Discretization of Poincaré map

We analytically study the relationship between the Poincare map and its one step discretization. Error estimates are established depending basically on the right hand side function of the investigated ODE and the given numerical scheme. Our basic tool is a parametric version of a Newton–Kantorovich type methods. As an application, in a neighborhood of a non-degenerate periodic solution a new type of step-dependent, uniquely determined, closed curve is detected for the discrete dynamics.


Introduction
This paper is devoted to the precise analytical derivation of the numerical/discretized Poincaré map of an ordinary differential equation possessing a periodic orbit.We have been motivated by papers [11,19], where numerical tools are used for computing the Poincaré map.On the other hand there is a nice theory studying dynamics of numerical approximations of ODE, see for instance [6-9, 17, 18].This paper is a contribution to this direction.
The continuous Poincaré map P for the smooth ODE with a 1-periodic orbit γ is a well understood topic and is contained in almost every textbook on continuous dynamical systems (e.g.[14]).In order to define the discretized version of Poincaré map, designated by P m , for the discrete dynamical system obtained from the one-step discretization procedure ψ we have chosen a method originated in [11] (m is the number of steps realized by the discretization scheme).Our goal is to give a precise analytical meaning of P m and to establish various error bounds between P and P m .It has to be noted that there are various possibilities how to define P m .Our approach is in some sense a natural one, it can be loosely summed up as: applying recurrently ψ with a constant step-size until the resulting elements are located on the "one side" of the Poincaré section and then establishing the suitable step-size needed to hit by ψ exactly that section.Precise setting and the corresponding analysis are treated in Section 2 and 3 (there arises a slight complication forcing us to assume p ≥ 2 for the order p of ψ -see Remark 2 in Section 3).Error bounds related to |P − P m | are given in a form C m q for m large enough and for a constant C essentially dependent on the right hand side of the ODE and the numerical scheme ψ (to be more precise, q = p in C 0 and q = p − 1 in C 1 norm estimates).Achieved results, as we have anticipated, correspond to [8] where the author examined the C j -closeness, j ≥ 0, between the flow and its numerical approximation.Our approach uses the techniques of a moving orthonormal system (introduced rigorously in [10] and then used successfully in [1,2,17]) and the Newton-Kantorovich type theorem (cf.[13,15,20]).Hence, P m is not unique but naturally depending on the choice of the Poincaré section and consequently on the corresponding tubular neighbourhood of the periodic orbit created by the mentioned moving orthonormal system.Sections 2, 3 and 4 are devoted to this topic.
In the last Section 5 we give an application of the previously developed results.It is a slight completion of [4], where two closed curves were found in a neighborhood of γ for the discrete dynamical system.The first one was found basically under the nondegeneracy of γ (that is when the trivial Floquet multiplier 1 of γ is simple).This curve is the set of m-periodic points x, where the step h of the scheme depends on x and is close enough to 1/m.The second, the maximal compact invariant set of the scheme in a neighborhood of γ, was derived under the hyperbolicity of γ, for any sufficiently small step (this is a historically well-known topic, it was treated for example in [1,2,5,16]).We also show using the nondegeneracy of γ that in a small neighborhood of γ the set of those points, which return into themselves under the action of P m , forms another new type of closed curves for any m large and h close enough to 1/m.Of course this curve in general differs from the compact maximal invariant set and depends on P m and the chosen tubular neighborhood.Hence, it might be considered as somewhat artificial.However, at the end of the paper, we show a simplification which leads us to the natural curve of m-periodic points depending only on the choice of the discretization mapping.We conclude Section 5 by a short remark on spectral properties of our detected curve, which is undoubtedly an interesting application of our achieved results about the numerical Poincaré map.
Finally we note that this paper is a starting point for our future study of discretized bifurcations near periodic orbits of parametrized ODEs.

General settings and tools
Assumptions made here are going to be valid for the whole paper.Let us have . Some technical reasons cause that we are forced to assume also p ≥ 2 (see below Remark 2 for more details).Let γ(s) := ϕ(s, ξ 0 ) be a 1-periodic solution for fixed ξ 0 ∈ R N .Then there is a system {e i (s)} N −1 i=1 of vectors in R N for any s ∈ R such that where i, j ∈ {1, . . ., N − 1}, δ i,j is a Kronecker's delta and Using the Implicit Function Theorem finite number of times we get that there is a δ tr > 0 such that is a C 3 -diffeomorphism between its domain and range (cf. the moving orthonormal system along γ in [10, Chapter VI.I., p.

