THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE

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1 MOSCOW MATHEMATICAL JOURNAL Volume 1, Number 1, January March 2001, Pages THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE J. GUCKENHEIMER AND YU. ILYASHENKO Abstract. Slow-fast systems on the two-torus T 2 provie new effects not observe for systems on the plane. Namely, there exist families without auxiliary parameters that have attracting canar cycles for arbitrary small values of the time scaling parameter. In orer to emonstrate the new effect, we have chosen a particularly simple family, namely ẋ = a cos x cos y, ẏ =, a (1, 2) being fixe. There is no oubt that a similar effect may be observe in generic slow-fast systems on T 2. The propose paper is the first step in the proof of this conjecture Math. Subj. Class. Primary 34A26, 34E15. Key wors an phrases. Slow-fast systems on the torus, canar solution, evil s staircase, Poincaré map. Generic slow-fast systems in the plane 1. Introuction ẋ = f(x, y, ), ẏ = g(x, y, ), (x, y) R 2 (1.1) exhibit a particularly simple escription of the shape of orbits as 0. Namely, the orbits of (1.1) ten to the orbits of a egenerate system constitute by alternating segments of phase curves y = const of the fast system an stable portions of the slow curve M: f(x, y, 0) = 0. In the presence of an auxiliary parameter, a new kin of limit behaviour of solutions was iscovere in [D] (also see [DR] an the references therein). Namely, for the two-parameter family, there is an exponentially narrow horn ẋ = y x 2 x 3, ẏ = (a y) (1.2) a (a 1 (), a 2 ()), 0 > > 0, a 2 () a 1 () < e c/ such that for (a, ) insie this horn solutions are close to an attracting cycle constitute by arcs passing not only near phase curves of the fast system an stable Receive September 27, 2000; in revise form February 2, First author supporte in part by the National Science Founation an the Department of Energy. Secon author supporte in part by RFBR uner grant , NSF uner grant DMS , an CRDF uner grant RM c 2001 Inepenent University of Moscow

2 28 J. GUCKENHEIMER AND YU. ILYASHENKO portions of the slow curve, but also near unstable portions of the slow curve, see Fig. 2 (a). Solution of this type are calle ucks or canars (ucks in French). The existence of canar solutions in two-imensional slow-fast systems was iscovere in ifferent situations, see [AAIS] an references there. But all the known examples containe an auxiliary parameter like a in (1.2). The system consiere in the present paper ẋ = a cos x cos y, ẏ =, (x, y) T 2 (1.3) has no auxiliary parameter: a in (1.2) is fixe, an one may replace a by, say, 1.1. There is a sequence of intervals C n { > 0}, C n 0 as n, such for any C n (1.3) has an attracting canar cycle, see Fig. 1. (a) (b) Figure 1. (a) Regular attracting cycle of system (1.3). (b) Canar cycle of the slow-fast system (1.3) on a torus. A great majority of solutions are attracte to this cycle. Therefore, a ranom solution in the slice containing the slow curve looks like the one shown in Fig. 1. We conjecture that there is an open set of slow-fast system in T 2 with the same property. However, we have chosen a particular example in orer to exibit the new effect an elaborate the tools for the stuy of the generic case. Canar solutions of slow-fast systems may be easily seen in the plane if we o not require that these solutions be attracting perioic. Inee, consier the classical picture of a jump near a fol point of the slow curve, Fig. 2 (b). A majority of solutions initiating to the left of the fol point approach the stable part of the slow curve, then gather in the exponentially narrow flow box of with e c/ an jump own near the fol point. This picture alreay contains the canar solutions; they may be seen if we pronounce the exorcism: reverse the time, see Fig. 2 (c).

3 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE 29 The initial conitions of positive semiorbits with canar behaviour form an exponentially narrow tube. It is so narrow that in computer simulations it is shown by a single curve, see Fig. 2 (c). (a) (b) (c) Figure 2. (a) Canar cycle. (b, c) Generation of canar solutions by time reversal: (b) a jump, (c) a reverse jump. System (1.3) is chosen to be reversible: the symmetry σ r : (x, y) (x, y), together with time reversal, brings system (1.3) to itself. The reason is twofol: reversibility facilitates the proofs; it is intimately relate with the canar solutions. Yet we believe that reversibility is not necessary for the existence of canars without auxiliary parameters on the two-torus.

