3 Discretized Poincar´ e map

Electronic Journal of Qualitative Theory of Differential Equations 2013, No.60, 1–33;http://www.math.u-szeged.hu/ejqtde/

Discretization of Poincar´e map

Michal Feˇckan^∗†

Department of Mathematical Analysis and Numerical Mathematics Comenius University, Mlynsk´a dolina, 842 48 Bratislava, Slovakia

and Mathematical Institute, Slovak Academy of Sciences ˇStef´anikova 49, 814 73 Bratislava, Slovakia

Email: Michal.Feckan@fmph.uniba.sk S´andor Kelemen ^‡

Mathematical Institute, Slovak Academy of Sciences ˇStef´anikova 49, 814 73 Bratislava, Slovakia

Email: kuglof@gmail.com October 6, 2013

Abstract

We analytically study the relationship between the Poincar´e map and its one step discretization. Error estimates are established de- pending 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.

Keywords: discrete Poincar´e map; Newton–Kantorovich Theorem;

periodic solutions.

AMS Subject Classification: 37M99; 65P99.

∗Corresponding author

†Partially supported by Grants VEGA-MS 1/0507/11, VEGA-SAV 2/0029/13 and APVV-0134-10

‡Partially supported by Grant VEGA-SAV 2/0029/13

1 Introduction

This paper is devoted to the precise analytical derivation of the numeri- cal/discretized Poincar´e map of an ordinary differential equation possessing a periodic orbit. We have been motivated by papers [11, 19], where numeri- cal tools are used for computing the Poincar´e 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 Poincar´e mapP for the smooth ODE with a 1-periodic orbit γ is a well understood topic and is contained in almost every text- book on continuous dynamical systems (e.g. [14]). In order to define the discretized version of Poincar´e map, designated byP_m,for the discrete dy- namical system obtained from the one-step discretization procedure ψ we have chosen a method originated in [11] (m is the number of steps real- ized by the discretization scheme). Our goal is to give a precise analytical meaning ofP_m and to establish various error bounds betweenP 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 recurrentlyψwith a constant step-size until the resulting elements are located on the “one side” of the Poincar´e section and then establishing the suitable step-size needed to hit byψexactly 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 orderp of ψ – see Remark 2 in Section 3). Error bounds related to |P − P_m| are given in a form _m^Cq form 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 inC⁰ andq =p−1 inC¹ 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 Poincar´e 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 ofγ (that is when the trivial

Floquet multiplier 1 of γ is simple). This curve is the set of m-periodic pointsx, 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 ofγ the set of those points, which return into themselves under the action ofP_m, forms another new type of closed curves for anym large andh close enough to 1/m. Of course this curve in general differs from the compact maximal invariant set and depends onP_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 Poincar´e map.

Finally we note that this paper is a starting point for our future study of discretized bifurcations near periodic orbits of parametrized ODEs.

2 General settings and tools

Assumptions made here are going to be valid for the whole paper. Let us havef ∈C³(R^N), N ∈N\ {1}such that

ϕ:R×R^N →R^N is the global flow of ˙x=f(x).

For a numerical schemeψ: [0, h0]×R^N →R^N, h0 ∈(0,1) suppose for some p∈Nthat

ψ(h, x) =ϕ(h, x) + Υ(h, x)h^p+1. (2.1) Assume again ψ,Υ ∈ C³([0, h₀]×R^N,R^N). 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, ξ₀) be a 1-periodic solution for fixed ξ₀ ∈ R^N. Then there is a system{e_i(s)}^N_i=1⁻¹ of vectors inR^N for any s∈R such that

ei∈C³(R,R^N), ei(s+ 1) =ei(s), he_i(s), e_j(s)i=δ_i,j, he_i(s), f(γ(s))i= 0,

)

(2.2) where i, j ∈ {1, . . . , N −1}, δ_i,j is a Kronecker’s delta and h·,·i is the standard Euclidean scalar product. Introduce an N × (N − 1) matrix

E(s) = [e1, . . . , eN−1] (i-th column isei, i= 1, . . . , N−1). Let us also set a tubular coordinate functionξ(s, c) :=γ(s) +E(s)cfors∈R, c∈R^N−1.For standard Euclidean norm|c|₂ :=p

hc, cinote that|E(s)c|₂ =|c|₂, c∈R^N−1. For δ > 0 introduce the notation B_N^δ₋₁ :=

c∈R^N−1 : |c|₂ < δ . Using the Implicit Function Theorem finite number of times we get that there is aδ_tr>0 such that

ξ: [0,1)×B_N^δtr₋₁→R^N is a C³-transformation, in other words ξ|_[0,1)×Bδtr

N−1

is aC³-diffeomorphism between its domain and range (cf. the moving orthonormal system alongγ in [10, Chapter VI.I., p.

214–219]) . For values

h∈[0, h0], s∈R, c∈R^N−1,∆∈[0, h0],

X := (x¹, x², . . . , x^m−1)∈R^N(m−1), xⁱ ∈R^N, m∈N, m≥4, define the following useful functions

F_m(h, s, c, X,∆) :=(G_m(h, s, c, X), H_m(h, s, c, X,∆)),

Gm(h, s, c, X) := ψ(h, ξ(s, c))−x¹, ψ(h, x¹)−x², ψ(h, x²)−x³, . . . , ψ(h, x^m−2)−x^m−1

, Hm(h, s, c, X,∆) :=

ψ ∆, x^m−1

−γ(s), f(γ(s)) . X¯_m:= ¯X_m(h, s, c) := ¯x¹,x¯², . . . ,x¯^m−1

,

¯

x^j :=¯x^j(h, s, c) :=ϕ(jh, ξ(s, c)), j= 1,2, . . . , m−1.

