Invariant manifolds for nonsmooth systems

Physica D 241 (2012) 1895–1902

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Invariant manifolds for nonsmooth systems D. Weiss a , T. Küpper b,∗ , H.A. Hosham b a

Mathematical Institute, University of Tübingen, Auf der Morgenstelle 10, D-72076, Germany

b

Mathematical Institute, University of Cologne, Weyertal 86-90, D-50931, Germany

article

info

Article history: Available online 29 July 2011 Keywords: Invariant manifold Nonlinear piecewise dynamical systems Invariant cones Periodic orbits Generalized Hopf bifurcation

abstract For piecewise smooth systems we describe mechanisms to obtain a similar reduction to a lower dimensional system as has been achieved for smooth systems via the center manifold approach. It turns out that for nonsmooth systems there are invariant quantities as well which can be used for a bifurcation analysis but the form of the quantities is more complicated. The approximation by piecewise linear systems (PWLS) provides a useful concept. In the case of PWLS, the invariant sets are given as invariant cones. For nonlinear perturbations of PWLS the invariant sets are deformations of those cones. The generation of invariant manifolds and a bifurcation analysis establishing periodic orbits are demonstrated; also an example for which multiple cones exist is provided. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Within the frame of the standard theory of smooth dynamical systems, it is well-known that qualitative changes (bifurcations) are typically due to changes in lower dimensional subsystems. The classical approach to analyze the qualitative behavior takes advantage of a reduction of the original high dimensional system to a lower dimensional system carrying the essential dynamics of the full system via the center manifold theorem. This approach is based on information provided by the linearization and requires smoothness of the functions determining the dynamical process. In that way a bifurcation and stability analysis of complex systems can substantially be reduced to the study of low dimensional systems by retaining the relevant interactions. The investigation of nonsmooth dynamical systems and their bifurcations is a research topic of present interest. Following the classical approach it is a natural question to explore if similar reduction techniques are at hand without a requirement of smoothness. The standard center manifold approach is related to properties of a linearized system, especially involving the number of eigenvalues with vanishing real part determining the center space spanned by the corresponding eigenvectors. Due to a lack of differentiability the notion of a linearization is not at hand; likewise there is no way to define eigenvalues. Instead of the analytical description of eigenvalues crossing the imaginary axis

∗

Corresponding author. E-mail addresses: [email protected] (D. Weiss), [email protected] (T. Küpper), [email protected] (H.A. Hosham). 0167-2789/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2011.07.012

a rather geometric view turned out to be of help which for example can be explained in the case of Hopf bifurcation [1–4]. The analytical criterion giving rise to Hopf bifurcation is given by the transversal crossing of a pair of complex eigenvalues through the imaginary axis. Since eigenvalues are not defined in a proper way, the corresponding geometric feature of a change in the phase space from a stable focus to an unstable focus through a center turns out to be the appropriate mean to describe a process which can be carried over to nonsmooth systems as well. For planar systems this view has been explored in [5–9] in connection with generalized Hopf bifurcation. For the special case of planar systems there was of course no need for any reduction. Based on this experience an extension to higher dimensional systems has been developed in [10,11] for piecewise linear systems. Here the notion of an invariant cone appeared generalizing the focus to an object on a cone consisting of periodic orbits or orbits spiraling ‘‘in’’ and ‘‘out’’ of zero, respectively. In the case of smooth systems, the cone reduces to an object which can be regarded as a flat, degenerated cone. It is the key observation to view that cone as a generalized invariant ‘‘manifold’’ determining the dynamics. In fact it is the main result of this paper to establish that even for nonlinear perturbations of piecewise linear systems there is a cone-like invariant ‘‘manifold’’ carrying the essential dynamics of the full system under appropriate conditions. In that way a reduction procedure to a two-dimensional surface has been established for nonsmooth systems allowing a bifurcation and stability analysis of a reduced system. The similarity with smooth systems already suggests to understand this procedure as a generalization of a center manifold reduction. The connections will be supported by the mechanisms employed to proof this result which make use of a Poincaré map with eigenvalue close to 1. We emphasize that the role of the cone

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D. Weiss et al. / Physica D 241 (2012) 1895–1902

as a key element in the reduction procedure has been developed within this approach; the existence of invariant cones consisting of periodic orbits, though, has already been observed in [1,12,2] in connection with piecewise linear system. The first results in the construction of invariant cones have been obtained for piecewise linear systems PWLS [11]. In that case the Poincaré map is (at least formally) explicitly given. Due do piecewise linearity it pertains linear behavior in one way, on the other hand a strong nonlinear component enters via the intersection times. Different scaling properties of those times in various directions offer a key to understand an extension to nonlinear perturbations of PWLS. As main result we will show that for nonlinear perturbations of homogeneous PWLS (PWNS) there is an invariant cone-like surface tangent at the origin to a cone of a basic PWLS. Although this result seems obvious at a first glance the proof requires subtle estimates involving different directions. The appropriate set to construct the Poincaré map for the PWNS is given by a small sector defined around the generating vector of the cone of the PWLS. The Poincaré map is then defined as sum of the Poincaré map of the PWLS and a nonlinearity. Using the Hadamard graph transform the existence of an invariant generating curve is established. Once existence of the invariant and attractive surface has been established it can be used for bifurcation analysis. We illustrate this process by an example for which the Poincaré map can explicitly be worked out. Consequently it is possible to determine the leading coefficients of the function generating the invariant surface. For simplicity we restrict our attention to piecewise smooth systems consisting of 2 components separated by a hyperplane M := {ξ ∈ Rn |eT1 ξ = ξ1 = 0}, where ei denotes the ith unit vector:

 f (ξ ), ξ˙ = + f− (ξ ),

ξ1 > 0, ξ1 < 0

(1)

Fig. 1. Different dynamics on cones, µ < 1, µ = 1 and µ > 1, respectively.