4 30 J. GUCKENHEIMER AND YU. ILYASHENKO System (1.3) has a global cross-section Γ efine by y = ±π, an hence a Poincaré map P which is a iffeomorphism of Γ with rotation number ρ(). The rotation number varies continuously with. In generic families of iffeomorphisms of the circle, the function ρ() is a Cantor function whose graph is often calle a evil s staircase [M]. The steps of the evil s staircase occur at rational values of ρ for generic families of circle maps. The lengths of these steps an the intervening gaps satisfy universal scaling properties that are intimately relate to the continue fraction expansions of the rotation numbers. One of our main concerns is to examine this geometry for the Poincaré map of (1.3). The limit 0 of (1.3) is singular. As 0, the rotation number ρ() increases without boun. At = 0, the zeros of the function a cos(x) cos(y) are equilibrium points of the system (1.3) an the flow no longer has a global cross-section. The singular nature of the limit 0 affects the scaling of the steps of the evil s staircase. In particular, there is a set of ominant steps for which ρ() is an integer an the Poincaré map has fixe points. We give an analysis here emonstrating that the gap lengths between these ominant steps in the evil s staircase are very short: there is a constant c > 0 so that the gap lengths ten to 0 faster than e c/. The vector fiel (1.3) has a time reversing symmetry σ r given by σ r (x, y) = ( x, y). Since σ r reverses time, it maps stable perioic orbits of (1.3) into unstable perioic orbits of (1.3) an vice-versa. Perioic orbits that contain one of the fixe points (0, 0), (π, 0), (π, 0) or (π, π) of σ r must be neutrally stable. (If a perioic orbit contains one fixe point of σ r, then it also contains a secon one.) Such orbits occur for parameters at the enpoints of C n. For the Poincaré map P, we have P 1 ( x) = P (x), implying that the graph of P is symmetric with respect to the reflection σ p : T 2 T 2, (x, y) ( y, x) in the circle x+y = 0. Thus P (x) = (P 1 ) ( x). If x+p (x) = 0, then P (x) = x an P (x) = 1. The neutral fixe points of P (if they exist) are 0 an π. The graph of the Poincaré map of (1.3) has very small slope at most points. The slope grows rapily insie a small interval an ecays just as quickly. The graph is approximate by vertical an horizontal circles that intersect on the fixe point circle x + y = 0 of σ p. The Poincaré map has exactly two fixe points, one with small slope an one with large slope, unless this intersection point is close to 0 or ±π. Characterizing the Poincaré map where its erivative changes from very small to very large is a critical part of our analysis. We ecompose the Poincaré map in this region into a composition of maps prouce by the flow across ifferent vertical slices of the torus. In any close cyliner forme by the orbits of the fast system that o not intersect the slow curve, the slow-fast system is orbitally smoothly equivalent to a system for which the Poincaré map is a mere shift. The normalizing coorinates ten to a smooth limit as 0. (Theorem 2). Near any close arc in the stable (or unstable) part of the slow curve, the slowfast system is orbitally smoothly equivalent to a linear one with respect to the fast variable. (Theorem 3).

5 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE 31 The Poincaré map near these parts of the slow curves is a pure linear contraction (an expansion) with the coefficient exponentially tening to zero (respectively, to infinity) as 0. Transitions from the unstable slow curve to the stable slow curve, or across the critical points of y on the slow manifol are short enough to play an inconsequential role in etermining the erivative of the Poincaré map. The ominant contribution to the erivative of the Poincaré map comes from the relative length of time that a trajectory spens close to the slow manifol. If the trajectory spens more time close to the stable slow curve than to the unstable slow curve, the slope of the Poincaré map is very small. If the trajectory spens more time close to the unstable slow curve s + than to the stable slow curve s, the slope of the Poincaré map is very large. The Poincaré map has a slope that is O(1) only if the trajectory crosses from s + to s near y = 0. Thus the bifurcations of the perioic orbits ajoin the region where the stable perioic orbits contain canars. The main results of the paper are state in Theorem 1 below. The outline of the proof is given in 2, using three lemmas that are prove in 4. Section 3 contains the proofs of Theorems 2 an 3. There are five aitional supporting results that we call propositions. The propositions are prove where state an serve to structure a portion of the proof of a theorem or lemma. 2. Canar cycles on the torus an the escription of the Poincaré map 2.1. Statement of the theorem. Fix an arbitrary cross-section J of the unstable part of the slow curve inepenent of. We will call a solution of (1.3) that crosses J a canar solution. The motivation is that in orer to reach J the solution must contain a segment close to an unstable arc of the slow curve whose length is inepenent of. Later on we will specify the choice of J. Theorem 1. There exists a sequence of intervals R n = (α n, β n ) an two sequences of isjoint intervals C n ± R n with the properties: 1. R n = O(e cn ). 2. α n = O(1/n) an such that: 3. For small R n, ρ() = 0 (mo Z). There are exactly two hyperbolic perioic orbits of (1.3), the unstable solution containing canars. 4. For any C ± n system (1.3) has exactly two perioic orbits, both hyperbolic canar cycles. One cycle is stable an the other is unstable The Poincaré map. Let, as before, Γ = {y = π}, P : Γ Γ be the Poincaré map along the orbits of (1.3). Let γ S 1 S 1 be the graph of P. The shape of the graph γ tens to the union of a horizontal an a vertical circles as 0. This is formalize in the following lemma: Lemma 1 (Shape Lemma). For any small > 0 there exists an interval D with the following properties: 1. D = O(e c/ ).

6 32 J. GUCKENHEIMER AND YU. ILYASHENKO 2. P S 1 \D = O(e c/ ). 3. In the notation D = D, Π = D S 1, Π = S 1 D, we have γ Π Π. The same statements hol for the inverse Poincaré map P 1 interchange. with D an D Lemma 1 is prove in 4.1. The next lemma formalizes the statement that the graph γ of the Poincaré map moves monotonically to the upper left as 0. To formulate it, we nee to specify a cross-section J from the beginning of 2.1. Let τ = cos 1 (a 1) be the maximal value of y on the slow curve M = {a cos(x) cos(y) = 0}. Pick 0 < δ < τ; let α = τ δ; in what follows, δ will be chosen small. Consier the following cross-sections: Γ ± = S 1 { α}, J + = {(x, y) Γ + x [0, π]}, J = {(x, y) Γ x [ π, 0]}. Solutions that cross J + will be calle canar solutions; these solutions are mentione in Theorem 1. For two points a, b on the oriente circle that split it in two nonequal arcs, let a < b if a is the left en (in the sense of the orientation of S 1 ) of the shortest arc with ens a an b. Denote by (a, b) the shortest arc that connects a an b oriente from a to b; the orientation may be opposite to that of S 1. For any a, b as above, enote by P a,b to y = b along the orbits of (1.3) passing over the arc (a, b). In Lemma 1, choose the Poincaré map of the cross-section y = a D = P α, π (J + ), D = [, + ], < +. (2.1) By efinition, all the orbits of (1.3) that cross D are canar solutions. Denote by A ± () the points on the graph γ lying above ± : A ± () = ( ±, P ( ± )). Let = {x+y = 0}, {B ± ()} = γ, B ± () = (b ± (), b ± ()), b () < b + (). The circle is the mirror of the symmetry σ p, so P (b ± ()) = 1. Hence, b ± () D. Lemma 2 (Monotonicity Lemma). Let ±, Ā ± (), B± () be the liftings to the universal covers of ±, A ± (), B ± (), which continuously epen on. Then 1. ±, (x y)(c()) as 0+ for any of the choices: C() = Ā+ (); Ā (); B+ (); B (). 2. Equation = πn has the solution = n = O(1/n). Now we can prove Theorem 1.