Further letBbe a compact set such thatγ(R) is contained in the interior ofB. Hence there is a constantR >0 such that

x∈R^N : min

s∈R

{|x−γ(s)|} ≤R ⊂B. (2.3) We mean by| · |the standard maximum norm|v|:= max{|v_i| : i= 1, . . . , l}

forv∈R^l, l∈N.Notation| · |is used also for linear operatorsA:R^l1 →R^l2 defined as|A|:= max_v∈

R^l1,|v|=1|Av|.Further byL(X, Y) for Banach spaces X, Y we mean the Banach space of continuous and linear operatorsA:X→ Y,in the case X=Y we set L(X) :=L(X, X). In general| · |_X will denote the norm in a Banach space X, however in most of the cases there are no arising confusions so we use again simply | · |.An open ball will be denoted asB(x, %) :={y ∈X : |y−x|< %} for any x∈X and % >0.

Several times we will use the following well-known result.

Lemma 2.1 (Neumann’s Inversion Lemma). Suppose that X is a Banach space andA∈ L(X)is invertible. Then forB ∈ L(X)such that|A⁻¹B|<1 we have (A+B)⁻¹∈ L(X),and

(A+B)⁻¹ =X

n≥0

(A⁻¹B)ⁿA⁻¹,

(A+B)⁻¹

≤ |A⁻¹| 1− |A⁻¹B|.

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 spacesX, Y, Z and open nonempty setsU ⊂X, V ⊂Y.Lety¯:U →V be any function such that

B(¯y(x), %)⊂V for everyx∈U and for some % >0.

Let us have a function F ∈C^r(U ×V, Z) for r≥1.Suppose that D_yF(x,y(x))¯ ⁻¹∈ L(Z, Y),

|F(x,y(x))| ≤¯ α, |D_yF(x,y(x))¯ ⁻¹| ≤β for everyx∈U and for some α, β >0.Let

|D_yF(x, y₁)−D_yF(x, y₂)| ≤l|y₁−y₂|, x∈U, y₁, y₂ ∈B(¯y(x), %) (2.4) hold for some l≥0.For constants α, β, l, % finally suppose

βl% <1, (2.5)

αβ < %(1−βl%). (2.6)

Then there is a unique function y :U →V such that

|y(x)−y(x)| ≤¯ % and F(x,y(x)) = 0 for all x∈U.

Moreover

|y(x)−y(x)|¯ < %, DyF(x,y(x))⁻¹ ∈ L(Z, Y) for allx∈U with an estimate

DyF(x,y(x))⁻¹

≤ β 1−βl%.

We also gety∈C^r(U, V) if we additionally assume the continuity of y.¯

3 Discretized Poincar´ e map

At first we state a lemma about the continuous Poincar´e map, the proof can be found in the Appendix.

Lemma 3.1 (Poincar´e’s time return map). There is an ε^? ∈(0,1/2) such that for every ε∈(0, ε^?]there is δ_re=δ_re(ε)∈(0, δ_tr]and a C³-function

τ :R×B_N−1^δre(ε) →(1−ε,1 +ε)

such that for t∈(1−ε,1 +ε), s∈R andc∈B_N−1^δre(ε) we have

z(t, s, c) = 0 for z(t, s, c) :=hϕ(t, ξ(s, c))−γ(s), f(γ(s))i (3.1) if and only if t=τ(s, c). In addition τ(s+ 1,·) =τ(s,·), s∈R.

In this context the usual Poincar´e map is defined as P(s, c) :=ϕ(τ(s, c), ξ(s, c)).

Further for admissible values of (h, s, c) using τ from the above lemma in- troduce

∆¯_m := ¯∆_m(h, s, c) :=τ(s, c)−(m−1)h.

To get the exact meaning of P_m mentioned informally in the introduc- tion we have to solve the equationFm(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 followingpositive constants

CΥ≥ max

h∈[0,h0], x∈B, k∈{0,1,2,3}

|D^[k]Υ(h, x)| ,

C_ϕ≥max (

h∈[0,hmax₀], x∈B, k∈{1,2,3}

|D^[k]ϕ(h, x)|},

h∈[0,3/2]max

|ϕ⁰_x(h, x)| , max

h∈[0,h0]

|ϕ^[4]_txxx(h, x)|

) , C_min≤ min

x∈γ(R){|f(x)|²₂}, Cτ ≥ max

s∈[0,1], k∈{1,2}, c∈B^δ_N−1^re(ε?)/2

n

|D^[k]τ(s, c)|o ,

C_E ≥max{|E⁰(s)|, s∈[0,1]}, Cψ ≥ max

h∈[0,h0], x∈B k∈{1,2,3}

|D^[k]ψ(h, x)| .































































(3.2)

Here D^[k] is the k-th Fr´echet 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 haveδ >0, µ∈(0,1) and introduce

dm :=dm(p, δ, µ) := µ−_m^Cp−1^τδ

m(m−1), for

m≥m0(p, δ, µ) := max (

2 h0

,

&

δ δre(ε^?)