Differentiating this identity with respect to ξ gives (j)

(j)

t− (λξ )λj = t− (ξ ),

0<λ<∞

(5)

for j ≥ 1, indicating possible difficulties for λ → 0. On the other hand we find that all derivatives of t− when applied in direction of the ray vanish. (j)

− Lemma 1. For j ≥ 1 we get t− (λξ )ξ = 0 for ξ ∈ W< and η ∈ W> .

Proof. Differentiating equation (4) with respect to λ gives the result for j = 1. The statement for j > 1 then follows by induction − and by differentiating 0 = eT1 eA t− (ξ ) ξ with respect to ξ , because − of the transversality guaranteed by η ∈ W> .  In the same way t+ (η) and W> can be defined, where similar results hold for t+ . For initial data ξ ∈ W< and η ∈ W> we define P− (ξ ) := − + et− (ξ )A ξ , P+ (η) := et+ (η)A η and the Poincaré map for PWLS (3) with P− (ξ ) ∈ W> by P (ξ ) := P+ (P− (ξ )) [13,14]. An invariant cone C is then generated by an ‘‘eigenvector’’ ξ¯ ∈ W< of the nonlinear eigenvalue problem P (ξ¯ ) = µξ¯

(6)

with constant matrices A and nonlinear C -parts g± (ξ ) = o(∥ξ ∥), k ≥ 1. Here we restrict our attention to trajectories with immediate transition between the halfspaces Rn− := {ξ ∈ Rn | ξ1 < 0} and Rn+ := {ξ ∈ Rn | ξ1 > 0}. In the following we refer to system (1) as ⊖-system and ⊕-system in the half-spaces Rn− and Rn+ respectively. Define

with some real positive ‘‘eigenvalue’’ µ. The orbits with initial value λξ¯ , λ > 0, form the invariant cone C in the phase space Rn anchored in 0 ∈ Rn . The surface of the cone is smooth except possibly at the intersection of C and M . We recall that for continuous PWLS (A+ ξ = A− ξ , ξ ∈ M ) the cone is C 1 even at M . The value µ > 0 determines the dynamic on the cone. For µ > 1 and µ < 1 solutions spiral out and in, respectively. If µ = 1, then the cone consists of periodic orbits, Fig. 1. In order to study the attractivity of the cone C we consider the eigenvalues of the Jacobian P ′ (λξ¯ ), which are independent of λ > 0 due to the homogeneity of the system: since the intersection times t± are constant on half-rays, we get the identity

W<± := {ξ ∈ M | eT1 A± ξ < 0},

P (λξ ) = λP (ξ ),

W>± := {ξ ∈ M | eT1 A± ξ > 0}.

Differentiating this identity with respect to ξ yields

with smooth functions f+ , f− : Rn → Rn ; we further assume that f+ (ξ ) = A+ ξ + g+ (ξ )

(2a)

f− (ξ ) = A ξ + g− (ξ ) −

(2b) ±

k

− Then for any initial value ξ ∈ W< ∩ W<+ the trajectory ϕ(t , ξ ) = tA− e ξ of the homogeneous PWLS

 + ˙ξ = A− ξ , A ξ,

ξ1 > 0, ξ1 < 0,

(3)

enters Rn− immediately in forward time due do the fact that both quantities eT1 A± ξ have negative sign. Assume that ϕ(t , ξ ) reaches M again for the first time t− (ξ ) at η := ϕ(t− (ξ ), ξ ) ∈ M ; hence there exists the intersection time t− (ξ ) = inf{t > 0 |

W<−

− eT1 eA t

P ′ (λξ ) = P ′ (ξ ), which confirms independency of λ. By differentiating with respect to λ we obtain P ′ (λξ¯ )ξ¯ = P (ξ¯ ) = µξ¯ . Hence µ is an eigenvalue of P ′ (ξ¯ ) with eigenvector ξ¯ . For the remaining n − 2 eigenvalues µi , i = 1, . . . , n − 2, we assume

|µi | ≤ α < min{1, µ}.

(7)

ξ = 0}.

W<+

Let W< ⊂ ∩ be the set on which t− is defined. We know − t− is smooth in ξ ∈ W< with η ∈ W> and constant on half-rays (see [11]), i.e. t− (λξ ) = t− (ξ ),

0 < λ < ∞.

0 < λ < ∞.

(4)

Remark 1. The attractivity condition (7) guarantees that all solutions with initial values close to C are attracted to the cone. In case of contracting spiraling on C itself these solutions converge faster to the cone than to the origin. These statements will be discussed in more detail in Section 2.

D. Weiss et al. / Physica D 241 (2012) 1895–1902

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In the following we assume that the corresponding PWLS possesses an attractive invariant cone generated by ξ¯ ∈ W< (see (6)), which is transversal to M , i.e. eT1 A+ ξ¯ · eT1 A− ξ¯ > 0,

(8)

eT1 A+ η¯ · eT1 A− η¯ > 0, with η¯ := P− (ξ¯ ) ∈ W> . The main result is the following:

Fig. 2. ε -sector of the cone in M .