214-219]) .
For values define the following useful functions Further let B be a compact set such that γ(R) is contained in the interior of B. Hence there is a constant R > 0 such that Lemma 2.1 (Neumann's Inversion Lemma).Suppose that X is a Banach space and A ∈ L(X) is invertible.Then for B ∈ L(X) such that |A −1 B| < 1 we have (A + B) −1 ∈ L(X), and Our central tool will be the following lemma.We also give a short proof in the Appendix.Lemma 2.2 (Newton-Kantorovich method).Let us have Banach spaces X, Y, Z and open nonempty sets U ⊂ X, V ⊂ Y. Let ȳ : U → V be any function such that B(ȳ(x), ) ⊂ V for every x ∈ U and for some > 0.

Let us have a function
for every x ∈ U and for some α, β > 0. Let hold for some l ≥ 0. For constants α, β, l, finally suppose βl < 1, (2.5) Then there is a unique function y : for all x ∈ U with an estimate We also get y ∈ C r (U, V ) if we additionally assume the continuity of ȳ.

Discretized Poincaré map
At first we state a lemma about the continuous Poincaré map, the proof can be found in the Appendix.
To get the exact meaning of P m mentioned informally in the introduction we have to solve the equation F m (h, s, c, X, ∆) = 0 near ( X, ∆).Here comes the first application of Lemma 2.2.Before this let us introduce some technicalities, at first the following positive constants Here D [k] is the k-th Fréchet differential.Note that an upper bound of a type C ψ could be given simply using (2.1) and constants C ϕ , C Υ .Next, let us have δ > 0, µ ∈ (0, 1) and introduce where x := min{k ∈ Z : k ≥ x} and x := − −x for any x ∈ R. Further The simple goal of these complicated assumptions is that for (h, s, c) ∈ H m it is straightforward to show Fix δ > 0, then for every m large, µ small enough and Moreover the functions X m , ∆ m are C 3 -smooth in their arguments and Proof.The proof is divided into several steps.Two main parts are the following ones: We solve H m (h, s, c, X m (h, s, c), ∆) = 0 for ∆ near ∆m .These parts are handled using Lemma 2.2 and contain four steps.
Step 1.2.We show that for any µ 1 ∈ (0, 1) and m is large enough (the main point is of course that the lower threshold of m-s depends also on µ 1 , its limit is ∞ as µ 1 → 0 + -from now on we omit remarks of this type).Using (2.1) again we get EJQTDE, 2013 No. 1, p. 8 Now AY = Z is solvable.Straightforward computation shows and (3.9) we arrive at the statement).
Next we also obtain in a moment |BY | ≤ C Υ h p+1 ((3.9) is used again).Now using we get and so we have 1 implies the invertibility of A + B and also that and we have arrived at (3.10).
Step 1.3.We show that for any µ 2 > 0 we have At first notice that from we have For m large enough we have that This follows from the following considerations.The condition δ/m p ≤ min{R/2C ϕ , δ re (ε )/2} is fulfilled for m large enough, this implies that |x j − γ(jh + s)| < R/2 (similar considerations as we obtained (3.9)).Now Using (3.14) and (3.2) we obtain Note again that (3.12) is valid, therefore for every m large enough we have and we have obtained exactly (3.13).
Step 1.4.Now the final step of the first part is coming.To fit into the framework of Lemma 2.2 with an equation G m (h, s, c, X) = 0 set It has to be noted that for large m, C X /m p ≤ R is valid and so (2.4) holds on B(ȳ(x), ).Conditions (2.5) and (2.6) have to be fulfilled.For (2.5) pick µ 3 ∈ (0, 1), then for m large enough we get Further using (3.12) we get , so (2.6) in this setting will be valid if According to the assumption C X < C X and that (1+µ) p+1 (1−µ 1 )(1−µ 3 ) → 1 + as µ, µ 1 , µ 3 → 0 + , there are always such suitably small parameters µ, µ 1 , µ 3 ∈ (0, 1) that (3.17) is valid.Therefore Lemma 2.2 can be used (the remaining assumptions are trivially satisfied) and it gives a unique element We show that for any µ 4 > 0 we have for all (h, s, c) ∈ H m and m large enough.At first note that where the first term vanishes because of Lemma 3.1.From (3.4) we infer ∆m ∈ (0, h 0 /2) for m large enough.Next For m large enough Step 2.2.We show for any µ 5 > 0 that where (h, s, c) ∈ H m and m is large enough.Straightforward computation yields where