7 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE 33 Π Π (a) (b) A B + B B (c) () B + A + (e) (f) Figure 3. (a) Annuli in the Shape Lemma. (b f) Application of the Monotonicity Lemma. Location of : (b) between R n 1 an R n ; (c) onto C n ; () between C + n an C n ; (e) onto C + n ; (f) between R n an R n+1.

8 34 J. GUCKENHEIMER AND YU. ILYASHENKO 2.3. Existence an lifespan of canar cycles. The proof of Theorem 1 is illustrate by Figure 3. Remark. Figures 3 (b), (c), (e), (f) correspon to the case ρ() = 0 (mo 1). All the rest of the evil s staircase is passe as ranges between C n + an Cn, see Fig. 3 (). Proof of Theorem 1. Define the segments R n in the following way: R n = { πn + } := [α n, β n ], R = R n. (2.2) By statement 1 of Lemma 2, R n is inee a segment for large n. By statement 2 of Lemma 2, α n = O(1/n). By statement 1 of Lemma 1, + = O( c/ ). By statement 1 of Lemma 2, this implies: R n = O( c/ ). This proves statements 1 an 2 of Theorem 1 for R n etermine by (2.2). To prove statement 3, enote by K the square D D, D = [, + ]. Let be the iagonal x = y (mo 2π) in T 2. The following proposition implies the thir part of the theorem. Proposition 1. Let K =. Then ρ() = 0 (mo Z) an P has exactly two hyperbolic fixe points. Proof. On S 1 \D, the graph γ of P has slope smaller than one. The enpoints of this graph lie on the left an right bounaries of K at the points A () an A + (). If we connect these points by a segment insie K, we obtain a close curve γ h on T 2 that has homotopy type (1, 0) because outsie K its slope is everywhere smaller than one, an it intersects each vertical circle in exactly one point. Consequently, γ h intersects at a unique point p s. The point p s is not in K because K =. Therefore, it is a fixe point of P. It is stable because P (p s ) < 1. Applying the symmetry σ r, we get a fixe point p u = σ r (p s ), which is unstable. This proves the proposition. By (2.2), for / R, 0, π / D ; hence, K =. An application of Proposition 1 proves statement 3 of Theorem 1. We can now prove the first part of statement 4, namely, the existence of canar cycles. Let C n = { R n (y x)( B ()) < 2πn < (y x)(ā ()) }, C + n = { R n (y x)(ā+ ()) < 2πn < (y x)( B + ()) }. The Monotonicity Lemma implies that Cn an C n + are intervals for n large. Obviously, Cn C n + R n. Let Cn ; case C n + is treate in the same way. The points A () an B () lie on an arc λ of the graph γ an λ is containe in K. By the efinition of Cn, these points lie on the ifferent sies of in K. Hence, λ crosses ; enote the intersection point by S. Below we prove that it is the unique stable point of P. The point S = (s, s ) correspons to a canar cycle, because s D ; hence, the cycle crosses J +. The symmetric point U = S γ correspons to another perioic solution, which is, in fact, unstable hyperbolic.

9 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE Uniqueness an stability of canar cycles. For any egree one smooth map P : S 1 S 1, between any two fixe points there exists a point at which P = 1. This follows from Rolle s theorem applie to the perioic function P (x) x. Consequently, the number of fixe points of P is boune from above by the number of solutions of the equation P (x) = 1. Lemma 3 gives us information about this equation. Lemma 3. The set {x S 1 P (x) [1/2, 2]} consists of two arcs containe in D. On the left arc P increases, on the right one it ecreases. Lemma 3 is prove in 4.2. Lemma 3 implies that there are always exactly two solutions of the equation P (x) = 1 on S 1. It also implies that P (S ) < 1 at the fixe point S of P constructe in 2.3. Therefore, the canar cycle constructe in 2.3 is stable. This finishes the proof of the fourth an last statement of the theorem. Note that if there is a fixe point x of P with P (x) = 1, then this is the only fixe point of the Poincaré map Nonlinear translations. 3. Normalization Theorem 2. Consier a vector fiel on the cyliner x S 1 = R/Z, y R efine by ẋ = f(x, y, ), ẏ = g(x, y, ), where f > 0, g > 0. If a, b R, a < b, an > 0, then the Poincaré map P ab Γ a : y = a to Γ b : y = b has the form of a cross-section P a,b : x G 1 (G 2 (x) + T ()), (3.1) where T () + an G 1,2 G 1,2 ; the maps G 1, G 2 are still iffeomorphisms. G 1,2 are iffeomorphisms of the circle. As 0, we have Proof. Let us ivie the vector fiel by f. We obtain a system with the same phase curves an the same Poincaré map: ẋ = 1, ẏ = w(x, y, ), w > 0. (3.2) Proposition 2. System (3.2) is smoothly equivalent to the system ẋ = 1, ẏ = v(y, ). (3.3) Proof. Consier the time-one Poincaré map F : {x = 0} {x = 0}. It is smooth in y, an has the form F : y y + F (y, ). (3.4) Lemma 4. The smooth family (3.4) on any y-segment on which F is positive is smoothly equivalent to the family embeable in the flow ẏ =. y y +