Theorem 1. Assume that the conditions (2), (3) and (6)–(8) hold for the corresponding PWLS and g± . Then, there exists a sufficiently small δ and a C 1 -function h: [0, δ) → M satisfying h(0) = 0 and ∂ h(0) = ξ¯ such that ∂u

where the eigenvalues of the matrix As are exactly the µi . Note that on the one hand M remains the separating plane, on the other the transformation of the last (n − 1) components is independent of ξ1 . We decompose

{h(u) | 0 ≤ u < δ}

P =

is locally invariant and attractive under the Poincaré map of system (1). For k = 2 the function h is C k in case of µ ≥ 1 and C min(k,j) in case of µ < 1 and α < µj .

according to the blocks in (9), so that y is a scalar and z ∈ Rn−2 . Due to assumption (7) we can choose a norm on Rn−2 such that

 

We decompose the Poincaré maps of PWLS and PWNS into a linear part and a nonlinearity. The Poincaré map P for PWNS will be written using properties of the Poincaré map of PWLS, P. 2. The piecewise linear system We first decompose P using the derivative at ξ¯ and an appropriate nonlinear term Q as P (ξ ) = P ′ (ξ¯ )ξ + Q (ξ ). Using the properties of P we immediately obtain Q (ξ¯ ) = 0, Q ′ (ξ¯ ) = 0 and Q (λξ ) = λQ (ξ ), Q ′ (λξ ) = Q ′ (ξ ),

0 < λ < ∞.

Hence, the function Q ′ is constant on half-rays. Differentiating the second equation with respect to ξ gives Q (j+1) (λξ )λj = Q (j+1) (ξ ),

0 < λ < ∞,

for j ≥ 1, again indicating possible difficulties for λ → 0. On the other hand we find vanishing derivatives of the return time t− applied in direction of the ray lead to corresponding results for derivatives of Q . Lemma 2. For j ≥ 1 we get Q P− (ξ ) ∈ W> .

(j+1)

(λξ )ξ = 0 for ξ ∈ W< and

′ (ξ )ξ Proof. The statement follows due to Q (j+1) (ξ ) = P (j+1) (ξ ), P− = P− (ξ ) and (j+1)

(λξ )ξ = 0,

(j+1)

(P− (λξ ))P− (ξ ) = 0,

P−

Pc Ps

,

ξ=

  y z

∥As ∥ =: α < min{1, µ}. On Rn−1 we define a norm by

∥ξ ∥ = max{|y|, ∥z ∥}. Remark 2. Due to the properties of the derivative of Q we are able to obtain an estimate for Q ′ in a neighborhood of the vector ξ¯ :

∥Q ′ (ξ )∥ ≤ Lε , ξ Sε (ξ¯ ), Sε (ξ¯ ) := {(y, z )T ∈ M | y > 0, ∥z /y∥ ≤ ε}, for some constant Lε with Lε → 0 for ε → 0 (see Fig. 2). We now use the property of the sector Sε (ξ¯ ) to obtain an estimate relating relations of Pc and Ps , hence the approximation property mentioned in Remark 1. For ξ ∈ Sε (ξ¯ ) we know ∥z ∥/y ≤ ε and hence ∥ξ ∥ = y for ε < 1. According to Remark 2 and Q (y, 0) = 0 we know

∥Ps (ξ )∥ ≤ ∥As z ∥ + ∥Q (ξ )∥ ≤ (α + Lε )εy, Pc (ξ ) ≥ (µ − Lε ε)y.

(10)

Combining these two estimates we see

∥Ps (λξ )∥ ≤ Pc (λξ )

α+Lε ε µ−Lε ε

<ε

(11)

for sufficiently small ε . Additionally we get α + Lε < 1 for small values of ε . Hence (10) shows the local attractivity of the cone C , whereas (11) guarantees that in case of contracting spiraling on C solutions close to the cone converge faster to the cone than to the origin. 3. The piecewise nonlinear system

which is guaranteed by Lemma 1 and the corresponding properties of t+ . 

As we are interested to prove the existence of a local manifold we use the usual techniques of ‘‘cut-off and scale’’ and without restriction we end up with a piecewise nonlinear system of the form

To simplify matters we linearly transform the coordinates of system (1) by a constant matrix

 + ˙ξ = A− ξ + g+ (ξ ), A ξ + g− (ξ ),

P+



1 0

0 T



T ∈ R(n−1)×(n−1)

,

to get ξ¯ = e2 ∈ Rn and

 µ ¯ P (ξ ) = ′

0

0 As



,

(9)

ξ1 > 0, ξ1 < 0,

(12)

where the nonlinear perturbations g± are C k -maps, k ≥ 1, defined on the whole phase space Rn with supp g± ⊂ {ξ ∈ Rn | ∥ξ ∥ ≤ δ} and g± = o(∥ξ ∥), ∥ξ ∥ → 0. Obviously we find a constant ′ o(1) depending on the scaling parameter δ with ∥g± ∥ + ∥g± ∥ ≤ o(1), δ → 0. A global invariant manifold of system (12) gives a local invariant manifold of system (1).

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D. Weiss et al. / Physica D 241 (2012) 1895–1902

3.1. The Poincaré map The Poincaré maps P− , P+ and P = P+ (P− (ξ )) will be defined on sectors Sϵ (ξ¯ ) resp. Sϵη (η) ¯ as long as ε, εη and δ are sufficiently small. We decompose the Poincaré map of system (12) using the Poincaré map of the PWLS:

P (ξ ) = P (ξ ) + R(ξ ),

R(ξ ) := P (ξ ) − P (ξ ).