1/p' ,

$ C_τδ

µ

_p−1¹ % + 1

) , wheredxe:= min{k∈Z:k≥x} and bxc:=−d−xe for any x∈R.Further

I_m :=I_m(p, δ, µ) :=

1

m −d_m, 1 m +d_m

, B_m:=B_m(p, δ) :=B_N−1^δ/mp,

H_m :=H_m(p, δ, µ) :=I_m×R× B_m,











(3.3)

also for m≥m0.

The simple goal of these complicated assumptions is that for (h, s, c) ∈ H_m it is straightforward to show

d_m>0, I_m ⊂(0, h₀], c∈B_N^δre₋₁^(ε?),

and 1−µ

m <∆¯_m < 1 +µ

m . (3.4)

Theorem 3.2. Choose any constants C_X,C_∆ such that C_X >C_X := C_ϕC_Υ, C_∆>C_∆:= NC³_ϕC_Υ

C_min . (3.5) Fixδ >0,then for everym large,µsmall enough and(h, s, c)∈ H_m(p, δ, µ) there exists a unique pair (X_m,∆_m) = (X_m(h, s, c),∆_m(h, s, c)) such that

F(X_m,∆_m) =F_m(h, s, c, X_m(h, s, c),∆_m(h, s, c)) = 0 and

|X_m−X¯m|<CX/m^p, |∆_m−∆¯m|<C∆/m^p. (3.6) Moreover the functions X_m,∆_m are C³-smooth in their arguments and

(X_m,∆_m)(h, s+ 1, c) = (X_m,∆_m)(h, s, c), (h, s, c)∈ H_m. (3.7)

Proof. The proof is divided into several steps. Two main parts are the following ones:

Part 1. The solution X_m close to ¯X_m of G_m(h, s, c, X) = 0 is found.

Part 2. We solve Hm(h, s, c, Xm(h, s, c),∆) = 0 for ∆ near ¯∆m. These parts are handled using Lemma 2.2 and contain four steps.

Step 1.1. We show that

|G_m(h, s, c,X¯_m)| ≤C_Υh^p+1 (3.8) is valid for all (h, s, c) ∈ H_m and m large enough. From (2.1) we have for j= 1, . . . , m−1 if m is large enough that

|(G_m(h, s, c,X¯_m))^j|=|(ψ(h,x¯^j−1)−ϕ(h,x¯^j−1))|

≤h^p+1|Υ(h,x¯^j−1)| ≤CΥh^p+1

where ¯x⁰ :=ξ(s, c). Indeed, noting that δ/m^p ≤min{R/C_ϕ, δ_re(ε^?)/2} and jh≤(m−1) _m¹ +dm

≤ ³₂ are valid for mlarge enough we get using (3.2) that

|¯x^j −γ(jh+s)|=

Z 1 0

ϕ⁰_x(jh, γ(s) +ϑE(s)c)E(s)cdϑ

≤C_ϕ|E(s)c| ≤C_ϕ|E(s)c|₂= C_ϕ|c|₂≤C_ϕδ/m^p≤R.

Hence using (2.3) we have

¯

x^j =ϕ(jh, ξ)∈B forj= 0,1, . . . , m−1, (3.9) and so |Υ(h,x¯^j−1)| ≤C_Υ and we are done.

Step 1.2. We show that for anyµ1∈(0,1) D_XG_m(h, s, c,X¯_m)⁻¹

≤ Cϕm

1−µ₁ (3.10)

holds if (h, s, c) ∈ H_m, and m is large enough (the main point is of course that the lower threshold ofm-s depends also onµ1,its limit is∞asµ1 →0⁺ – from now on we omit remarks of this type).

Using (2.1) again we getDXGm(h, s, c,X¯m)[Y] =AY +BY where AY := −y¹, ϕ⁰_x(h,x¯¹)y¹−y², ϕ⁰_x(h,x¯²)y²−y³, . . .

. . . , ϕ⁰_x(h,x¯^m−2)y^m−2−y^m−1 ,

BY := 0, h^p+1Υ⁰_x(h,x¯¹)y¹, h^p+1Υ⁰_x(h,x¯²)y², . . . , h^p+1Υ⁰_x(h,x¯^m−2)y^m−2 .

NowAY =Z is solvable. Straightforward computation shows y¹ =−z¹,

y^j =−z^j−

j−1

X

r=1

ϕ⁰_x(rh,x¯^j−r)z^j−r, j= 2, . . . , m−1.











(3.11)

Therefore|A⁻¹Z| ≤C_ϕm(because (3.11) implies|y^j| ≤(1 + (m−2)C_ϕ)|Z|

forj = 1, . . . , m−1, noticing Cϕ ≥1 and (3.9) we arrive at the statement).

Next we also obtain in a moment|BY| ≤C_Υh^p+1 ((3.9) is used again). Now using

h < 1

m +dm< 1 +µ

m (3.12)

we get

|A⁻¹B| ≤CϕmCΥh^p+1 < C_ϕC_Υ(1 +µ)^p+1 m^p

and so we have|A⁻¹B| ≤µ₁ < 1 ifm is large enough. Lemma 2.1 implies the invertibility ofA+B and also that

(A+B)⁻¹

≤ |A⁻¹|

1− |A⁻¹B| ≤ C_ϕm 1−µ₁ and we have arrived at (3.10).

Step 1.3. We show that for anyµ2>0 we have

|D_XGm(h, s, c, X1)−DXGm(h, s, c, X2)| ≤ (1 +µ)Cϕ+µ2

m |X₁−X2| (3.13) for allX₁, X₂ ∈B( ¯X_m, R/2),(h, s, c)∈ H_m and m large enough.

