Reduction to invariant cones for non-smooth systems

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1 Available online at Mathematics and Computers in Simulation 8 () Reduction to invariant cones for non-smooth systems T. Küpper, H.A. Hosham Mathematisches Institut, Universität zu Köln, Weyertal 86-9, Germany Available online October Abstract The reduction of smooth dynamical systems to lower dimensional center manifolds containing the essential bifurcation dynamics is a very useful approach both for theoretical investigations as well as for numerical computation. Since this approach relies on smoothness properties of the system and on the existence of a basic linearization the question arises if this approach can be carried over to non-smooth systems. Extending previous works we show that such a reduction is indeed possible by using an appropriate Poincaré map: the linearization will be replaced by a basic piecewise linear system; a fixed point of the Poincaré map generates an invariant cone which takes the role of the center manifold. The occurrence of nonlinear higher order terms will change this invariant manifold to a cone-like surface in R n containing the essential dynamics of the original problem. In that way the bifurcation analysis can be reduced to the study of one-dimensional maps. IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Non-smooth dynamical systems; Invariant cones; Periodic orbits; Generalized Hopf bifurcation. Introduction The center manifold approach provides a powerful tool to reduce high-dimensional parameter dependent dynamical systems to lower dimensional systems carrying the essential dynamics responsible for example for bifurcation processes. In this paper we will continue to investigate if a similar approach is available for non-smooth systems. First we briefly recall the key facts for smooth systems. Let ξ = f (ξ, λ), (ξ R n,λ R p ), () denote a smooth dynamical system with stationary solution ξ =. Using linearization and transformations according to the structure of the eigenvalues of the linearization A:=( f/ ξ)( ξ)off, Eq. () can be stated in the following form with ξ =(x, y, z) T and an accordingly arranged matrix A A = A A A +, Corresponding author. addresses: kuepper@math.uni-koeln.de (T. Küpper), hbakit@math.uni-koeln.de (H.A. Hosham) /$36. IMACS. Published by Elsevier B.V. All rights reserved. doi:.6/j.matcom...4

2 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () where the submatrices A, A and A + correspond to the eigenvalues λ i in the spectrum σ(a)=σ(a ) σ(a ) σ(a + ) of A with negative, vanishing and positive real part, respectively: x A x g (x, y, z, λ) y = A y + g (x, y, z, λ), z A + z g + (x, y, z, λ) here g, g and g + collect terms of higher order in x, y and z. Since the stationary solution is already unstable if σ(a + ) /=, we assume for simplicity that σ(a + )=, hence Eq. () is equivalent to ẋ = A x + g (x, y, λ) ẏ = A y + g (x, y, λ). () The center manifold approach performs a locally equivalent reduction to a system defined in the center space, i.e. there exists a function h, defined in a neighborhood of ȳ = in the center space mapping into the stable space satisfying h()=,( h/ y)() = such that the reduced equation: ẏ = A y + g (h(y),y,λ), (3) is locally equivalent to (). The advantage of this approach relies on the fact that usually in relevant applications n := dim y n, typically n =orn =. Once (3) has been established the dynamics, stability and bifurcation behavior of () can be obtained by studying (3). The underlying center manifold approach essentially depends on smoothness properties of the original problem using the properties of the linearized problem. Hopf bifurcation serves as a typical example which can be studied via the center manifold approach. The bifurcation of periodic orbits from a stationary solution is triggered by a crossing of exactly one pair of eigenvalues of the linearization A through the imaginary axis. For non-smooth systems linearization is not at hand due to a lack of smoothness. A review of recent results concerning nonsmooth problems is given by [ 5,7,,]. The growing interest to investigate non-smooth problems of high dimensions involving many parameters with regard to stability and bifurcation has stimulated the question if destabilization and bifurcation for non-smooth systems arises as well through a change in low dimensional terms and if suitable reduction techniques can be developed. Since there is no linearized equation defined for non-smooth systems criteria by use of eigenvalues are not at hand. In previous studies [9,,3 6] restricted to planar systems it has been shown that the analytical criterium based on the eigenvalues crossing the imaginary axis can be substituted by an equivalent interpretation: Hopf bifurcation is due to a change in phase space from a stable focus to an unstable focus via a center. This is considered as the key observation that this situation can be mimicked for piecewise linear systems, and for that reason it has been suggested in [8,4,6] to replace the linearized problem by a basic piecewise linear problem. In various papers it has indeed been shown that the occurrence of periodic orbits in terms of generalized Hopf bifurcation can be achieved. For planar systems there is of course no need for any reduction of the system. Based on a first approach in [8] we now continue to set up methods to reduce a high dimensional non smooth system to a low dimensional one.. N-dimensional piecewise systems To describe the reduction we consider the simple situation of a non-smooth problem given by two smooth problems defined in half-spaces separated by a hyperspace M with appropriate transition rules motivated by examples described