Due to the compact support of the perturbations g± depending on

δ , we know R(ξ ) = 0 for ∥ξ ∥ ≥ const · δ , const sufficiently large. In this section we will study further properties of the remaining term R. A crucial step in the definition of P− , P+ , relies on the definition and properties of the intersection times τ± of PWNS (12), which in case of the ⊖-system is given by (we omit the (−)-indices)

τ− (ξ ) = inf{τ > 0 | F (τ , ξ ) = 0},    t F (t , ξ ) = eT1 eAt ξ + eA(t −s) g (y(s, ξ ))ds , 0

eT1 A

e

= eT1



The statements about τ− and τ−′ are given by (13), (14) and (5) in case of k = 1. Let k ≥ 2. By (14), Remark 3 and Lemma 1 we find τ ′ (λξ )ξ = O (1). The statement for higher derivatives follows inductively by differentiating F (τ (ξ ), ξ ) = 0 with respect to ξ , by the transversality condition (8) and by the observation ∂∂t F (τ (λξ ), λξ ) = O (λ).  Lemma 4. The remaining term R is C k in Sε (ξ¯ ). Furthermore there is a constant Kδ independent of ε with Kδ → 0 for δ → 0 and

∥R(ξ )∥ + ∥R′ (ξ )∥ ≤ Kδ , ξ ∈ Sε (ξ¯ ). Let ξ ∈ Sε (ξ¯ ) with ∥ξ ∥ = 1. In case of k = 2 we get R′′ (λξ ) = O (1),

where y(t , ξ ) is the solution of y˙ = Ay + g (y), y(0) = ξ . Applying Gronwall’s Lemma it is obvious, that we have y(t , ξ ) = eAt ξ + o(∥ξ ∥) for t ∈ [0, T ]. The existence of C k -functions τ± for initial values close to the cone is guaranteed by the Implicit Function Theorem due to the transversality condition (8) and the hypothesis on the perturbations g± (see proof of Lemma 3). Furthermore we know that τ± (ξ ) is ‘‘close’’ to t± (ξ ) for small perturbations g± : − t ∗ A−

Proof. Let ε and δ be sufficiently small, so that transversality of the perturbed vector fields is still given. Application of the Implicit Function Theorem implies the existence of τ− (ξ ) for ξ = (1, z )T , ∥z ∥ ≤ ε. The existence on Sε (ξ¯ ) can be concluded applying the Contraction-Mapping Theorem to the operator t → t − − 1 F (t , λξ ) for λ > 0, where eT1 A− eτ− (ξ )A ξ = β > 0. λβ

Proof. Since the intersection times τ± are C k the same holds for τ ± ± the remaining term R. Using P± (ξ ) = eA τ± ξ + 0 ± eA (τ± −s) g± (y± (s, ξ ))ds we get

 + − + −  R(ξ ) = eA τ+ eA τ− − eA t+ eA t− ξ + eA  +

 ξ t− (ξ ) − τ− (ξ ) 

τ− (ξ )

τ+

τ−



+τ +

− eA (τ− −s) g− (y− (s, ξ ))ds

0 + eA (τ+ −s) g+ (y+ (s, P− (ξ )))ds,

0

τ− = τ− (ξ ), τ+ = τ+ (P− (ξ )), t− = t− (ξ ), t+ = t+ (P− (ξ )).

− eA (τ− (ξ )−s) g− (y− (s, ξ ))ds

Obviously the Lemma is true for the last two terms. For the first term we write equivalently

0

with intermediate time t ∗ . Due to the transversality condition (8) and the hypothesis on g− we find

τ− (ξ ) − t− (ξ ) = o(1)

0 < λ < ∞.

 − + − + −  + −  eA τ+ eA τ− − eA t+ eA t− ξ = eA τ+ eA τ− − eA t− ξ



 + +  − + eA τ+ − eA t+ eA t− ξ .

(13)

for δ → 0 or ∥ξ ∥ → 0. Differentiating the equations defining t− (ξ ) and τ− (ξ ) with respect to ξ and using (8) and the properties of g− we find τ ′ (ξ ) = O (∥ξ ∥−1 ) and with (13) additionally

By differentiating [eA τ− − eA t− ]ξ with respect to ξ and using (13) and (14) and Remark 3 it is easy to conclude that the Lemma holds for this term and thus for R. 

τ−′ (ξ ) − t−′ (ξ ) = o(∥ξ ∥−1 )

3.2. Hadamard’s graph transform

(14)

for δ → 0 or ∥ξ ∥ → 0.

−

−

Using the explicit form of P leads to the composition of P which we can use to define Hadamard’s graph transformation:

Remark 3. In case of k ≥ 2 we assume without loss of generality g± = O (∥ξ ∥2 ). Thus we can replace the o-terms in (13) and (14) by O (∥ξ ∥) and O (1) respectively. Similarly we conclude

P (ξ ) =



µ 0

0 As



ξ + R(ξ ),

τ−′′ (ξ ) − t−′′ (ξ ) = O (∥ξ ∥−1 ).

R(ξ ) := Q (ξ ) + R(ξ ). Obviously the remaining term R is C k in Sε (ξ¯ ) and we get

All o- and O -terms are independent of ε .

∥R′ (ξ )∥ ≤ Lε,δ := Lε + Kδ ,

Corresponding to Lemma 1 and (5) we get Lemma 3. The intersection time τ− is C k in Sε (ξ¯ ), ε suitably small. Let ξ ∈ Sε (ξ¯ ) with ∥ξ ∥ = 1. For 0 ≤ j ≤ k we get

τ−(j) (λξ ) = O (λ−j ),

0 < λ.

In case of k ≥ 2 we gain one power of λ in direction of the ray ξ :

τ−(j+1) (λξ )ξ = O (λ−j ),

0 < λ.