At first notice that from

ϕ(h, x) =ϕ(0, x) + Z 1

0

∂

∂η(ϕ(ηh, x))dη

=x+h Z 1

0

ϕ⁰_t(ηh, x)dη we have

ϕ⁰⁰_xx(h, x) =h Z 1

0

ϕ⁰⁰⁰_txx(ηh, x)dη which readily implies (cf. (3.2))

|ϕ⁰_x(h, x₁)−ϕ⁰_x(h, x₂)| ≤hC_ϕ|x₁−x₂| (3.14)

for allx1, x2 such thatx1+ϑ(x2−x1)∈B, ϑ∈[0,1].

Form large enough we have that

∀X₁, X₂ ∈B( ¯X_m, R/2) : x^j₁+ϑ(x^j₂−x^j₁)∈B, j= 1, . . . , m−1. (3.15) This follows from the following considerations. The condition δ/m^p ≤ min{R/2C_ϕ, δre(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

|x^j₁+ϑ(x^j₂−x^j₁)−γ(jh+s)|

≤(1−ϑ)|x^j₁−x¯^j|+ϑ|x^j₂−x¯^j|+|¯x^j−γ(jh+s)|

<(1−ϑ)R 2 +ϑR

2 +R 2 =R

so from (2.3) we havex^j₁+ϑ(x^j₂−x^j₁)∈B which is exactly (3.15).

For such anX1, X2 using (2.1) we derive that

(DXGm(h, s, c, X1)−DXGm(h, s, c, X2)) [Y]

=

0, ϕ⁰_x(h, x¹₁)−ϕ⁰_x(h, x¹₂)

y¹, ϕ⁰_x(h, x²₁)−ϕ⁰_x(h, x²₂) y², . . . . . . , ϕ⁰_x(h, x^m−2₁ )−ϕ⁰_x(h, x^m−2₂ )

y^m−2 +

0, h^p+1 Υ⁰_x(h, x¹₁)−Υ⁰_x(h, x¹₂)

y¹, h^p+1 Υ⁰_x(h, x²₁)−Υ⁰_x(h, x²₂) y², . . . . . . , h^p+1 Υ⁰_x(h, x^m−2₁ )−Υ⁰_x(h, x^m−2₂ )

y^m−2

.

Using (3.14) and (3.2) we obtain

|D_XG_m(h, s, c, X₁)−D_XG_m(h, s, c, X₂)| ≤h(C_ϕ+h^pC_Υ)|X₁−X₂|.

Note again that (3.12) is valid, therefore for everym large enough we have h(Cϕ+h^pCΥ)< (1 +µ)Cϕ+^(1+µ)_m^p+1p ^CΥ

m ≤ (1 +µ)Cϕ+µ2

m 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 equationG_m(h, s, c, X) = 0 set

U :=H_m, V :=R^N(m−1), x= (h, s, c),y(x) := ¯¯ X_m(h, s, c), α:= CΥ

m^p+1, β := C_ϕm 1−µ1

, l:= (1 +µ)C_ϕ+µ₂

m , %:= CX

m^p.







(3.16)

It has to be noted that for largem, CX/m^p ≤R is valid and so (2.4) holds onB(¯y(x), %).Conditions (2.5) and (2.6) have to be fulfilled. For (2.5) pick µ₃∈(0,1), then for m large enough we get

βl%= (1 +µ)C²_ϕ+µ2CXCϕ

m^p ≤µ₃ <1.

Further using (3.12) we get αβ

%(1−βl%) < CϕC_Υ(1 +µ)^p+1 C_X(1−m₁)(1−µ₃), so (2.6) in this setting will be valid if

CϕCΥ

(1 +µ)^p+1

(1−µ1)(1−µ3) <CX. (3.17) According to the assumption CX < CX and that _(1−µ^(1+µ)p+1

1)(1−µ3) → 1⁺ as µ, µ₁, µ₃ →0⁺, there are always such suitably small parametersµ, µ₁, µ₃ ∈ (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 elementXm(h, s, c)∈ B( ¯X_m,C_X/m^p) such that

Gm(h, s, c, Xm(h, s, c)) = 0.

Moreover Xm isC³-smooth, |X_m−X¯m|<C_X/m^p and

|D_XG_m(h, s, c, X_m)⁻¹| ≤ β

1−βl% ≤ Cϕm (1−µ₁)(1−µ₃). Step 2.1. Set

z(h, s, c,∆) : =H_m(h, s, c, X_m(h, s, c),∆)

=

ψ(∆, x^m−1_m )−γ(s), f(γ(s)) .

)

(3.18) We show that for anyµ₄>0 we have

|z(h, s, c,∆¯_m)| ≤ NC²_ϕC_X+µ₄

m^p (3.19)

for all (h, s, c)∈ H_m and m large enough. At first note that z(h, s, c,∆¯m) =

ϕ( ¯∆m,x¯^m−1)−γ(s), f(γ(s)) +

ϕ( ¯∆_m, x^m−1_m )−ϕ( ¯∆_m,x¯^m−1) + ¯∆^p+1_m Υ( ¯∆_m, x^m−1_m ), f(γ(s))

where the first term vanishes because of Lemma 3.1. From (3.4) we infer

∆¯_m ∈(0, h₀/2) form large enough. Next

|ϕ( ¯∆m, x^m−1_m )−ϕ( ¯∆m,x¯^m−1)| ≤Cϕ|x^m−1_m −x¯^m−1|<CϕCX/m^p,

|∆¯^p+1_m Υ( ¯∆m, x^m−1_m )| ≤ (1 +µ)^p+1CΥ

m^p+1 . From|ha, bi| ≤N|a||b|andϕ⁰_t(0, x) =f(x) we obtain

|z(h, s, c,∆¯_m)| ≤

NC_ϕ

C_ϕC_X+ ^(1+µ)_m^p+1CΥ

m^p .