3 98 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () in [8].InR n we take M ={ξ R n e T ξ = } as separation manifold and assume that the dynamical system is given by f + (ξ, λ), e T ξ>, ξ = (4) f (ξ, λ), e T ξ<, where f +,f : R n R n are smooth functions and e R n. Since we are particularly interested in trajectories crossing M we have to define rules for trajectories reaching M by employing the limiting vector field. For fixed λ set, W ± > ={ξ M et f ± (ξ) > }, W ± < ={ξ M et f ± (ξ) < }, W ± ={ξ M et f ± (ξ) = }. Note that for ξ W< W < + or ξ W > W > +, there is direct crossing from one half-space to the other via M, for ξ W> W < +, the flow starting in ξ is restricted to M in forward time by the reduced equation: ξ = [ (e T e T [f (ξ) f + f (ξ))f + (ξ) (e T f + (ξ))f (ξ) ]. (ξ)] For initial values ξ W < W + < the trajectory under the flow ϕ (t, ξ) of ξ = f (ξ, λ) leaves M immediately so that the return time: t (ξ) = inf{t > e T ϕ (t, ξ) = }, is well defined. If t (ξ)< and η:=ϕ (t (ξ),ξ) W> W > +, we can define t + (η) = inf{t > e T ϕ + (t, η) = }, in a similar way and in case ϕ + (t + (η),η) W < W + <, set up a Poincaré map by P (ξ):= ϕ (t (ξ), ξ), P + (η):= ϕ + (t + (η), η) and P(ξ):= P + (P (ξ))... The basic piecewise linear problem For the piecewise linear problem A + ξ, e T ξ ξ>, = A ξ, e T ξ<, (5) the Poincarè map of the flow is formally given as P (ξ):=e t (ξ)a ξ, P + (η):=e t+ (η)a + η, P(ξ):=P + (P (ξ)) = e t+ (η)a + e t (ξ)a ξ. Note that t and t + are constant on rays in W < W + < and W > W + >. Although the maps are nonlinear due to the nonlinearity contained in the return times t (ξ), t + (η) they preserve useful linearity properties which we collect in:

