#erential Equations, 31,53---98.

....OE Gamma OE 0 and ffl 2 0:1 in case of bond angle bending. In the context of hyperbolic PDEs, the same expression is related to the high frequency modes in the Fourier spectrum of the solutions. Differential equations of the form (1) fall into the class of singularly perturbed systems of type [4] d dt z = 1 ffl f(z; ffl) 5) see Section 2 for more details) Solutions of (5) satisfy, in general, jz(t)j = O(1) and j d dt z(t)j = O(ffl Gamma1 ) i.e. they are bounded but vary rapidly in t. Thus the step size of a numerical integrator has, in general, to be of order O(ffl) This ....

....requirement is however that 2. M 0 is an exponentially stable manifold of the differential equation d dt z = f(z; 0) Under the Assumptions 2, one can show that there exists a family M ffl of smooth manifolds with M ffl=0 = M 0 such that M ffl is an exponentially stable invariant manifold of (5) [4]. Furthermore, the solutions on M ffl reflect the long time behavior of the general solutions of (5) with initial values in a ffi neighborhood of M ffl up to terms of order O(ffi ffl) Since the solutions on M ffl satisfy now dz=dt = O(1) time steps of order O(1) can be used in a numerical ....

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Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J.D.E., 31, 53--98, 1979.

....of , h and n with nh T . We note that these orders of approximation are the same as stated in Theorem 1. The way to circumvent the missing uniformity in the error bounds, is to make use of attractive invariant manifolds. As is known from the geometric theory of singular perturbation problems [14], 46] there is a manifold M = f(y; z) z = s (y)g (locally near the stationary point, which itself is on M ) such that solutions of (1) starting on M remain on M and are smooth in the above sense. Also the function s de ning the manifold has arbitrarily many derivatives bounded ....

N. Fenichel, Geometric singular perturbation theory for ordinary dierential equations. J. Di. Eq. 31 (1979), pp. 53-98.

....implemented with methods which are natural for pde s. Remark 2: Fiber representions are constructions from geometric singular perturbation theory which permit one to follow the long time fate of the full motion with an orbit totally restricted to a slow manifold. These were developed by Fenichel [65, 66, 67, 68] to provide a geometric understanding of singular perturbation methods, such as those of Howard and Kopell [110, 92] Recent descriptions of these brations, with explicit examples, may be found in [97, 145, 147] Remark 3: The argument is a shooting method , with the nal target the stable ....

N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J Di Eqns, 31:53-98, 1979.

....by introducing u = OE; u 0 = v; v 0 = w and 2 w 0 = z 8 : u 0 = v v 0 = w 2 w 0 = z 2 z 0 = w cv u N (u) 1. 5) Then, this 4 dimensional system is analysed by the geometric theory for singularly perturbed systems originally developed by Fenichel [11]. Here we use a form of this theorem 3 due to Jones [18] to show that, for sufficiently small, there exists a solution which connects the two fixed points (0; 0; 0; 0) and (1; 0; 0; 0) Also, we find that there exists a critical wavespeed c such that for c c the solution is monotonic ....

....connecting the stationary points u = 1 with u = 0. We now state precisely the results on invariant manifolds which allow us to study (2.4) and (2.5) for ffl small but nonzero. The first use of normally hyperbolic invariant manifolds to treat such singularly perturbed problems was due to Fenichel [11]. We use a form of this theorem due to C. Jones [18] The following theorem is a combination of Theorems 1,2 and 3 of [18] Theorem 2.4 Given a C 1 vector field of the form x 0 = f(x; y; ffl) 2.10) y 0 = fflg(x; y; ffl) such that when ffl = 0, the system of equations has a compact, ....

N. Fenichel (1979) Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq. 31, 53-98.

.... the following result, originally obtained in [1, 2] if x ( is a family of asymptotically stable equilibria of (2) then any solution of (1) starting in a suciently small neighbourhood of x (0) will, after a short transient, track the curve x ( t) at a distance of order (see e.g. [3, 4, 5] for various developments of this result) For the ageing device, this implies that we need not worry as long as the nominal equilibrium x ( remains asymptotically stable. This naturally raises the question of what happens if the equilibrium x ( undergoes a bifurcation at = 0 . ....