10 36 J. GUCKENHEIMER AND YU. ILYASHENKO This lemma is a global version of Theorem 7 from [I]. For the sake of completness we reprouce here the proof of the above theorem with the minor changes neee for passing from germs to maps of segments. Proof. Let us first exten (3.4) to the whole line R, replacing F by a smooth function equal to F on the given segment, an to one outsie some larger segment. Consier the vector fiel F (y, 0) y on the line, F 0. It is smoothly equivalent to y. In the rectifying chart, still enote by y, the globalize map (3.4) has the form: F : y y f(y, ) with f 0 outsie some segment. We want to conjugate this map with S : z z+, using the coorinate change H : y y + h(y, ) = z smooth in y an 0. The functional equation H F = S H takes the form: h(y f, ) h(y, ) = f(y, ). We will solve this equation by successive approximations, the Borel theorem an the homotopy metho. Let f = n f n (y) +..., here an below the ots replace higher orer terms in. Let h = n h n (y) The functional equation implies h n = f n. Let f(y, ) = fn (u) n 1 be a semiformal series (formal in with functional coefficients in x) for the iscrepancy. The successive approximation metho prouces a semiformal series y + h, h = hn (y) n that conjugates y f with y + as semiformal series. The Borel theorem provies a smooth map y + h that conjugates F with F = y + + g(y, ), where g is flat on the line { = 0}. The last step is to prove that the map F is C embeable. To this en consier the map G: (y,, t) (G,, t), t [0, 1], G = y + + tg. It is sufficient to fin a vector fiel (X, 1) in R (R, 0) [0, 1] that is G-invariant. The corresponing equation is G y X X G = g. The solution of this equation is given by the formula X = (G n g) G 1 ; 0 here G is regare as a one-imensional map in y epening on, t as parameters. Note that g has a compact support because f has the same property. For a flat g on the plane { = 0} with compact support this formula gives for 0 a vector fiel X that amits a smooth extension by 0 through the line = 0.

11 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE 37 We omit the etaile estimates. Note only that the number of nonzero terms in the sum above is of orer 1/ for fixe, while the erivatives of G n for n of orer 1/ ten to zero faster than any power of. Lemma 4 is prove. We have constructe system (3.3) which has the same Poincaré map as (3.2). Let V t an W t be the time t flow transformations for systems (3.3) an (3.2), respectively. For t = 1 they bring the line x = 0 into itself, an coincie with the Poincaré map on it. The map H : (x, y) V x W x (x, y) transforms the flow W t to V t. Since the Poincaré maps F for (3.2) an (3.3) coincie, H(1, y) = (1, F F 1 (y)) = (1, y). Hence, H is well efine on the whole cyliner. This proves the proposition. The map H epens on an tens to the ientity as 0. The Poincaré map P a,b : {y = a} {y = b} for system (3.3) is a rotation: P a,b : x x + T (), T () = 1 b a y v(y, ). (3.5) We complete the proof of Theorem 2 by calculating the map P a,b in the normalizing chart that brings (3.2) to (3.3). Let (x, a) be on the section Γ a = {y = a}. Denote w a (x) = x 0 w(ξ, a, 0)ξ, ω(a) = w a (1). Note that in (3.3), v(a, 0) = ω(a). Let G a : S 1 S 1 be a iffeomorphism such that x = G a (x) implies V x W x (x, a) {y = a}. The iffeomorphism G a epens on, but we o not inicate this by abuse of notation. The point p = (x, a) Γ a is mappe by H to q = (x, y(x, a)). Let ϕ be the orbit of (3.2) passing through p, an ψ be the orbit of (3.3) passing through q. Then H(ϕ) = ψ. The orbit ψ crosses Γ a at the point (G a (x), a). The Poincaré of (3.3) takes this point to (G a (x) + T (), b) ψ. The corresponing point on Γ b lying on ϕ is obtaine by the application of G 1 b. Hence, map P a,b P a,b (x) = G 1 b (G a (x) + T ()). The map G a for = 0 may be easily foun by using the variation equation with respect to the parameter. For = 0, (3.2), (3.3) imply that y x = 0. From Proposition 2 it follows that in (3.3) v is at least C 1 -smooth in y,. Hence, On the other han The efinition of x implies Hence, V x (0, a) = (x, a + x ω(a) + o()). W x (0, a) = (x, a + w a (x) + o()). x ω(a) = w a (x) + o(1). G a (x) w a(x) as 0 ω(a) an the limit is a iffeomorphism of S 1. This proves Theorem 2.