4 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () Lemma.. (i) Forˆξ W < + W <, and allλ >: a) λˆξ W < + W <, t (λˆξ) = t (ˆξ) andp (λˆξ) = λp (ˆξ). b) t is differentiable inˆξ and t (ˆξ).ˆξ =. c) P is differentiable inˆξ and( P / ξ)ˆξ = e t (ˆξ)A [A ˆξ t (ˆξ) + I]. (ii) For ˆη W > + W >, and eachλ >: a) λˆη W > + W >, t+ (λˆη) = t + (ˆη) andp + (λˆη) = λp + (ˆη). b) t + is differentiable inˆη and t + (ˆη) ˆη =. c) P + is differentiable inˆη and( P + / η)ˆη = e t+ (ˆη)A + [A + ˆη t + (ˆη) + I]. (iii) Forˆξ W < + W <,ˆη W > + W >, and allλ >: a) P(ˆξ) = P + (P (ˆξ)) is differentiable inˆξ and( P/ ξ)(ˆξ) = e t+ (ˆη)A + [A + ˆη t + (ˆη) + I] e t (ˆξ)A [A ˆξ t (ˆξ) + I]. b) ( P / ξ)(ˆξ) ˆξ = P(ˆξ). c) P(λˆξ) = λp(ˆξ). If there exists ξ W < + W < and μ >such that P( ξ) = μ ξ, then ξ generates an invariant cone under the flow of (5) due to P(λ ξ) = λp( ξ) = λ μ ξ. Further P ξ ( ξ) ξ = P( ξ) = μ ξ, hence ξ is an eigenvector of ( P/ ξ)( ξ) with eigenvalue μ. If we assume that the remaining (n ) eigenvalues λ,...,λ n of ( P/ ξ)( ξ) satisfy λ j < min{, μ}, (j =,...,n ), (6) then the invariant cone is attractive under the flow of (5) while the dynamics on the cone is determined by μ <, μ = or μ>. In that way the investigation of the dynamical behavior of the original problem can be reduced to the dynamics on a two-dimensional surface. It is our key point that this results carries over to nonlinear perturbations of (5) with the cone replaced by a cone-like two-dimensional surface. Consequently the dynamics stability properties and bifurcation of the n-dimensional problem can be investigated by studying a one-dimensional map. 3. The nonlinear problem We now assume that the PWLS (5) has been derived from (4) by considering linearized problems in each half-space; i.e.: f + = A + ξ + g + (ξ), f = A ξ + g (7) (ξ), where g +,g C k (R n, R n )(k ) satisfying g ± () = and ( g ± / ξ)() =. We assume that the corresponding PWLS possesses an attractive invariant cone generated by ξ and μ with the eigenvalues of ( P/ ξ)( ξ) satisfying (6). We further assume that (7) is already given in a truncated version; i.e for same r > g ± (ξ), ( ξ r). The Poincaré map P associated to (6) is defined close to the ray R ={y ξ y } and satisfies P(ξ) = P(ξ), (ξ R, ξ r). The following theorem states that the invariant ray R for the (PWLS) is replaced by an invariant curve R ={h(y) y} tangent to R in y =. (8)

5 984 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () Theorem 3.. Under the previous hypotheses on A ± and g ±, there exists a smooth function h : [, ) M satisfying h() =,( h/ y)() = ξ such that R := {h(y) y}, is invariant under P. Note that due do truncation h(y) = y ξ for y sufficiently large. Details concerning the proof of theorem will be given elsewhere [6]. Here we use that the flow applied to R generates an invariant surface in R n and that the bifurcation analysis for (6) close to ξ = can be reduced to the study of P restricted to R. For example a fixed point of P restricted to R corresponds to a periodic solution of (7). 3.. Generation of invariant cones for PWLS The occurrence of invariant cones plays a similar role for PWLS as the presence of purely imaginary eigenvalues of the linearization of smooth systems with respect to the generation of bifurcation of periodic orbits for nonlinearly perturbed systems. Seen from the perspective of non-smooth systems the periodic orbits of a smooth linear system might be seen as a degenerate (flat) cone. It appears as a natural approach to investigate how a (flat) cone for a smooth system develops under non-smooth system perturbations. Motivated to study the generation of invariant cones out of smooth systems we consider systems dependent on two parameters: A ± = A + αb ± + βc ±, (9) where we assume that the matrix A has exactly two purely imaginary eigenvalues ±iw with corresponding eigenvectors ξ and ξ. We further assume that span Re ξ, Im ξ intersects M transversally, hence the periodic orbits corresponding to ξ, ξ cross M. Without restriction we may assume that ξ:=ξ + ξ M, clearly ξ is a fixed point of e ta generating a flat cone. In [8] we have already set up an extended system to determine fixed points of P in a neighborhood of α =,β =. Using the implicit function theorem and the bordering lemma, we obtain that there are functions ξ = ξ(β), t = t (β), t + = t + (β) and α = α(β) satisfying ξ() = ξ, t () = t + () = (π/w), α() = such that F( ξ(β),t (β),t + (β),α(β),β) =, hence there is an invariant cone defined by ξ(β) due to the transversality of the basic periodic orbit through ξ(), further we also have ξ(β) W + < W <, for β sufficiently small. Since W < + W < may also depend on β it is possible that ξ(β) leaves the admissible range so that sliding motion may occur. As specific example we choose A, B ± and C ± in the following form w c ± 3 A = w B ± = b ± C ± = c ±. μ c 3 ± As specific values of the coefficient we choose: (a) w =, μ =., b ± =, c+ 3 = c+ 3 =.5, c 3 = c 3 =, and c± =. Newton s Method is used to compute the roots of F(X(β), β)=.fig. a. shows an attractive invariant cone consisting of periodic orbits.