N. Fenichel, Geometric singular perturbation theory for ordinary dierential equations, J. Di. Eq. 31:53-98 (1979).

....of dynamical systems dx ds = f(x; 2) where is considered as a xed parameter. Equation (1) can be viewed as a version of (2) in which the parameter = s = t is made slowly time dependent. The existence of a relation between solutions of (1) and (2) is con rmed by the following result [1, 2, 3, 4]: Assume that x ( is a family of hyperbolic equilibria of (2) Then (1) admits a particular solution x(t) such that k x(t) x (t)k 6 c ; uniformly for in compact intervals. If x ( is asymptotically stable, x(t) attracts nearby solutions exponentially fast. If x ( has ....

N. Fenichel, Geometric singular perturbation theory for ordinary dierential equations, J. Di. Eq. 31:53-98 (1979).

....ffi 0:1 in case of bond angle bending. In the context of hyperbolic PDEs, the same expression is related to the high frequency modes in the Fourier spectrum of the solutions. Smoothed Dynamics 4 Differential equations of the form (1) fall into the class of singularly perturbed systems of type [6] d dt z = 1 ffl f(z; ffl) 8) Solutions of (8) satisfy, in general, jjz(t)jj = O(1) and jj d dt z(t)jj = O(ffl Gamma1 ) i.e. they are bounded but vary rapidly in t. Thus the step size of a numerical integrator has, in general, to be of order O(ffl) This implies a significant amount ....

....requirement is however that 2. M 0 is an exponentially stable manifold of the differential equation d dt z = f(z; 0) Under the Assumptions 2, one can show that there exists a family M ffl of smooth manifolds with M ffl=0 = M 0 such that M ffl is an exponentially stable invariant manifold of (8) [6]. Furthermore, the solutions on M ffl reflect the long time behavior of the general solutions of (8) with initial values in a oe neighborhood of M ffl up to terms of order O(oe ffl) Since the solutions on M ffl satisfy now dz=dt = O(1) time steps of order O(1) can be used in a Smoothed ....

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Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J.D.E., 31, 53--98, 1979.

....the many structures present in the invariant center manifold in a near future. Similar ideas were used in [dlLW89] We note that the use of methods based in normal hyperbolicity to deal with systems with two scales of time in a geometric way has been successfully used for a long time (See e.g. [Fen79]) We want to draw attention to [BT98] which presents another method to obtain similar results without using the theory of normally hyperbolic manifolds. They construct the transition chain relying in standard KAM theory and Poincar e Melnikov theory. Of course, the methods used in [Mat96] are ....

N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations, 31(1):53-- 98, 1979.

....x = P 0 (E) E) 4) and the procedure itself is known as adiabatic approximation (elimination) in nonlinear optics. The analogous technique in kinetic theory is called the local equilibrium approximation. While the above approximation can be made precise in an ODE problem (see, e.g. Fenichel [8]) its validity is a subtle question for partial di erential 2 equations. In geometric terms, adiabatic elimination assumes the presence of an invariant manifold near P = P 0 (E) in the in nite dimensional phase space of (3) on which the leading order evolution is governed by (4) The problem is ....

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Di. Eqns, 31 (1979), pp. 53-98.

....is far from the invariant space Pi and the slow time = p t when the phase point is near a resonant torus on Pi. For this reason, we study, in section 4, the center stable and center unstable manifolds of the space Pi, by means of a geometric singular perturbation theory developed in [13]. Using Fenichel s results, it is shown that there exist for small perturbed center, center stable and center unstable manifolds with the same codimension. In section 4, we compute the Mel nikov function and using the implicit function theorem, we establish that the existence of simple zeros ....

....and the Liouville Arnold Jost theorem [8] guarantees the existence of an open set Pi on which we can introduce canonical action angle variables (I ; OE) 2 IR Theta S 1 . In order to study smoothness in and persistence of invariant manifolds, we consider as a variable (see Fenichel, [13]) and rewrite the system (1.1) as: q k = Gammai q k 1 Gamma 2q k q k Gamma1 h 2 Gamma ijq k j 2 (q k 1 q k Gamma1 ) 2i 2 q k Gamma i ae aq 2 k (a bq k )fjq k j 2 k h 2 ln( k )g oe = 0 (3.1) Now, parameterizes the phase space N as follows N = N Theta ....

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Fenichel, N., Geometric Singular Perturbation Theory of Ordinary Differential Equations J. Diff. Eq. 31, pp. 53--98, 1979.