Elementary considerations show that
therefore for m large enough we obtain This shows (3.20) and we are done.
Step 2.4.Finally we solve z(h, s, c, ∆) with Lemma 2.2 (see (3.18)).Set is valid for any µ 6 ∈ (0, 1) if m is sufficiently large which fulfills (2.5).Now Because of C ∆ < C ∆ and the already proven part of our theorem -that is C X can be chosen arbitrarily close to C X for m large enough -we conclude that (3.23) can be fulfilled (with sufficiently small µ, µ 4 , µ 5 , µ 6 > 0).Now Lemma 2.2 gives a unique element are valid and the proof is finished ((3.7) is a straightforward consequence of the 1-periodicity of G m , Xm , H m , z, ∆m in the variable s, and the uniqueness parts of the steps 1.4.and 2.4.).EJQTDE, 2013 No. 1, p. 13 Remark 1.In the framework of Theorem 3.2 a natural approximation of P is Notice that 2) and (3.15)).In addition from (3.4) and (3.6) we have Hence for any fixed µ 7 > 0 we have ∆ p+1 m Υ(∆ m , x m−1 m ) ≤ µ 7 m p for every m sufficiently large.
Putting all this together we arrive at where κ > κ := C ϕ (C X + C ∆ ) is an arbitrary constant, m is sufficiently large and µ, µ 7 are small enough (c.f.(3.5)).
Remark 2. With minor modifications in our settings p ≥ 1 would be possible until now (basically to tackle the additional case p = 1 we would need: the extension ψ to be a function defined on [−h 0 , h 0 ] × R N ; enlarging constants in (3.2) by replacing [0, h 0 ] with [−h 0 , h 0 ]; suitable changes in the definitions of d m , m 0 , I m , B m ).The fundamental difference in the case p = 1 would be that the natural requirement 0 < ∆ m < 2h is generally not satisfied, even for m large.So the last step-size is inappropriate.Possible correction would EJQTDE, 2013 No. 1, p. 14 be to find the right number of iterations of ψ(h, •) to ensure that the next iteration with a step ∆ near h (at least satisfying 0 < ∆ < 2h) we hit the Poincaré section.This procedure does not fit to our approach based on Lemma 2.2 therefore we are not going to specify the details.

Closeness of differentials
Now we would like to get an upper bound in the spirit of ( Then we are able to extend the results of Lemma 2.2 by an estimate where Proof.From the equations F (x, y(x)) = 0 and F (x, ȳ(x)) = ϑ(x) after differentiation we infer for x ∈ U that y From now we omit (x, y(x)) and (x, ȳ(x)), the superscript ¯above F will indicate the substitution of (x, ȳ(x)), otherwise we substitute (x, y(x)).We have from which we get exactly (4.2) (using (4.1) and the assumptions and results of Lemma 2.2) and the proof is finished.

EJQTDE, 2013 No. 1, p. 15
Adopting the notations of Theorem 3.2 and applying the previous lemma we may obtain the following statement, which is a continuation of Theorem 3.2.Theorem 4.2.There are constants C V,v for V ∈ {X, ∆} and v ∈ {h, s, c} such that where δ > 0 is an arbitrary constant, m is large enough, µ is sufficiently small and (h, s, c) ∈ H m (p, δ, µ).
Proof.To be able to apply Lemma 4.1 twice with frameworks described in (3.16) and (3.22) we have to find additional constants (for the sake of (4.1)) for all V ∈ {X, ∆}, v ∈ {h, s, c}.This will be a bit sweating task.Part 1.1 -about α 1 [X, v] for v ∈ {h, s, c}.After differentiation we get for j = 1, 2, . . ., m − 1. Therefore (using (3.2) and that where µ 9 > 0 is an arbitrary parameter and m is large enough (C ϕ C E δ/m p ≤ µ 9 is valid for m large enough).So This implies .
for any fixed µ 10 > 0, every m large enough and µ sufficiently small.This yields for Part 1.4 -determining C X,v for v ∈ {h, s, c}.Now we are ready to apply Lemma 4.1 in a setting (3.16) extended with (4.5),(4.6)and (4.7).From (4.2) we obtain exactly (4.4) EJQTDE, 2013 No. 1, p. 17 for every m large enough.Indeed, for example in the case v = h (others are treated similarly) we get from (4.3) for µ 11 > 0 that for m large and µ small enough (we have also used (3.5) from Theorem 3.2).Part 2.1 -about α 1 [∆, v] for v ∈ {h, s, c}.We easily get Therefore where So at first we handle terms D v w m for v ∈ {h, s, c}.Straightforward computation shows that where Let us have µ 12 > 0, then computations as in the previous parts show that for m large and µ small enough we have For the remaining parts of the right side of (4.9) we have upper bounds in (4.5), (4.8) and in the already proved case of (4.4) (c.f.Part 1.4).Putting this together we get for any µ 13 > 0 that where m is sufficiently large, µ is small enough and )/m p .Therefore we can finish this step with the following choices where µ 14 > 0 is an arbitrary parameter, m is large and µ is small enough.