6 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () a Attractive invariant cone ξ ξ Repulsive invarint cone b ξ ξ Fig.. Attractive/repulsive invariant cone consists of periodic orbits: (a) β =. α =.453, t = , t + = 3.43 and fixed pint ξ = [,.953,.33] T and (b) β =. α =.75, t = t + =.89 and fixed point ξ = [,.4939,.8695] T. We also take an example of the form treated in [4] where we choose C ± as c ± C ± = c ±. c 3 ± As specific values of the coefficient of C ±, we choose (b) w =, μ =., b ± =, c+ = 3., c+ = 5.6, c+ 3 = 75.3, c =., c =.8 and c 3 =.64, Fig. b. shows a repulsive invariant cone consisting of periodic orbits. 4. 3D-Example To illustrate the previous results we consider a parameter dependent example in R 3 of the form: f + (ξ, λ), e T ξ>, ξ = f (ξ, λ), e T ξ<. ()

7 986 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () Since we want to study systems switching between the half-spaces e T ξ > and e T ξ <, we assume that there are complex eigenvalues with non-vanishing imaginary part; i.e. λ ± + iω ± and μ ±. Further we assume that the eigenvectors corresponding to the complex eigenvalues are not both contained in the hyperplane M = {ξ e T ξ =}. Without restriction we can assume that one of the linear parts can be transformed to Jordan normal form; i.e.: λ + w + A + = w + λ + μ + Note that the separating manifold M (given by e) needs to be transformed simultaneously to a hyperplane which we again denote by M = {ξ e T ξ =} with possibly transformed vector e. Transversality of the rotating orbits is guarantied if {e, e } / M Once the transformation of the system has been fixed there is no restriction on A besides the desired rotation through M. To be specific we assume that A is obtained through a similarity transformation of a suitable Jordan normal form incorporating the desired properties of the eigenvalues: A = S A N S where λ w A N = w λ μ and α S = γ δ with suitable parameters α, γ, δ, hence a a a 3 A = a a a 3 a 3 a 3 a 33 λ + w (αδ ) αγμ λ w + (λ + w )(αδ ) αδμ α(λ μ + w (αδ )) = w ( αγ) (λ + w )( αγ) αw ( αγ). αγ γ(μ λ ) + w (γ δ) γw δ(λ + w μ ) w α(γ δ) λ αγ + μ Although this choice does not yet represent the most general form it offers sufficiently many choices to illustrate the plurality of occurring phenomena. In case δ = the system is continuous although not smooth; the invariant half-planes in the left hand and the right space meet in the line y = z =, hence there is an invariant two-dimensional surface in R 3.Forδ /= the half-planes separate; the parameter δ may be considered to measure the jump. In case α = the return time is constant, hence the computation of fixed points is simplified. In the following we fix e = e and γ =.

8 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () For the system we obtain W > + ={ξ R3 /ξ =,ξ < }, W < + ={ξ R3 /ξ =,ξ > }, hence half-planes, constant t + (η) = π/w + and P + (η) = e λ+ (π/ω +) η. e μ+ (π/ω +) η 3 For the system the sectors W>,W < are given by W< ={ξ R3 /ξ =,a ξ + a 3 < }, W> ={ξ R3 /ξ =,a ξ + a 3 > }. () The flow of the system for initial values in W< is determined by ξ(t ) = e λ t {[ȳ + α z][cos(ω t )S e + sin(ω t )S e ] + ȳ[cos(ω t )S e sin(ω t )S e ]} + [δȳ + z]e μ t S e 3, hence t (ξ) is determined as smallest positive root of = e T ξ(t) for ξ W <. Using the special form of A this amounts to: = ξ [c + s( + αδ) s + c( + αδ) αδe ] + α [c + s( + αδ) E ], () or = c + s( + αδ) s + c( + αδ) αδe ξ α c + s( + αδ) E =: H(t ), (3) where we have set c = cos(w t ), s = sin(w t ), E = e μ t λ t. Note that ξ W < W + < requires: >a ξ + a 3 and ξ >, or < a if a 3 >, ξ a 3 > a if a 3 <, ξ a 3 for ξ W< with t (ξ)< the map P (ξ)isgivenby ( P (ξ) = e λ t c + s αs δ(s + c) + δe αδs + E hence with F:=e λ+ t + +λ t, E + :=e μ+ t + λ + t + P(ξ) = P + (P (ξ)) = F ( c s )( ξ ), αs E + δ(s + c E ) E + (αδs E ) )( ξ ), (4) An admissible fixed point ξ W < W + < is determined by Eq. () for t ( ξ) and by the nonlinear system P( ξ) = ξ (5)