12 38 J. GUCKENHEIMER AND YU. ILYASHENKO 3.2. Normalization along slow curves. Theorem 3. Consier a system ẋ = f(x, y, ), ẏ = g(x, y, ) with a curve of hyperbolic fixe points of the fast system, > 0 small an g > 0. In the neighborhoo of the curve of fixe points, the system is orbitally smoothly equivalent to a family linear in a fast variable. The normal form is: ẋ = a(y, )x, ẏ =. (3.6) Remark. For system (1.3), the normal form near a segment x = s(y, ) of the true slow curve is ẋ = (sin s(y, ))x, ẏ = (3.7) Proof. When > 0 is small, near a curve γ 0 of hyperbolic fixe points of the fast system there exists an invariant curve γ, the true slow surface, close to γ 0 by the Fenichel theorem [F]. Therefore the system may be written in the form ẋ = xf 1 (x, y, ), ẏ = g 1 (x, y, ), g 1 > 0. Diviing by g 1, we obtain a vector fiel with the same trajectories an ẋ = xa(y, ) + O(x 2 ), ẏ =. (This step is not neee for system (1.3).) For = 0 we obtain a family of oneimensional systems with hyperbolic singular point 0. This family may be linearize by a smooth transformation of x. We fin ẋ = xa(y, ) + x 2 b(x, y, ), ẏ =. (3.8) We assert that system (3.8) may be transforme by a finitely smooth coorinate change x = X + h(x, Y, ), y = Y (3.9) to (3.6) in which x is replace by X. To o so, we exten the family (3.8) to a single system by aing the equation = 0. This equation has a 2-imensional invariant manifol x = 0. We will prove that many lower nonlinear terms in x may be kille by the coorinate change (3.9). Then (3.8) will be equivalent to ẋ = xa(y, ) + x N b 1 (x, y, ), ẏ =, N 1. (3.10) The jets of orer N 1 of systems (3.6) an (3.10) coincie on x = 0. Hence, the systems are finitely smoothly equivalent by the Belitski Samovol theorem [IL]. It remains to prove the equivalence of systems (3.8) an (3.10). For this, we use the metho of successive approximations an conjugate system (3.10) with a similar system in which N is replace by N + 1. The composition of these conjugations gives the esire transformation. So it remains to prove the following: Proposition 3. The system ẋ = xa + x N b(y, ) + O(x N+1 ), may be transforme by the coorinate change x = X + X N h(y, ), y = Y ẏ =

13 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE 39 to Ẋ = Xa + X N+1 b 1 (Y, ) + O(x N+1 ), Ẏ =. Proof. An elementary calculation gives the following equation for h: Equation (3.11) is equivalent to the system h Y = a(1 N)h + b. (3.11) ḣ = a(1 N)h + b, Ẏ =. (3.12) For = 0, (3.11) has a smooth solution h = b/a. This is the slow curve for (3.12) with = 0. The true slow curve for (3.12) with > 0 is the esire solution h(y, ) of (3.11). Hence, (3.11) has a smooth solution. This proves Proposition 3 an completes the proof of Theorem The Poincaré map In this section Lemmas 1, 2 an 3 are prove Distortion: Proof of Lemma 1. Let α = τ δ, σ = [ α, α]. Let s + an s be the unstable an stable portions of the slow curve over σ. Let, as before, J + = Γ + {x [0, π]}, J = Γ {x [ π, 0]}. Denote, for brevity, Q = P π, α : Γ Γ +, an let D = (Q ) 1 (J + ). Proposition D = O(e c(δ)/ ), c(δ) 0 as δ For any y 0 [ π, α] the erivative of the map along the orbits P π,y0 : Γ Γ y0 = {y = y 0 } is no greater than O(e c1(δ)/ ), c 1 (δ) = O(δ 3/2 ). Proof. Consier a ecomposition of Q 1 into three factors. Take Γ 1 = {y = τ + δ 2 }; Γ 2 = {y = τ δ 2 } an let Q 1 : Γ + Γ 1, Q 2 : Γ 1 Γ 2 an Q 3 : Γ 2 Γ be the mappings along the orbits of system (1.3) with time reverse. Then Q 1 = Q 3 Q 2 Q 1. To stuy Q 1, apply the normal form (3.7). Let x = x(y, ) be the true slow curve for system (1.3) regare over the segment [ τ +δ 2, τ δ 2 ] where x = x(y, 0) > 0 is the graph of the unstable portion of the slow curve. The function a(y, ) = sin x(y, ) has the form a(y, ) = (C(a) + o(1)) y + τ + O δ (), where o(1) 0 as δ 0; O δ () is of orer for any fixe δ. The constant C(a) may be calculate explicitly, but its value is of no importance. The mapping Q 1 is linear in its normal form coorinate x 1 : Q 1 : x 1 Λ()x 1, Λ() = e δ3/2 (C(a)+o(1)+Oδ()) < e δ3/2 ( 3 4 C(a)) for small δ an < 0, where 0 epens on δ. Let us now estimate Q 2 in terms of its initial coorinate x. (The transition function from x to x 1 has boune erivative uniformly in, so it may be neglecte in future calculations.) The variational