....In this case, the relation lim 0 x( x ( 8 0 (9) implies that we may indeed take the formal limit 0 directly in (4) Theorem 1 has a long history. In the asymptotically stable case, points 1. and 5. were originally proved in [12, 13] A di erent approach has been used in [14]. The exponential bounds in the analytic case are a result of an iterative scheme in [8] an alternative approach can be found in [15] The periodicity of solutions is a consequence of the implicit function theorem, and the computation of invariant manifolds in the hyperbolic case is explained in ....

N. Fenichel, Geometric singular perturbation theory for ordinary dierential equations, J. Di. Eq. 31 (1979), 53-98.

....u 2 2 F (u) see [Ter85] In the limit of strong damping, # # #, system (1.1) reduces to the gradient flow u = f(u) 1.3) Indeed, we may rescale time (t # #t) and rewrite (1.1) as a system u # = v # 2 v # = v f(u) 1. 4) Standard geometric singular perturbation theory ([Fen79], Wig94] then identifies a slow manifold v = f(u) O(# 2 ) 1.5) in the phase plane (u, v) # X which is normally hyperbolic with exponential rate # # 2 of attraction. Inside this inertial manifold , being a graph over u, the flow is given by u # = f(u) O(# 2 ) 1.6) 2 The ....

....a local span and W s 0 (E) is again understood for the slow flow (4.9) in S 0 . In the repelling case of a negative sign in (4.14) the roles of the stable and unstable manifolds are reversed, and in particular i(E) i S# (E) 1. For a proof we refer to geometric singular perturbation theory [Fen79]; see also [Wig94] The appearance 15 of # in (4.7) together with regularity condition (4.13) ensure, in fact, normal hyperbolicity in the sense of [HPS77] with exponential normal contraction expansion rate of the order O(1 #) 5 Proof of theorem 1.1 We give the proof, postponing three ....

N. Fenichel. Geometric singular perturbation theory for ordinary di#erential equations. J. Di#. Eq., 31:53--98, (1979).

....sI has n Gamma i positive real eigenvalues and i negative real eigenvalues. Figure 5.1 may be helpful in keeping track of the notation and geometry of this section. It is supposed to show the case n = 2, but u space and w space are drawn as if they were one dimensional. By results of Fenichel [5], each E i , i = 1; n Gamma 1 (the compact ones) has local stable and unstable manifolds that fiber over E i . This means: through each point (u; 0; s) in E i there 14 SCHECTER L u d e 1 s u w u 1 (s) 2 u (s) E E E 0 1 2 u= u= s=l 1 (u) s=l 2 (u) g Figure 5.1. Notation and ....

N. Fenichel, Geometric Singular Perturbation Theory for Ordinary Differential Equations, J. Differential Eqs. 31 (1979), 53--98.

....Hopf Bifurcations N. Berglund Weierstra# Institut f#r Angewandte Analysis und Stochastik Mohrenstra#e 39, D 10117 Berlin, Germany March 30, 1999 Abstract The slow passage through a Hopf bifurcation leads to the delayed appearance of large amplitude oscillations. We construct a smooth scalar feedback control which suppresses the delay and causes the system to follow a stable equilibrium branch. This feature can be used to detect in time the loss of stability of an ageing device. As a by product, we obtain results on the slow passage through a bifurcation with double zero eigenvalue, ....

....D 10117 Berlin, Germany March 30, 1999 Abstract The slow passage through a Hopf bifurcation leads to the delayed appearance of large amplitude oscillations. We construct a smooth scalar feedback control which suppresses the delay and causes the system to follow a stable equilibrium branch. This feature can be used to detect in time the loss of stability of an ageing device. As a by product, we obtain results on the slow passage through a bifurcation with double zero eigenvalue, described by a singularly perturbed cubic Li#nard equation. 1991 Mathematics Subject Classication. 34E15, 58F14, ....

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N. Fenichel, Geometric singular perturbation theory for ordinary dioeerential equations, J. Dioe. Eq. 31:5398 (1979).