Similar results hold for τ+ .

(15)

so that Lε,δ can be made as small as necessary by setting ε and δ suitably small. We will prove the existence of a smooth function H: [0, ∞) → Rn−2 with H (0) = 0, which satisfies the invariance condition H (Pc (y, H (y))) = Ps (y, H (y))

(16)

for y ≥ 0 using Hadamard’s Graph Transform T : D → D defined by

[TH ](ζ ) := Ps (y, H (y)),

ζ ≥ 0,

(17)

and ζ = Pc (y, H (y)). Obviously, a fixed point of the operator T vanishes at y = 0 and fulfills the invariance condition (16).

D. Weiss et al. / Physica D 241 (2012) 1895–1902

In case of k = 1 we define D as a set of maps H: [0, ∞) → Rn−2 , satisfying H (0) = 0, ∥H ∥∞ ≤ ε , graph(H ) ⊂ Sε (ξ¯ ), i.e. (y, H (y)) ∈ Sε (ξ¯ ) for all y ≥ 0 and ∥H (y1 )− H (y2 )∥ ≤ ε|y1 − y2 | for y1 , y2 ≥ 0. Due to Remark 2, Lemma 4 and the following Lemma 5 the existence of a fix-point of the operator T can be proved quite similar to [15]. We only have to make use of Q (ξ¯ ) = 0 and to ˜ ) ⊂ Sε (ξ¯ ) for H˜ = TH, which can easily be seen: guarantee graph(H We will show

P (Sε (ξ¯ )) ⊂ Sε (ξ¯ ) which holds for ε and δ sufficiently small. For ξ ∈ Sε (ξ¯ ) we get similar to (10)

∥Ps (ξ )∥ ≤ ∥As z ∥ + ∥Q (ξ )∥ + ∥R(ξ )∥ ≤ (α + Lε + Kδ ε −1 )εy, Pc (ξ ) ≥ (µ − Lε ε − Kδ )y. Combining these two estimates we get for sufficiently small ε and δ as in (11)

∥Ps (ξ )∥ α + L ε + Kδ ε − 1 ε ≤ ε. ≤ Pc (ξ ) µ − o(ε) − Kδ Lemma 5. For each ζ ≥ 0 and each H ∈ D there is a unique y = ω(ζ , H ) with

Pc (y, H (y)) = ζ . Furthermore the function ω( . , H ) is Lipschitz-continuous with constant 1/(µ − Lε,δ ). Proof. Since

|Rc (y1 , H (y1 )) − Rc (y2 , H (y2 ))| ≤ Lε,δ max{|y1 − y2 |, ∥H (y1 ) − H (y2 )∥} ≤ Lε,δ |y1 − y2 |,

(18)

the function Pc ( . , H ( . )) given by Pc (y, H (y)) = µy+Rc (y, H (y)) ≥ 0 is strictly monotonically increasing as long as Lε,δ < µ. Hence there exists such a function ω( . , H ). Using (18) a second time we find |ζ1 − ζ2 | ≥ µ|y1 − y2 | − Lε,δ |y1 − y2 |.  The differentiability of the fix-point H = TH can be shown as in [16]. To prove H ′ (0) = 0 we use Lemma 5 and the invariance condition (16) to conclude lim sup ζ →0

• ∥H ∥1,∞ := max{∥H ∥∞ , ∥H ′ ∥∞ } ≤ ε , • ∥H ′ (y1 ) − H ′ (y2 )∥ ≤ L′ |y1 − y2 | for all y1 , y2 ≥ 0, where the Lipschitz-constant L′ will be determined later. D is a Banach space with respect to the norm ∥ · ∥1,∞ . Additionally to Lemma 5 we need. Lemma 6. The function y = ω( . , H ) is continuously differentiable with (19)

∂ for y′i := ∂ζ ω(ζi , H ), where the constant K is independent of L′ for (ε + Lε,δ )L′ ≤ const. Furthermore for yi := ω(ζ , Hi ) and ∂ y′i := ∂ζ ω(ζ , Hi ) we find

for some constant Kε,δ with Kε,δ → 0 for ε, δ → 0.

1 = µy′ + Rc′ (y, H (y))





1 y′ . H ′ (y)

Setting y′i := y′ (ζi ), i = 1, 2, and using the abbreviations Rj′ := R′ (yj , H (yj )), Hj′ := H ′ (yj ), j = 1, 2, we estimate

       ′ 1  1 ′ ′  µ|y1 − y2 | ≤ R1 y1 − R2 y′2   H1′ H2′          ′ 1    0 ′ ′  ′ ′   ≤ R1 ′ (y1 − y2 ) + R1 ′ ′ y2  H1 H1 − H2      ′   1 ′ ′ + ′ y2  ,  R1 − R2 ′

′

H2

where the first term can be estimated by Lε,δ |y′1 − y′2 |, the second term by Lε,δ L′ |y1 − y2 ∥ y′2 | and the third using Lemmas 2 and 4:

    ′  [R − R′ ] 1′  ≤ K |y1 − y2 |. 2  1  H2

More precisely we get

         ′    1 1 [R − R′ ] 1′  = R′′  |y1 − y2 | ′ , ′ i,2  i,1   i,∗  H2

H∗

H2

with intermediate value y∗ , where Ri is the ith component of R and Ri′′,∗ := Ri′′ (y∗ , H (y∗ )), H∗′ := H ′ (y∗ ). Finally

      ′′  1 1 Q  = O (ε)L′ , ,  i,∗  H∗′ H2′       ′′  1 1 R  = O (1), ,  i,∗  H∗′ H2′ prove the first statement. We now set yi := ω(ζ , Hi ), i.e.