14 40 J. GUCKENHEIMER AND YU. ILYASHENKO equation for the erivative of the solutions of (1.3) (with time reverse) with respect to its initial conitions has the form X = 1 sin x(y)x, (4.1) where x = x(y) is the orbit of (1.3). The time of motion from Γ 1 to Γ 2 is equal to 2δ 2. Hence, In particular, Q 2 e 2δ2 Q 1Q 2 e δ3/2 Λ 1 (). for small. The map Q 3 is very simple. By Theorem 2, it is a rotation in a specific chart combine with transition functions that have boune erivatives. Therefore, C(a) 2 (Q 1 ) J + e c/, c = O(δ 3/2 ). This proves statement 1 of Proposition 4 an, consequently, statement 1 of Lemma 1. The same argument shows that for any y 0 [ π, α] (P π,y0 ) e c1/, c 1 = O(δ 3/2 ) (4.2) with another constant c 1. The proposition is prove. Statement 2 of Lemma 1 follows from Theorem 3, (4.2) an reversibility. Inee, P = Q P + Q, where Q : Γ Γ +, P+ : Γ + Γ an Q : Γ Γ are maps along the orbits of (1.3). Let σ r be the time reversing symmetry σ r (x, y) = ( x, y). Note that σ r (Γ + ) = Γ. By reversibility, Q = Q 1 σ r. Hence, Q J is contracting, by (4.2). We have Q (Γ D ) = Γ + J + by the efinition of D. As is shown below, P+ (Γ + J + ) J, an Q is contracting on J. Orbits starting on Γ + J + enter a fixe neighborhoo of the stable portion of the slow curve over the segment σ in a time O(1). By Theorem 3, the erivative of the map along the orbits efine on a segment close to Γ + in this neighborhoo is contracting with the coefficient Λ e C/. Hence, P+ (Γ + J + ) J. Now, in the ecomposition for P, only the first factor is expaning, but this expansion is strongly ominate by the contraction of P+. (We stuy P+ more thoroughly in 4.2.) This proves statement 2 of Lemma 1. Statement 3 of Lemma 1 follows from statement 2 an reversibility. Reversibility implies that the graph γ of P is symmetric with respect to the iagonal x+y = 0. The symmetry σ p interchanges Π 1 an Π 2. If there is a point p = (x, y) γ outsie Π 1 Π 2, then the symmetric point q = ( y, x) γ also lies outsie Π 1 Π 2. By

15 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE 41 statement 2, the slope of γ at p an q is very small. By symmetry, the prouct of these slopes is 1, a contraiction. This completes the proof of Lemma Convexity: proof of Lemma 3. In this section we analyze the erivative of the Poincaré map P. We prove that {x Γ P (x) [1/2, 2]} consists of two isjoint arcs, with P increasing on one arc an ecreasing on the other. Since log is an increasing function, we work with the function log P instea of P in proving these monotonicity statements. The strategy that we use is similar to the strategy use to prove Lemma 1. We ecompose the Poincaré map P as above: P = Q P+ Q. By the chain rule Hence, log(p ) = log Q P + Q + log(p + ) Q + log Q. (4.3) P (u) P (u) = u log Q P + Q (u) + u log(p + ) Q (u) + u log Q (u). (4.4) We will prove that the secon term ominates the two others. First of all let us escribe the orbits x = x(y, u) with initial conition u such that P (u) [1/2, 2]. The epenence on is omitte in the notation. We will prove below that these orbits stay near the unstable part s + of the slow curve over the arc [ α, y + ] with y + close to 0. Then some orbits jump ownwar (in negative irection along the x-circle), the others jump upwar to the neighborhoo of the stable part s of the slow curve, an remain there when y ranges in [y, α], where y is O()-close to y +. For the orbits uner consieration that jump ownwar, we have P > 0; for those jumping upwar, P < 0. Note that the variational equation implies: u log(p a,b ) (u) = 1 a,b log(p ) (u) = 1 where x(y; u, a) is a solution with b a b a sin x(y; u, a)y, (4.5) cos x(y; u, a)x(y; u, a)y, (4.6) x(y; u, a) x(a; u, a) = u; X(y; u, a) =. u Let x = x ± (y, ) be the equation of the unstable (for +) an stable (for ) parts of the true slow curve of (1.3) over the segment σ. Let s ± (y) = x ± (y, 0). Let (x 1, y), (x 2, y) be normalizing charts from Theorem 3 for equation (1.3) near s + an s, respectively. From now on we suppose that u Γ satisfies P (u) [1/2, 2], (4.7) Let U, S be neighborhoos of s +, s efine by x 1 < b, x 2 < b respectively in the normalizing coorinates. By Lemma 1, the graph of the solution x(y; u, π) intersects J + an J. Let this solution leave U with y = y + an enter S with y = y. Let us prove first that y + 3δ C for small. (4.8)

16 42 J. GUCKENHEIMER AND YU. ILYASHENKO Inee, by (4.5), log(p + ) = 1 ( y+ α ) α sin s + (y)y + sin s (y)y + O(1). y The symmetry of the vector fiel implies that s + (y) = s ( y), yieling log(p + ) = 1 y+ y sin s + (y)y + O(1). On the other han, the x-component of the vector fiel of (1.3) is boune away from zero outsie any neighborhoo of the slow curve. Hence, y + = y + O(). Therefore, log(p + ) > c y + with some c epening on a in (1.3). We have prove in 4.1 that log Q < δ, log Q < δ. Together with (4.4) an (4.7) this yiels (4.8). Proposition 5. Let u Γ satisfy (4.7). Let Then for small, I = α α sin s + (y)y. (4.9) u log (P + ) Q (u) > exp I 5. (4.10) Proof. Let us use Theorem 3 applie to equation (1.3). In the normalizing coorinates (x 1, y), (x 2, y) near the unstable an stable parts of the slow curve, respectively, (1.3) has the form x 1 y = 1 ϕ(y, )x 1, x 2 y = 1 ψ(y, )x 2, (4.11) where ϕ(y, ) = sin x + (y, ), ψ(y, ) = sin x (y, ), y σ = [ α, α]. By the reversibility of (1.3), ψ(y, ) = ϕ( y, ). (4.12) The function a cos x cos y is even in y. Hence, ϕ =0 is even as well. Therefore, ψ(y, ) = ϕ(y, ) + ϕ 1 (y, ). (4.13) Consier the primitive functions of ϕ an ψ with respect to y: Φ = ϕ, Φ( α, ) = 0; Ψ = ψ, Ψ(α, ) = 0. Then, by (4.12), Φ( y, ) = Ψ(y, ). (4.14) Let I be the same as in (4.9). Then Φ( y, 0) = I Φ(y, 0), Φ( y, ) = I Φ(y, ) + k 1 (y, ), (4.15) where k 1 is a smooth function for all near 0, y σ. The functions k j that appear below are also smooth for all near 0 an the same y.