....a curve v f(u) Gamma1) i 1 k g(u) 0 and the two equilibria are also connected by a heteroclinic orbit. Remark: The heteroclinic orbits are part of the slow manifold, an invariant manifold that exists for small 0 near the singular curve C except in a neighborhood of the extrema, cf. [1]. Proof of Theorem 1.1: Without restriction we assume that f 0 (u) 0 for all u 2 [u 1 ; u l ] the case f 0 0 can be treated similarly (by reversing time and the use of negatively invariant regions instead of positively invariant ones) We proceed indirectly and suppose that there is a ....

N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J.Diff.Eq., 31:53--98, 1979.

....at Buffalo Buffalo, NY 14214 October 10, 1997 Research supported in part by NSF Grant DMS 9302970 y Research supported in part by Polish Scientific Grant (KBN) No 0449 P3 94 06. z Research supported in part by the National Science Foundation 1 1 Introduction Following N. Fenichel s paper [4] the ideas of dynamical systems have been applied to problems involving singular perturbations resulting in a technique often referred to as the theory of geometric singular perturbations (see also the earlier works on invariant manifolds [5, 3] The fundamental concepts involve invariant ....

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, JDE, 31, 1979, 53--98.

....D E a homoclinic bifurcation and E F a saddle node bifurcation of periodic orbits. can be influenced by the choice of C. This equation happens to be a codimension four unfolding of the singular vector field (y; Gammax 2 y Gamma x 3 ) which has been studied in detail, see [KKR, VT] and references therein. The bifurcation diagram in the section fl = ffi = 0 has already been studied in [Ta] it is shown in Fig. 4. We now consider the time dependent version of (37) dz d = f(z; b u(z; 45) It can be shown that similar transformations as above yield the equation dx d = y ....

.... fl( x 2 ffi( xy Gamma x 2 y Gamma x 3 O(kzk 4 ) R(x; y; 46) where R(0; 0; 0) is directly related to the drift d d z ( of the nominal equilibrium. The dynamics of (46) depends essentially on the path ( a( through the bifurcation diagram of Fig. 4, the effect of fl( and ffi( is small in a neighbourhood of the 11 a b x x c d x x Figure 5. Solutions of equation (46) in the case fl = ffi j 0, R j 1, a( 2 and different functions ( a) Gamma0:2: We traverse the bifurcation diagram of Fig. 4 from region A to region B. The system ....

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N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq. 31:53--98 (1979).

.... Moreover, connecting our method of proof (which is analytical although we use geometric results) with recent results about the geometric nature of dae s by Reich [22] and Rabier Rheinboldt [21, 20] leads to the open problem of generalizing the geometric singular perturbation theory by Fenichel [3]. The (hopefully positive) answer will lead to a complete geometric treatment of the convergence problem which, in our opinion, is the essence of singular perturbations in dae s. In order to be more precise, let Q denote a projection of IR m onto the nullspace N (A(x, t) and P = I Q. We ....

....# I 0 0 0 # . The outline of the paper is as follows: In Section 2 we consider a semiexlicit index 2 system and derive the underlying singular perturbation problem. In Section 3 we recall the necessary notations and the convergence result of a geometric singular perturbation theory developed in [3]. As an application we are now in a position to present the main convergence result for the semiexplicit system of Section 2 in Section 4. Section 5 contains the proof of Lemma 4.1, which lies at the heart of our approach. In Section 5 we show the same convergence result for the original problem ....

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N. Fenichel. Geometric singular perturbation theory for ordinary di#erential equations. J. Di#er. Equations, 31, 1979.

....are mostly of a geometrical nature, the problems when taking the next step will have a computational, but not a conceptual, character. In order to describe the homoclinic saddle node bifurcation to be studied in this paper, we first need to appeal to some fundamental results due to Fenichel ([7], 12] Since Gamma (1.2) is normally hyperbolic in the limit 0, both Gamma and its (two dimensional) stable and unstable manifolds, W s ( Gamma) and W u ( Gamma) persist under the perturbation (see section 2 for more details: we do not need to distinguish between Gamma 0 (i.e. 0) ....

....the fast flow counteracts the slow flow. The approach of the analysis in this paper is mainly geometrical and does not rely on a specific form of the model system. This includes the quantitative results (Theorems 4.1, 5. 1) these are also based on a geometrical singular perturbation theory ([7], 15] 13] 12] This approach enabled us to unravel much of the complex structure of W s ( Gamma) W u ( Gamma) in (1.1) Due to the nature of the analysis in this paper, it is likely that a similar complexity can be observed in a more general class of singularly perturbed systems. The ....