ζ = Pc (yi , Hi (yi )) = µyi + Rc (yi , Hi (yi )). Hence

and therefore

For k = 2 we define D as a set of C 1 -maps H: [0, ∞) → Rn−2 , satisfying the additional conditions

|y1 − y2 | + |y′1 − y′2 | ≤ Kε,δ ∥H1 − H2 ∥1,∞

Proof. Obviously the function ω( . , H ) is C 1 together with P and H. Differentiating ζ = Pc (y, H (y)) gives

µ|y1 − y2 | ≤ ∥R(y1 , H1 (y1 )) − R(y2 , H2 (y2 ))∥   ≤ Lε,δ |y1 − y2 | + ∥H1 − H2 ∥∞

∥H (ζ )∥ 1 ∥As H (y) + Rs (y, H (y))∥ ≤ lim sup |ζ | µ − Lε,δ y→0 |y| α + Lε,δ ∥H (y)∥ ≤ lim sup . µ − Lε,δ y→0 |y|

|y′1 − y′2 | ≤ K |ζ1 − ζ2 |

1899

|y1 − y2 | ≤

Lε,δ

µ − Lε,δ

∥H1 − H2 ∥∞ .

∂ For y′i := ∂ζ ω(ζ , Hi ) and Rk,i := R(yi , Hk (yi )), Hk,i = Hk (yi ) we get

       ′  1 1 ′ ′ µ|y′1 − y′2 | ≤  R y − R y′2  2 ,2 H ′  1,1 H1′ ,1 1  2 ,2        ′  1 1 ′ ′ ′ ≤ R1,1 H2′ ,1 y1 − R2,2 H2′ ,2 y2  Lε,δ

+ ∥H ′ − H2′ ∥∞ µ − Lε,δ 1        ′  1 1 ′ ′  ≤ R2,1 y1 − R2,2 y′2   H2,1 H2,2     ′  1  ′ ′  |y | +  R1,1 − R2,1  1 H2,1

+

Lε,δ

µ − Lε,δ

∥H1′ − H2′ ∥∞ ,

1900

D. Weiss et al. / Physica D 241 (2012) 1895–1902

where the first term is already estimated above. For the second term we find

˜ i = THi and yi , y′i as in Lemma 6. For any H1 , H2 ∈ D we define H Then by definition (17) and Lemma 6 we find

    ′  1  ′ R  − R i,2,1  i,1,1 H2,1        ′′  0 1 ∗  = R ( y , z ) ,  i 1  H1,1 − H2,1 H2,1

∥H˜ 1 (ζ ) − H˜ 2 (ζ )∥ ≤ ∥As ∥∥H1,1 − H2,2 ∥ + ∥R1,1 − R2,2 ∥ ≤ α(ε|y1 − y2 | + ∥H1 − H2 ∥∞ )   + Lε,δ |y1 − y2 | + ∥H1 − H2 ∥∞ ≤ [α + o(1)]∥H1 − H2 ∥1,∞

≤ [O (ε) + O (δ)]∥H1′ − H2′ ∥∞ with intermediate value z ∗ . More precisely:

      ′′  0 1 Q (y1 , z ∗ )  ,  i  H1,1 − H2,1 H2,1 ≤ O (ε)∥H1′ − H2′ ∥∞       ′′  0 1 R (y1 , z ∗ )  ,  i  H1,1 − H2,1 H2,1 for y1 = O (δ).

′



In the following we will prove that T : D → D is a contraction. ˜ := TH for H ∈ D gives Defining H

˜ (ζ ) = Ps (y, H (y)), H

y = ω(ζ , H ).

˜ is a continuously differentiable function with H˜ (0) = 0. Clearly, H Further we get with Q (y, 0) = 0, Remark 2 and Lemma 4 ∥H˜ (ζ )∥ ≤ ∥As ∥∥H (y)∥ + ∥R(y, H (y))∥ ≤ (α + Lε + Kδ ε −1 )ε. Using Q (y, 0) = 0, Lemmas 2 and 4 we find

    ′  ′ 1  |y | ∥H˜ ′ (ζ )∥ ≤ ∥As ∥∥H ′ (y)∥|y′ | +  R ( y , H ( y )) ′  H (y)  α + O (ε) + Kδ ε −1 ε. ≤ µ − Lε,δ Eventually we can guarantee for ε and δ sufficiently small. ˜ ′ is Lipschitz continuous: For ζ1 , ζ2 ≥ 0 we define Indeed, H y1 := ω(ζ1 , H ) and y2 := ω(ζ2 , H ). Then

∥H˜ ′ (ζ1 ) − H˜ ′ (ζ2 )∥ ≤ ∥As ∥∥H1′ y′1 − H2′ y′2 ∥        ′ 1  1 ′ ′  + R1 y1 − R2 y′2  . H1′ H2′ The first term of the right-hand side can be estimated by

∥H1′ y′1 − H2′ y′2 ∥ ≤ ∥H1′ (y′1 − y′2 )∥ + ∥H1′ − H2′ ∥|y′2 | ≤ ε|y′1 − y′2 | + L′ |y1 − y2 ∥ y′2 | whereas the second term is already estimated in the proof of Lemma 6. We then arrive at

∥H (ζ1 ) − H (ζ2 )∥ ≤ (αε + Lε,δ )|y1 − y2 | + (α L′ + Lε,δ L′ + O (1))|y1 − y2 ∥ y′2 |, ′