17 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE 43 The orbit of (1.3) passing from (y, x 1 ) = ( α, ξ) J + an jumping ownwar to (y, x 2 ) = (α, η(ξ)) J first intersects the line x 1 = b, then x 2 = b. The first intersection point is (y +, b), the secon one is (y, b), y ± epen on ξ. Note that the function ξ η(ξ) gives the mapp+ in the normalizing charts. Obviously, y + = y + k 2 (y +, ), (4.16) Solutions of equations (4.11) passing through p, have the form (y, x 1 )(p) = ( α, ξ), q = P + (p), (y, x 2 )(q) = (α, η(ξ)) x 1 (y, ) = ξ exp 1 Φ(y, ), x 2(y, ) = η(ξ) exp 1 Ψ(y, ). Then b = ξ exp 1 Φ(y +(ξ), ), b = η(ξ) exp 1 Ψ(y (ξ), ), η(ξ) = ξ exp Φ(y +, ) Ψ(y, ). By (4.14), (4.15), (4.16): (4.17) Φ(y +, ) Ψ(y, ) = Φ(y +, ) Φ( y, ) = Φ(y +, ) Φ( y +, ) + k 3 (y +, ) On the other han, Hence, ( 2Φ(y+, ) η(ξ) = ξ exp = 2Φ(y +, ) I + k 4 (y +, ). exp Φ(y +, ) = b ξ. I ) ( + k 4(y +, ) = b2 ξ exp I ) + k 4(y +, ). We will show that this function behaves like Cξ 1 for ξ < 0, hence, is concave. In orer to stuy η as a function of ξ, let us express y + in terms of ξ by making use of (4.17). We have: Φ(y +, ) = (log b log( ξ)). The function Φ is monotonic in y + ; let z be the inverse function: z (Φ(y +, )) y +. Then y + = z ((log b log( ξ)). Hence, b 2 exp k 4 (y +, ) = k 5 ((log b log( ξ)), ); k 5 (u, ) is smooth in u,. Let for brevity k 5 = k. Then η (ξ) = k((log b log( ξ)), ) + k u((log b log( ξ)), ) ξ 2 log η (ξ) = 2 log( ξ) + log(k + k u ) I, ( exp I ), ξ log η (ξ) = 2 ξ + k u + k uu + O() = 2 > 1 k + k u ξ ξ ξ, ξ < 0.

18 44 J. GUCKENHEIMER AND YU. ILYASHENKO Hence, the function η(ξ) is concave on the ξ-segment that we stuy. Let us estimate this concavity. For this let us estimate 1/ ξ from below. By (4.17) By (4.8), ξ = b exp Φ(y +, ). Φ(y +, ) > I 2 y + max ϕ > I σ α 3 for δ small. Hence, ξ log η (ξ) > 1 ξ > b 1 exp I 3. (4.18) When we pass from the normalizing coorinates ξ, η to the initial coorinate x boune terms only will be ae; so the following estimate hols for P+ : x log(p + ) (u) > b 1 exp I 4, provie that u satisfies (4.7), an the corresponing orbit jumps ownwars from U to S. By (4.2), u log(p + ) Q = ( x log(p + ) )Q (u) b 1 exp I O(δ3/2 ). 4 This proves Proposition 5 for orbits jumping ownwar. For the orbit jumping upwar the same evaluations go with the only ifference that (4.17) shoul be replace by b = ξ exp Φ(y +, ), b = η(ξ) exp Ψ(y, ), an (4.18) becomes ξ log η (ξ) < 2 + O() < 1 ξ ξ < b 1 exp I 3 ; ξ > 0. The erivative in u is estimate as above. This conclues the estimate of the secon term in (4.4) an proves Proposition 5. Now let us estimate the two other terms in (4.4). By (4.6), u log Q (u) 1 α X(y; u π)y. By Statement 2 of Proposition 4, X(y; u, π) = (P π,y Hence, u log Q (u) On the other han, for y [α, π], X(y; u, π) = u P π,y π ) (u) O(e O(δ 3/2 ) ). O(δ 3/2 ) O(e ). P = (P π,y ) P (u) P (u).

19 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE 45 The last factor is in [1/2, 2] by assumption of the lemma. The first one is O( O(δ3/2 ) ) by reversibility an Proposition 2. Hence, the thir term in (4.4) amits the same estimate as the first one. Therefore, u log P has the same sign as the secon term in (4.4) for δ small. This proves Lemma Parameter variations: proof of Lemma 2. Lemma 2 (Monotonicity Lemma). Let ±, Ā± (), B ± () be liftings to the universal covers of ±, A ± (), B ± () which continuously epen on. Then 1. ±, (x y)(c()) as 0+ (4.19) for any of the choices: C() = Ā+ (); Ā (); B+ (); B (). 2. Equation = πn has a solution = n = O( 1 n ). Proof. Let us first prove (4.19) for + an B + (). The proof for, Ā ± (), B () requires a few aitional arguments provie below. By efinition, D = Q 1 (J + ), where Q is the map Γ Γ + along the phase curves of (1.3). Here Γ = {y = π}, Γ + = {y = α}, α = τ δ. We have: + = Q 1 (π). Let Q 3 Q 2 Q 1 be the ecomposition of Q 1 efine in 4.1. Denote R = Q 2 Q 1, now stressing its epenence on. By (3.1), Q 3 (x) = x + T () + ϕ (x). (4.20) Here ϕ is an -epenent x perioic function uniformly boune in the C 1 -norm as 0; T () is provie by (3.5): Now we have Hence, T () = 1 π τ δ 2 y v(y, ). v(y, ) = v(y, 0) + v 1 (y, ), v(y, 0) = T () = C + T 1(), C = π τ δ 2 2π 0 x a cos x cos y. y v(y, 0) < 0 (4.21) with T 1 smooth in. Note that C < 0 since π < τ δ 2. Differentiating (4.21), we fin T () + as 0 +. (4.22) The erivative of R (π) in is obviously positive. Inee, outsie the omain M : a cos x cos y 0 boune by the slow curve M, the orbit of system (1.3) is a curve that is transverse to the orbits of the same system with replace by a larger µ: < µ. Denote the first system by (1.3 ), an the secon one by (1.3 µ ). The map R is the map along the orbits of (1.3 ) with time reverse. The vectors of