Form large enough ^NCϕ(1+µ)_m ^p+1CΥ ≤µ4 is valid, therefore (3.19) holds.

Step 2.2. We show for anyµ₅ >0 that

|D_∆z(h, s, c,∆¯m)⁻¹| ≤ 1 +µ5

Cmin

(3.20) where (h, s, c) ∈ H_m and m is large enough. Straightforward computation yieldsD_∆z(h, s, c,∆m) =|f(γ(s))|²₂+wm(h, s, c) where

w_m(h, s, c) :=D

f(ϕ( ¯∆_m, x^m−1_m ))−f(ϕ( ¯∆⁰_m,x¯^m−1,0)) + ¯∆^p+1_m Υ⁰_h( ¯∆m, x^m−1_m ), f(γ(s))

E ,

∆¯⁰_m := ¯∆m(h, s,0) = 1−(m−1)h,

¯

x^m−1,0 :=¯x^m−1(h, s,0) =γ(s+ (m−1)h).

Elementary considerations show that

|w_m| ≤ NC²_ϕδ(C_X+δ(C_τ+√ NC_ϕ))

m^p ,

therefore form large enough we obtain

|D_∆z(h, s, c,∆¯_m)| ≥ |f(γ(s))|²₂

1 +µ₅ ≥ C_min 1 +µ₅. This shows (3.20) and we are done.

Step 2.3. We have that

|D_∆z(h, s, c,∆1)−D_∆z(h, s, c,∆2)| ≤NCϕC_ψ|∆₁−∆2| (3.21)

is valid for all (h, s, c)∈ H_m,∆1,∆2∈[0, h0] and m large. We easily derive that

D∆z(h, s, c,∆1)−D∆z(h, s, c,∆2)

=

ψ⁰_h(∆₁, x^m−1_m )−ψ⁰_h(∆₂, x^m−1_m ), f(γ(s))

= Z 1

0

ψ_hh⁰⁰ (∆2+ϑ(∆1−∆2), x^m−1_m )dϑ, f(γ(s))

(∆1−∆2) which immediately yields (3.21).

Step 2.4. Finally we solvez(h, s, c,∆) with Lemma 2.2 (see (3.18)). Set U :=H_m, V := (0, h₀), x:= (h, s, c),y(x) := ¯¯ ∆_m(h, s, c),

α:= NC²_ϕCX +µ4

m^p , β := 1 +µ₅ Cmin

, l:=NCϕCψ, %:= C∆/m^p.







(3.22)

Note (3.4) again, so B( ¯∆_m, %)⊂V holds for mlarge enough. Now βl%= (1 +µ₅)NC_ϕC_ψC_∆

m^p ≤µ6<1

is valid for any µ6∈(0,1) ifm is sufficiently large which fulfills (2.5). Now αβ

%(1−βl%) ≤ (NC²_ϕCX +µ4)(1 +µ5) C∆(1−µ6) , therefore (2.6) holds if

(NC²_ϕCX+µ4)(1 +µ5)

C∆(1−µ6) <1. (3.23) Because of C∆<C∆and the already proven part of our theorem – that is CX

can be chosen arbitrarily close to CX formlarge enough – we conclude that (3.23) can be fulfilled (with sufficiently smallµ, µ₄, µ₅, µ₆>0). Now Lemma 2.2 gives a unique element ∆_m ∈ B( ¯∆_m,C_∆/m^p) with z(h, s, c,∆_m) = 0.

Moreover

|∆_m−∆¯m|<C∆/m^p, |D_∆z(h, s, c,∆m)⁻¹| ≤ β

1−βl% ≤ 1 +µ5

Cmin(1−µ4) are valid and the proof is finished ((3.7) is a straightforward consequence of the 1-periodicity ofGm,X¯m, Hm, z,∆¯m in the variables, and the uniqueness parts of the steps 1.4. and 2.4.).

Remark 1. In the framework of Theorem 3.2 a natural approximation ofP is

P_m(h, s, c) :=ψ ∆_m(h, s, c), x^m−1_m (h, s, c) . Now

|P(s, c)− P_m(h, s, c)| ≤

ϕ(τ, ξ)−ϕ(∆m, x^m−1_m ) +

∆^p+1_m Υ(∆m, x^m−1_m ) . Notice that

ϕ(τ, ξ)−ϕ(∆m, X_m^m−1) =

ϕ( ¯∆m,x¯^m−1)−ϕ(∆m, x^m−1_m )

≤

ϕ( ¯∆_m,x¯^m−1)−ϕ(∆_m,x¯^m−1) +

ϕ(∆_m,x¯^m−1)−ϕ(∆_m, x^m−1_m )

≤ Z 1

0

|ϕ⁰_t(∆m+ϑ( ¯∆m−∆m),x¯^m−1)|dϑ|∆¯m−∆m| +

Z 1 0

|ϕ⁰_x(∆m, x^m−1_m +ϑ(¯x^m−1−x^m−1_m ))|dϑ|¯x^m−1−x^m−1_m |,

therefore |ϕ(τ, ξ)−ϕ(∆m, X_m^m−1)| ≤Cϕ(C_X + C_∆)/m^p (we used (3.2) and (3.15)). In addition from (3.4) and (3.6) we have

|∆_m| ≤ |∆¯m|+|∆_m−∆¯m| ≤ 1 +µ m + C_∆

m^p so

∆^p+1_m Υ(∆m, x^m−1_m ) ≤

1 +µ+_m^Cp−1^∆

p+1

CΥ

m^p+1 .