′

where the O -term is independent of L′ . Using Lemma 6 we eventually get

α L′ + Lε,δ L′ + O (1) |ζ1 − ζ2 |, ∥H (ζ1 ) − H (ζ2 )∥ ≤ (µ − Lε,δ )2 ˜′

˜′

where the O -term is still independent of L′ . Choosing L′ sufficiently large and ε, δ small we end up with

∥H˜ ′ (ζ1 ) − H˜ ′ (ζ2 )∥ ≤ L′ |ζ1 − ζ2 | which proves T (D) ⊂ D.

and the second term is already estimated in the proof of Lemma 6:

        ′ 1 1 ′ ′ R y − R y′2  ′ ′ 2 ,2 H  1,1 H1,1 1  2,2 ≤

Lε,δ

µ − Lε,δ

∥H1′ − H2′ ∥∞ + o(1)∥H1 − H2 ∥1,∞

∥H˜ 1′ (ζ ) − H˜ 2′ (ζ )∥ ≤

α + Lε,δ + o(1) ∥H1 − H2 ∥1,∞ µ − Lε,δ

for ε, δ → 0. Since α < min{1, µ} we can guarantee that T is a contraction for ε, δ sufficiently small, hence by the Contraction-Mapping Theorem there is a fixed point defining the invariant graph. Remark 4. Theorem 1 holds even in case of k = 3. The proof depends crucially on Lemma 3, which guarantees

∥H˜ ∥1,∞ ≤ ε

˜′

∥H1′ ,1 y′1 − H2′ ,2 y′2 ∥ ≤ ∥H1′ ,1 ∥|y′1 − y′2 |   + ∥H1′ ,1 − H1′ ,2 ∥ + ∥H1′ ,2 − H2′ ,2 ∥ |y′2 |  ′  1 ≤ ε|y′1 − y′2 | + L |y1 − y2 | + ∥H1′ − H2′ ∥∞ µ − Lε,δ

for ε, δ → 0. Finally we end up with

′

˜′

∥H˜ 1′ (ζ ) − H˜ 2′ (ζ )∥ ≤ ∥As ∥∥H1′ ,1 y′1 − H2′ ,2 y′2 ∥        ′  1 1 ′ ′ + R y − R y′2  ′ ′ 2 ,2 H  1,1 H1,1 1 , 2,2 where the first term can be estimated by

≤ O (δ)∥H1 − H2 ∥∞ ′

for ε, δ → 0. Furthermore we get

R′′′ (λξ )ξ = O (λ−1 ), R′′′ (λξ )ξ 2 = O (1),

0 < λ < ∞.

Remark 5. Once the existence of H (y) = a1 y + a2 y2 + · · ·, has been established, we can use H to determine the dynamics on {h(y) = (y, H (y)) | 0 ≤ y < δ}. For example to determine periodic solutions we consider the fixed point equation P (ξ ) = ξ , reduced to the first component

µy + Rc (y, H (y)) = y,

(y ≥ 0).

Dividing by y we obtain

µ − 1 = Rc (y, H (y))/y.

(20)

Solutions y > 0 of (20) then lead to periodic orbits. 4. Example We illustrate the result by an example where the piecewise linear system has been designed according to the choice provided in [11]. Further we have chosen a nonlinearity such that the solution is explicitly known for comparison. The ⊕-system is taken in normalized form; for the ⊖-system we have chosen the situation represented by δ = 0 in [11]. Hence we consider the system

ξ˙ = A± ξ + g± (ξ ),

±eT1 ξ > 0,

(21)

D. Weiss et al. / Physica D 241 (2012) 1895–1902

with µ1 = µ1 (t− (ξ¯ )), d = d(t− (ξ¯ )), µ = µ(t− (ξ¯ )); further t− (ξ¯ ) is determined as smallest positive root of

where



λ+



−ω+ λ+

A+ = ω+ 0

0 0 ,

 − − 0 = −2s¯ + α[¯c − s¯ − e(µ −λ )t− m,

µ+

0

λ− − ω − A =  ω−

−2 ω − λ− + ω−



−

0

with κ = λ − µ − ω− and −

 g+ (ξ ) = ρ+ 



0 0

ξ12 + ξ22

  ξ32 g− (ξ ) = ρ−  0  .

,

0

Note that the eigenvalues of A± are given by λ± ± iω± , µ± , and PWNS (21) has transversal crossing if ξ2 [−2ω− ξ2 + α(λ− − µ− − ω− )ξ3 + ρ− ξ32 ] < 0. − For ξ = (y, z )T ∈ W< the intersection time τ− (ξ ) = τ− is determined as smallest positive root of equation 0 = −2sye

λ− τ

+ α[(c − s)e

−

λ− τ

−

−e

µ− τ−

]z + G1 (τ− )z 2

(22)

Q (y, H (y)) = O (y3 ), R(y, H (y)) =

(µ− − κ)(s − c ) + 2ω− c λ− τ− e (λ− − 2µ− )2 + ω−2 2µ− − λ− − ω− − + ρ− − e2µ τ− , (λ − 2µ− )2 + ω−2

¯2 − b1 = − G 

b2 = −

G1 (τ− ) = ρ−

where we have used the abbreviations s := sin(ω− τ− ) and c := cos(ω− τ− ). Further the map P− is given by

P− (y, z ) =

(c + s)eλ

−τ

α seλ

−τ

−

−

  y z

µ − τ−

0

e

 +

G2 (τ− )z 0

 2

,

(λ− − 2µ− )s − ω− c λ− τ− e (λ− − 2µ− )2 + ω−2 ω− − + ρ− − e2µ τ− . − 2 − 2 (λ − 2µ ) + ω

y2 + O (y3 ).