20 46 J. GUCKENHEIMER AND YU. ILYASHENKO (1.3 ) with time reverse point to the left of the orbits of (1.3 µ ) with time reverse passing outsie M. Hence, for < µ, R (π) < R µ (π) = R (π) 0. Together with (4.20) an (4.22), this proves (4.19). For B + () the same argument works. Namely, (x y)( B + ()) = 2b + () = 2P 0, π (π). The corresponing orbit lies outsie M, because no orbit quits M in the omain {y [ π, 0]} an no orbit enters M in {y [0, π]}. Decompose P 0, π (π) = Q 3 R (π), R = P 0, τ δ2. We have: Q 3 + as 0 +, as prove previously. On the other han, R (π) > 0. The proof is the same as for R. To prove (4.19) for, Ā ± (), B (), this argument shoul be extene. It is well known [MR] that the true slow curve x 1 = 0 (x 2 = 0) in the initial coorinates is an -epening curve with x = s + (y) (x = s (y)), y [ β, β], β = τ δ 2, (4.23) s ± (y) = s ± (y) + s ± 1 (y) +... The erivative s ± 1 (y) may be easily foun from the assumption that (4.23) is an invariant curve of the system (1.3). Namely, s + 1 (y) = s+ (y) ϕ(y, 0), s 1 (y) = s (y) ψ(y, 0), where s ±, ϕ, ψ are the same as in 4.2. Hence, the ifference s + ( β) s + ( β) is positive of orer. On the other han, the maps P 0, β an Q 1 = P α, β are contracting on [0, π] with coefficient that tens to zero as exp( C/). This is prove for Q 1 = Q 1 in 4.1; the proof for P 0, β is the same. Hence, the points Q 1 (0), P 0, β (0) lie outsie M, an (P 0, β ) (0) 0 as 0. The monotonicity arguments above show that Q 2 > 0 outsie Γ 1 M. Hence, ( ) R (0) = Q 2 Q 1 (0) + Q 2 Q 1 Q 1(0). The first term is positive by the monotonicity arguments above; the secon one tens to zero. Hence, R (0) has a nonnegative lower limit as 0. The same arguments as for + prove (4.19) for an B (). To prove (4.19) for Ā± (), we must establish that y( ± ) = P ( ± ) has a nonpositive lower limit (in fact this limit is ). This is one by using the same arguments as for R. This proves statement 1 of Lemma 2.

21 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE 47 To prove statement 2, we must estimate the magnitue of R. Between the sections Γ + = {y = τ + δ} an Γ 1 = {y = τ + δ 2 }, the x coorinate of the trajectories oes not change by more than 2π because they are trappe by slow curves. The time of transition from Γ 1 to Γ 2 = {y = τ δ 2 } is 2δ 2 /. Therefore, the magnitue of R is ( ) δ 2 R (x) = O + O(1). Combining this estimate with (4.20), we see that n satisfies the equation nπ = C + O(δ2 ) + O() with C < 0. The solution of this equation is of magnitue O(1/n). This completes the proof of Lemma 2. References [AAIS] V. I. Arnol, V. S. Aframovich, Yu. S. Ilyashenko, L. P. Shilnikov, Bifurcation Theory, Dynamical Systems, vol. 5, Springer-Verlag, [D] M. Diener, The canar unchaine or how fast/slow ynamical systems bifurcate, The Mathematical Intelligencer 6 (1984), [DR] F. Dumortier an R. Roussarie, Canar cycles an center manifols, Mem. Amer. Math. Soc., vol. 121, no. 577, [F] N. Fenichel, Geometric singular perturbation theory for orinary ifferential equations, J. of Diff. Eq. 31 (1979), [I] Yu. Ilyashenko, Embeing theorems for local maps, slow-fast systems an bifurcations from Morse Smale to Smale Williams, Topics in singularities theory, V. I. Arnol s 60th anniversary collection, AMS Transl. ser. 2, vol. 180, Amer. Math. Soc., 1997, pp [IL] Yu. Ilyashenko, W. Li, Nonlocal bifurcations, Amer. Math. Soc, Provience, RI, [M] B. Manelbrot, Fractals in Geometry an Nature, Freeman, [MR] E. F. Mishchenko an N. Kh. Rozov, Differential equations with small parameters an relaxation oscillations, Plenum Press, New York Lonon, (Translate from the Russian by F. M. C. Goospee.) Mathematics Department, Cornell University, Ithaca, NY Inepenent University of Moscow, B. Vlasievsky per. 11, Moscow , Russia, Mathematics Department, Cornell University, Ithaca, NY 14853, Moscow State University, an Steklov Mathematical Institute aress: yulijs@mccme.ru, yulij@math.cornell.eu