Hence for any fixed µ₇ >0 we have

∆^p+1m Υ(∆_m, x^m−1_m )

≤ _m^µ7_p for every m sufficiently large.

Putting all this together we arrive at

|P(s, c)− P_m(h, s, c)| ≤κ/m^p, (3.24) where κ > κ := Cϕ(CX + C∆) is an arbitrary constant, m is sufficiently large and µ, µ7 are small enough (c.f. (3.5)).

Remark 2. With minor modifications in our settingsp≥1 would be possible until now (basically to tackle the additional casep= 1 we would need: the extensionψto be a function defined on [−h₀, h₀]×R^N; enlarging constants in (3.2) by replacing [0, h₀] with [−h₀, h₀]; suitable changes in the definitions ofdm, m0,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 formlarge. So the last step-size is inappropriate. Possible correction would

be to find the right number of iterations of ψ(h,·) to ensure that the next iteration with a step ˆ∆ near h (at least satisfying 0 < ∆ˆ < 2h) we hit the Poincar´e section. This procedure does not fit to our approach based on Lemma 2.2 therefore we are not going to specify the details.

4 Closeness of differentials

Now we would like to get an upper bound in the spirit of (3.24) but for various differentials |D_v[P(s, c)− P_m(h, s, c)]|for v ∈ {h, s, c}. At first we upgrade Lemma 2.2. Undoubtedly it is of its own interest in this abstract setting.

Lemma 4.1. Suppose all the assumption of Lemma 2.2. Moreover let us haveα₁, α₂, l₁ ≥0 such that

¯

y∈C¹(U, V) and |¯y⁰(x)| ≤α₁,

|ϑ⁰(x)| ≤α₂, x∈U, forϑ(x) :=F(x,y(x)),¯

|F_x⁰(x, y1)−F_x⁰(x, y2)| ≤l1|y₁−y2|forx∈U, y1, y2 ∈B(¯y(x), %).





 (4.1)

Then we are able to extend the results of Lemma 2.2 by an estimate

|y⁰(x)−y¯⁰(x)| ≤%1, x∈U, (4.2) where

%1 := β

1−βl%(l%α1+l1%+α2). (4.3) Proof. From the equations F(x,y(x)) = 0 and F(x,y(x)) =¯ ϑ(x) after dif- ferentiation we infer forx∈U that

y⁰(x) =−(F_y⁰(x,y(x)))⁻¹F_x⁰(x,y(x)),

¯

y⁰(x) = (F_y⁰(x,y(x)))¯ ⁻¹(ϑ⁰(x)−F_x⁰(x,y(x))).¯

From now we omit (x,y(x)) and (x,y(x)),¯ the superscript ¯ above F will indicate the substitution of (x,y(x)), otherwise we substitute (x,¯ y(x)).We have

y⁰−y¯⁰ = (F_y⁰)⁻¹ −F_x⁰ −F_y⁰y¯⁰

= (F_y⁰)⁻¹ ( ¯F_y⁰ −F_y⁰)¯y⁰−F¯_y⁰y¯⁰−F_x⁰

= (F_y⁰)⁻¹ ( ¯F_y⁰ −F_y⁰)¯y⁰+ ¯F_x⁰ −F_x⁰ −ϑ⁰ ,

from which we get exactly (4.2) (using (4.1) and the assumptions and results of Lemma 2.2) and the proof is finished.

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

D_v[V_m−V¯_m]

≤C_V,v/m^p, V ∈ {X,∆}, v ∈ {s, c},

Dh[Vm−V¯m]

≤C_V,h/m^p−1, V ∈ {X,∆},

)

(4.4) 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))

α₁=α₁[V, v], α₂=α₂[V, v], l₁ =l₁[V, v]

for allV ∈ {X,∆}, v∈ {h, s, c}.This will be a bit sweating task.

Part 1.1 – about α₁[X, v]for v ∈ {h, s, c}.After differentiation we get D_h(¯x^j) =f(¯x^j)j, D_s(¯x^j) =ϕ⁰_x(jh, ξ)(f(γ(s)) +E⁰(s)c),

D_c(¯x^j) =ϕ⁰_x(jh, ξ)E(s)

forj= 1,2, . . . , m−1.Therefore (using (3.2) and that|E(s)| ≤√ N)

|D_h( ¯X_m)| ≤C_ϕm, |D_s( ¯X_m)| ≤C²_ϕ+µ₉, |D_c( ¯X_m)| ≤C_ϕ√ N whereµ₉ >0 is an arbitrary parameter andmis large enough (C_ϕC_Eδ/m^p ≤ µ9 is valid form large enough). So

α1[X, h] := Cϕm, α1[X, s] := C²_ϕ+µ9, α1[X, c] := Cϕ

√

N . (4.5) Part 1.2 – about α2[X, v]for v ∈ {h, s, c}.Note that

G¯^j_m :=Gm(h, s, c,X¯m(h, s, c))^j =ψ(h,x¯^j−1)−ϕ(h,x¯^j−1)

=h^p+1Υ(h,x¯^j−1), j= 1,2, . . . , m−1.