Note that from a triangle inequality we have
Employing Newton-Leibniz formula straightforward computation implies that for any µ 15 > 0 and m large enough we have the choices for every m large, µ small.The proof is complete.Remark 3. Now as in the proof of Theorem 4.2 (see (4.9)) we get for v ∈ {h, s, c}, where From (3.4) we infer After a lengthy computation for µ 16 > 0 we get where m is large µ is small enough.Using in addition (4.4), (4.5) and (4.8) for remaining terms in (4.12) we finally obtain where One may wish to continue in this direction developing bounds for This is quite technical (computations rather for computer), therefore we only show the key equipment namely the natural extension of Lemma 2.2 to the next level in the spirit of Lemma 4.1.
Lemma 4.3.Suppose all the assumptions of Lemma 2.2 with X = X 1 × X 2 × X 3 (X i , i ∈ {1, 2, 3} are Banach spaces, and Let us have F ∈ C r (U × V, Z) for r ≥ 2 and also ȳ ∈ C 2 (U, V ).Suppose (like in (4.1)) that EJQTDE, 2013 No. 1, p. 21 for i ∈ {1, 2, 3}.Introduce also 1,i := β 1−βl (l α 1,i + l 1,i + α 2,i ) accordingly to (4.2).Further let us have for i, j ∈ {1, 2, 3}, i ≤ j and for all x ∈ U and y 1 , y 2 ∈ B(ȳ(x), ).Then where 2,i,j := Proof.Partial derivations with respect to x i into the direction δv ∈ X i of the equations F (x, y(x)) = 0 and F (x, ȳ(x)) = ϑ(x) gives (we use notation F from the proof of Lemma 4.1) Now differentiating once more with respect to x j into the direction δw ∈ X j we get Therefore as in the proof of Lemma 4.1 we infer (y Now using the symmetry of the second derivatives, switching to the norms and employing the assumptions of the theorem the final statement (4.16) follows and the proof is finished.Now we show a sketch of one possible application of Lemma 4.3.Let the equation G m (h, s, c, X) = 0 be in the role of F (x 1 , x 2 , x 3 , y) = 0 with a basic framework given in (3.16).We only deal with the case i = j = 3, when we are looking for a bound of |D 2 cc X m − D 2 cc Xm |.The proof of Theorem 4.2namely (4.5), (4.6) and (4.7) -using notations of (4.14) implies , l 1 = 0 needed in (4.14).Remaining constants in (4.15), skipping the details of the lengthy computation, are , α 5 = 0, Now application of Lemma 4.3 yields that for C X,cc > C X,cc , m large and µ small enough we have where C X,cc : Similarly it is possible to handle the equation z(h, s, c, ∆) = 0 in a setting (3.22).From (4.8), (4.10) and (4.11) we get EJQTDE, 2013 No. 1, p. 23 Omitting again the details we get for µ 17 > 0, m large and µ small enough that for m large enough where C ∆,cc > C ∆,cc := N (p+1)pCϕC Υ C min .Now as in the Remark 1 it would be possible to derive for some constant C. Instead of this we show a weaker result, namely that |D 2 cc P m (h, s, c)| is uniformly bounded for every m large enough (uniformity is related to m-s).

A closed curve for a discrete dynamics
The nondegeneracy condition of γ From (3.24) we get Picking up any µ 18 ∈ [0, 1) for every m large enough we obtain So from Lemma 2.1 we infer that g c (h, s, 0) is invertible with In the context of the setting of Lemma (2.2) we have derived and we have also = δ m p .Now for any µ 19 ∈ (0, 1) we get  Thorough study of the set of m-periodic points for discretized dynamics was done in [4].Our approach implies some results also to this direction.Theorem 5.2.Suppose all the assumptions of Theorem 5.1 and fix any η ∈ (0, 1).Then for m large enough we have for every s ∈ R a unique element h (s) ∈ I m such that ∆ m (h (s), s, ζ m (h (s), s)) = h (s).
This is satisfied for m large and κ − κ, µ, µ 18 , µ 19 small enough because of (3.24) and δ > √ N κa.Application of Lemma 2.2 gives ζ m with the desired properties and the proof is finished.