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N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eq., 31:53--98, 1979.

....2 q 2f(p) i 1 Gamma tanh 2 i 1 2 q 2f(p) jj : 2.2) Inside H [ H Gamma a family of invariant cylinders filled with periodic orbits exist. The planes M Sigma represent the solutions of (1.1) with trivial U state U = Sigma1. Since M Sigma are normally hyperbolic, Fenichel theory [11] guarantees, that any C 2 small perturbations preserve nearby hyperbolic slow manifolds M Sigma = M c Sigma O( for any compact M c Sigma ae M Sigma . A special choice of the perturbation g(u; v; p; q) might guarantee that M Sigma = M c Sigma : the manifolds M Sigma are ....

....for M Sigma that will be convenient to describe the slow flow in x2:2. For small 0 the manifolds M Sigma no longer consist of fixed points, but are slow manifolds. The slow flow on M Sigma is determined by the function k(u; p) We come back to this later. By Fenichel [11] M Sigma have stable and unstable manifolds, which we call W s (M Sigma ) and W u (M Sigma ) respectively, that are O( close to those of M c Sigma . From now on we omit the in the notation of M Sigma and define W s (M Sigma ) and W u (M Sigma ) as those parts ....

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N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eq., 31:53--98, 1979.

....the many structures present in the invariant center manifold in a near future. Similar ideas were used in [LW89] We note that the use of methods based in normal hyperbolicity to deal with systems with two scales of time in a geometric way has been successfully used for a long time (see, e.g. [Fen79]) 2 We want to draw attention to [BT99] which presents another geometric method to obtain similar results (In particular they also give a geometric proof of Mather s result. They construct a transition chain relying on standard KAM theory and the Poincar e Melnikov method and do not use ....

N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations, 31(1):53--98, 1979.

....only points for which v f(u) su has the same values. This is exactly the Rankine Hugoniot condition for waves propagating with speed s. Moreover the direction of the fast vector eld is in accordance with the entropy condition. Geometric singular perturbation theory in the spirit of Fenichel [4] is a strong tool in regions where the singular curve is normally hyperbolic, i.e. where the points on C s are hyperbolic with respect to the fast eld. Unfortunately, trajectories that pass near the top of the curve C s are not captured by this classical theory. However, in recent years, methods ....

....theory, parts of the singular curve C s that are normally hyperbolic, survive for small 0. Deleting a small neighborhood of the fold point from C s leaves two branches: One branch A 0 which is attracting for the fast dynamics and one branch R 0 which is repelling. By Fenichels theory [4], there will be two invariant curves A and R for 0 small. The dynamics on this branches is close to the slow dynamics on A 0 and R 0 , in particular, the equilibria u i and u i 2 , which survive for 0, will lie on A and R . Moreover, A must contain the unstable manifold of u i ....

N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J.Di.Eq., 31:53-98, 1979.

....points for which v f(u) su has the same values. This is exactly the Rankine Hugoniot condition for waves propagating with speed s. Moreover the direction of the fast vector eld is in accordance with the Oleinik entropy condition. Geometric singular perturbation theory in the spirit of Fenichel [2] makes precise statements how the slow and the fast equations together describe the dynamics of (3) for small 0. It is a strong tool in regions where the singular curve is normally hyperbolic, i.e. where the points on C s are hyperbolic with respect to the fast eld. The only non hyperbolic ....

N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J.Di.Eq., 31:53-98, 1979.

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N. Fenichel, Geometric singular perturbation theory for ordinary di#erential equations, J. Di#. Eq., 31, 53--98 (1979).

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N. Fenichel, Geometric singular perturbation theory for ordinary di#erential equations, J. Di#. Eq., 31, 53--98 (1979).

No context found.

N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J. Dierential Equations, 31(1):53-98, 1979.

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N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J. Di. Eqns., 31 (1979), 53-98.

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N. Fenichel. Geometric singular perturbation theory for ordinary dierential equations. J.Di.Eq., 31:53{ 98, 1979.

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N. Fenichel, Geometric singular perturbation theory for ordinary dierential equations. J. Dierential Equations, 31(1979), 53-98.

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N. Fenichel, Geometric singular perturbation theory for ordinary dierential equations. J. Dierential Equations, 31(1979), 53-98.

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