This implies

D_h( ¯G^j_m) =h^p[(p+ 1)Υ(h,x¯^j−1)

+h(Υ⁰_h(h,x¯^j−1) + Υ⁰_x(h,x¯^j−1)D_h(¯x^j−1))], Ds( ¯G^j_m) =h^p+1Υ⁰_x(h,x¯^j−1)Ds(¯x^j−1),

D_c( ¯G^j_m) =h^p+1Υ⁰_x(h,x¯^j−1)D_c(¯x^j−1).

Using Part 1.1. of this proof andh < ^1+µ_m for ¯Gm := ( ¯G¹_m,G¯²_m, . . . ,G¯^m−1_m ) we infer

|D_h( ¯G_m)| ≤ C_Υ(C_ϕ+p+ 1) +µ₁₀

m^p ,

|D_s( ¯G_m)| ≤ C_ΥC²_ϕ+µ₁₀

m^p+1 , |D_c( ¯G_m)| ≤ CΥCϕ

√

N +µ10

m^p+1 .

for any fixed µ10 >0, everym large enough and µsufficiently small. This yields

α₂[X, h] := C_Υ(C_ϕ+p+ 1) +µ₁₀

m^p , α₂[X, s] := CΥC²_ϕ+µ10

m^p+1 , α2[X, c] := CΥCϕ

√

N+µ10

m^p+1 .









 (4.6)

Part 1.3 – about l₁[X, v] for v ∈ {h, s, c}. We have in a moment that l1[X, v] = 0 forv∈ {s, c}. Further note at first that

D_hG_m(h, s, c, X_i) = (ψ_h(h, ξ), ψ_h(h, x¹_i), . . . , ψ_h(h, x^m−1_i ))

for Xi ∈ B( ¯Xm,C_X/m^p), i ∈ {1,2}. Now for x1, x2 such that x1+ϑ(x2− x₁)∈B for all ϑ∈[0,1] we have

|ψ_h(h, x1)−ψh(h, x2)| ≤ Z 1

0

|ψ⁰⁰_hx(h, x2+ϑ(x1−x2))|dϑ|x₁−x2|

≤C_ψ|x₁−x2|

which implies that l₁[X, h] := C_ψ is a good choice. Therefore

l₁[X, h] := C_ψ, l₁[X, s] := 0, l₁[X, c] := 0. (4.7) Part 1.4 – determiningCX,v forv∈ {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) in a caseV =X, v∈ {h, s, c} with

C_X,h>C_X,h:= C_ϕ[C²_ϕC_X + C_ψC_X + C_Υ(C_ϕ+p+ 1)], C_X,s>C_X,s := C³_ϕ[C_ϕC_X+ C_Υ],

C_X,c>C_X,c:=√

NC²_ϕ[C_ϕC_X + C_Υ]

for everym large enough. Indeed, for example in the casev=h (others are treated similarly) we get from (4.3) for µ₁₁>0 that

Dh(Xm−X¯m)

≤ β 1−βl%

h

l%α1[X, h] +l1[X, h]%+α2[X, h]

i

= C_ϕm

(1−µ1)(1−µ3)

"

(1 +µ)C_ϕ+µ₂ m

C_X

m^pCϕm+ Cψ

C_X m^p +C_Υ(C_ϕ+p+ 1) +µ₁₀

m^p

#

≤ C_X,h+µ₁₁ m^p−1

formlarge andµsmall enough (we have also used (3.5) from Theorem 3.2).

Part 2.1 – about α1[∆, v]for v∈ {h, s, c}.We easily get D_h( ¯∆_m) =−m+ 1, D_s( ¯∆_m) =τ_s⁰, D_c( ¯∆_m) =τ_c⁰. Therefore

α1[∆, h] :=m, α1[∆, s] := Cτ, α1[X, c] := Cτ. (4.8) Part 2.2 – about α2[∆, v]for v ∈ {h, s, c}. Lemma 3.1 implies (see also the definition (3.18))

z(h, s, c,∆¯m) =

ϕ( ¯∆m,x¯^m−1)−γ(s), f(γ(s))

+hw_m(h, s, c), f(γ(s))i

=hw_m(h, s, c), f(γ(s))i, where

w_m:=ϕ( ¯∆_m, x^m−1_m )−ϕ( ¯∆_m,x¯^m−1) + ¯∆^p+1_m Υ( ¯∆_m, x^m−1_m ).

Now

Dvz(h, s, c,∆m) =hD_vwm, f(γ(s))i, s∈ {h, c},

D_sz(h, s, c,∆_m) =hD_sw_m, f(γ(s))i+hw_m, f_x⁰(γ(s))f(γ(s))i.