eT3 A− η¯ eT1 A− η¯

eT2 A− η¯ eT1 A−

η¯

 ¯ 1 eλ+ t+ m2 , G

µ+ t

¯ 1e G

+

m2 + G3 η¯ 22 ,

¯ i = Gi (t− (ξ¯ )). where η¯ = P− (ξ¯ ) and G To determine the coefficient a2 we substitute H into the equation representing the invariance condition; hence H (P1 (y, H (y))) = P2 (y, H (y)), which leads to mµy + m(da2 + b1 )y2 + a2 µ2 y2

and thus

G2 (τ− ) = ρ−

+ For η = (y, z ) ∈ W> , we have t+ := of the nonlinearity g+ and therefore T

 P+ (y, z ) =

−e eµ

+t

+

λ+ t+

a2 =

τ+ (η) = π /ω independent +

y

,

z + G3 y2 +





+

µ1 (τ− )

d(τ− )

 +

y z

+

with − τ +λ+ t − +

,

− + d(τ− ) = −α seλ τ− +λ t+ ,

µ(τ− ) = eµ

− τ +µ+ t − +

.

We write P (y, z ) as

 µ1 0

d

µ

  y z

.

Finally we can use the expression for H (y) to study bifurcation of periodic orbits by determining fixed points of the reduced system

y∗ ≈

 

−G2 (τ− )eλ t+ z 2  2 − G3 ((c + s)y + α sz )eλ τ− + G2 (τ− )z 2

µ1 (τ− ) = −(c + s)eλ

µ2 − µ1

db2 − µ(1 − µ)b1

µ2 − µ1

y2 + O (y3 ).

Thus the fixed point is approximately given by



µ(τ− )

0

b2 − mb1

P1 (y, H (y)) := µy +



where G3 = ρ+ e2λ t+ − eµ t+ /(2λ+ − µ+ ). For P (y, z ) = P+ (P− (y, z )) we obtain

P (y, z ) =

b1 b2

= mµy + µa2 y2 + b2 y2 + O (y3 ),

where

P (y, z ) =

 

After lengthy computations we obtain

with



(23)

where we have used the abbreviations s¯ = sin(ω− t− (ξ¯ )), c¯ = cos(ω− t− (ξ¯ )), and the invariant ‘‘eigenvector’’ ξ¯ satisfying P (ξ¯ ) = µξ¯ is chosen as ξ¯ = (¯y, z¯ )T = (1, m)T with m = (µ − µ1 )/d. Note that we want to consider the situation that µ ≈ 1; attractivity of the cone is then guaranteed if |µ1 | < min{1, µ}. By Theorem 1, we know there exists a local invariant set tangent to the cone at 0 which is generated by a graph of the form H (y) = my + a2 y2 + O (y3 ). Using Q (ξ¯ ) = 0, Q ′ (ξ¯ ) = 0 and g± = O(∥ξ ∥2 ), we obtain

ακ αω−  µ− 

0

−

1901

+ R(y, z ),



1−µ db2 − µ(1 − µ)b1

(µ2 − µ1 ).

Using this general form various situations can be explicated by a special choice of parameters. The simulation is done with parameters set at λ+ = −0.5, λ− = 0.5, µ+ = 0.2, α = 0.5, t+ = π , w + = w − = 1.0, ρ− = −0.01, ρ+ = 0.1 and bifurcation + ¯ parameter µ− close to µ− 0 := −µ t+ /t− (ξ ) ≈ −1.0604, where t− (ξ¯ ) ≈ 0.59253 is determined by 0 = −2s¯ + α[¯c − s¯ − e−(µ

+ t + +λ t − (ξ¯ )) −

]m ,

where m = We get µ(µ0 ) = 1 and ∂µ∂ − µ(µ− 0 ) > 0. For this set of values in PWLS (i.e. ρ− = ρ+ = 0), the phase space contains two attractive invariant cones (Fig. 3), likewise there are two generalized center manifolds of PWNS (21), Fig. 4. These attractive cones are given by ξ¯ = (0, 1, 0)T and ξ¯ = (0, 1, m) with eigenvalues µ1 = 1, µ ≈ 0.067 and µ = 1, µ1 ≈ −0.388 respectively and are separated by a nonattractive cone.Recall, that the existence 1−µ1 . d

−

1902

D. Weiss et al. / Physica D 241 (2012) 1895–1902

Acknowledgment

0 –0.5

The research of the author (H.A. Hosham) was supported by the Department of Mathematics, Faculty of Science, Al-Azhar University of Assiut, Egypt.

z

–1 –1.5 –2 –2.5

References

–3 0.5 0

0.5

0

–0.5 –1

y

–0.5 x

–1

Fig. 3. Two attractive invariant cones of PWLS for µ− = µ− 0 . H(y)

0.5 0

z

–0.5 –1 –1.5 –2 H(y)

–2.5 0.5

0.5

0

0 – 0.5

–0.5 –1 –1

y

x

Fig. 4. Two generalized center manifolds of PWNS (ρ − = −0.01, ρ + = 0.1) for

µ− = µ− 0 .

–0.01

z

–0.015 –0.02 –0.025 –2 –1 0 × 10-3

2

1 2

–2 –4

3 x

4 –8

–6

0 × 10-3

y

Fig. 5. Stable periodic orbit of PWNS for µ− = −1.06 > µ− 0 .

of two attractive cones is not possible in continuous PWLS (see [1] Theorem 2). A periodic orbit on the manifold generated by Hopf bifurcation is shown in Fig. 5.

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