So at first we handle termsD_vw_m forv∈ {h, s, c}.Straightforward compu- tation shows that

Dvwm =(A1+A2)Dv∆¯m+ (A3+A4)Dvx¯^m−1

+ (A₅+A₄)D_v(x^m−1_m −x¯^m−1), v∈ {h, s, c}

)

(4.9)

where

A₁ :=ϕ⁰_t( ¯∆_m, x^m−1_m )−ϕ⁰_t( ¯∆_m,x¯^m−1),

A2 :=(p+ 1) ¯∆^p_mΥ( ¯∆m, x^m−1_m ) + ¯∆^p+1_m Υ⁰_h( ¯∆m, x^m−1_m ), A₃ :=ϕ⁰_x( ¯∆_m, x^m−1_m )−ϕ⁰_x( ¯∆_m,x¯^m−1),

A4 := ¯∆^p+1_m Υ⁰_x( ¯∆m, x^m−1_m ), A₅ :=ϕ⁰_x( ¯∆_m, x^m−1_m ).

Let us haveµ₁₂>0,then computations as in the previous parts show that form large andµ small enough we have

|A₁+A₂| ≤ C_ϕC_X + C_Υ(p+ 1) +µ₁₂

m^p ,

|A₃+A₄| ≤ CϕCX +µ12

m^p , |A₅+A₄| ≤C_ϕ+µ₁₂

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 µ13>0 that

|D_hwm| ≤ C1+µ13

m^p−1 , |D_swm| ≤ C2+µ13

m^p , |D_cwm| ≤ C3+µ13

m^p , wherem is sufficiently large,µis small enough and

C₁ :=C_ϕC_X + C_Υ(p+ 1) + C²_ϕC_X + C_X,hC_ϕ, C₂ := C_ϕC_X + C_Υ(p+ 1)

C_τ+ C³_ϕC_X + C_X,sC_ϕ, C₃ := C_ϕC_X + C_Υ(p+ 1)

C_τ+√

NC²_ϕC_X + C_X,cC_ϕ.

Furthermore, forC4 := CϕCX similar computations show also|w_m| ≤(C4+ µ₁₃)/m^p.Therefore we can finish this step with the following choices

α₂[∆, h] := NCϕC1+µ14

m^p−1 , α2[∆, s] := NCϕ(C2+C4) +µ14

m^p ,

α₂[∆, c] := NC_ϕC₃+µ₁₄

m^p ,















(4.10)

whereµ₁₄>0 is an arbitrary parameter,m is large and µis small enough.

Part 2.3 – about l₁[∆, v] for v ∈ {h, s, c}.For ∆∈ B( ¯∆_m,C_∆/m^p) dif- ferentiating yields

Dvz(h, s, c,∆) =hψ_x⁰(∆, x^m−1_m )Dvx^m−1_m , f(γ(s))i, v∈ {h, c}, D_sz(h, s, c,∆) =hψ_x⁰(∆, x^m−1_m )D_sx^m−1_m , f(γ(s))i

+hψ(∆, x^m−1_m )−γ(s), ϕ⁰⁰_tx(s, ξ0)i.

Note that from a triangle inequality we have

|D_hx^m−1_m | ≤ |D_hx¯^m−1|+|D_h(x^m−1_m −x¯^m−1)| ≤C_ϕm+ C_X,h/m^p,

|D_sx^m−1_m | ≤C²_ϕ+µ₉+ C_X,s/m^p, |D_cx^m−1_m | ≤√

NC_ϕ+ C_X,c/m^p. Employing Newton–Leibniz formula straightforward computation implies that for anyµ15>0 and m large enough we have the choices

l₁[∆, h] :=NC_ψC²_ϕm+µ₁₅, l₁[∆, s] :=NC_ψC_ϕ(1 + C²_ϕ) +µ₁₅, l₁[∆, c] :=N^3/2C_ψC²_ϕ+µ₁₅.

)

(4.11) Part 2.4 – determining C_∆,v for v ∈ {h, s, c}. As in Part 1.4 we apply Lemma 4.1 in a setting (3.22) extended with (4.8), (4.10) and (4.11). From (4.2) we obtain (4.4) in a caseV = ∆, v∈ {h, s, c}with

C_∆,h>C_∆,h:= NC_ϕ[C_ψC_∆(1 + C_ϕ) +C₁] Cmin

, C_∆,s >C_∆,s := NCϕ[CψC∆(Cτ + C²_ϕ) +C2+C4]

C_min ,

C_∆,c>C_∆,c:= NC_ϕ[C_ψC_∆(C_τ+√

NC_ϕ) +C₃] Cmin

for everym large,µ small. The proof is complete.

Remark 3. Now as in the proof of Theorem 4.2 (see (4.9)) we get D_vP(s, c)−D_vP(h, s, c) = ( ¯A₁−A¯₂)D_v∆¯_m+ ( ¯A₃−A¯₄)D_vx¯^m−1

−( ¯A₅+ ¯A₂)D_v(∆_m−∆¯_m)−( ¯A₆+ ¯A₄)D_v(X_m^m−1−x¯^m−1) (4.12) forv∈ {h, s, c},where

A¯₁ :=ϕ⁰_t( ¯∆_m,x¯^m−1)−ϕ⁰_t(∆_m, x^m−1_m ),

A¯2 :=(p+ 1)∆^p_mΥ(∆m, x^m−1_m ) + ∆^p+1_m Υ⁰_h(∆m, x^m−1_m ), A¯₃ :=ϕ⁰_x( ¯∆_m,x¯^m−1)−ϕ⁰_x(∆_m, x^m−1_m ),

A¯4 :=∆^p+1_m Υ⁰_x(∆m, x^m−1_m ),

A¯₅ :=ϕ⁰_t(∆_m, x^m−1_m ), A¯₆ :=ϕ⁰_x(∆_m, x^m−1_m ).