9 988 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () Since there are still many parameters involved we illustrate various situations for a special choice of parameters. In the case αδ = the equation to determine t (ξ) reduces to = ξ [ s] + α [c s E ], (6) and, in particular for α =to = sξ, i.e. constant t (ξ ) = π/w. We will treat the cases α = resp. δ = separately; for δ =,t ( ξ) has to be determined form (6). In both cases P reduces to a triangular matrix allowing easy access to eigenvalues and eigenvectors of P; but note that the map P depends through t (ξ) in a nonlinear way on ξ. 4.. Case I: α = In the simple case α =, there are two invariant half-planes for the and -system, respectively, spanned by e, e for the -system and e, for the -system. δ The Poincaré map P mapping ξ, -plane into itself is given by ( c s P = F E + δ(s + c E ) E + E )( ξ ). Since t is determined by the equation ξ s =, we obtain t (ξ) = (π/w ) for ξ >, hence P is independent of ξ on W + < W < ={(ξ,ξ ) ξ > }, there is no sliding motion. Fixed points of P are determined by the linear system Pξ = ξ. The eigenvalues of P are F(s + c) = F = e (λ+ /w + +λ /w )π, FE + E = e (μ+ /w + +μ /w )π. ( ) Note that ξ = is an eigenvector representing the z-axis which can be considered as a degenerated invariant cone although ξ is not contained in W< W < +. There are two cases to obtain an invariant cone of periodic orbits: Since an invariant cone requires that one eigenvalue of P equals. FE + E = (i.e. μ + /w + + μ /w = ) implies δξ =, and F ξ = ξ hence either ξ = or δ =, F =. The case δξ = corresponds to an invariant z-axis. Iterations of the Poincaré map starting in (ξ, ) W < W + < give ξ n+ = Fξ n = F n+ ξ, ξ n+ 3 = FE + δ( + E )ξ n + ξn 3,

10 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () hence ξ n+ = F n+ ξ, ξ n+ 3 = n i= FE+ δ( + E )ξ i +, = FE + δ( + E ) n i= ξi +, = FE + δ( + E ) n i= Fξ +, = FE + δ( + E )ξ n i= F i + ξ 3, hence, F < ξ n ξ, F =, F > FE + δ( + E )ξ ξ3 n F, F <., F Since t = π/w the relation F = is equivalent to λ + /w + + λ /w =. In any case there is no periodic cone. F =, (i.e.λ + /w + + λ /w ) = and FE + E /=. The corresponding eigenvector is calculated as ξ = δ E+ ( + E ) E + E, the corresponding cone is attractive if μ + /w + + μ /w < and repulsive if μ + /w + + μ /w >, hence stability is determined by the time spent in each half-space, which is measured by w and w +. For example see Figs. and 3. We further on concentrate on case (ii). For δ = (the smooth case) there is a degenerate (flat) cone within the (x, y)-plane, for δ /= a nontrivial cone tending to the positive z-axis for δ( E + E ) and to the negative z-axis for δ( E + E ) +. For fixed δ /=, variation of μ + for example from to corresponds to an attractive cone developing out of the ξ, ξ -plane approaching the -axis for μ + μ w /w + ; for μ + = μ w /w + there is no invariant cone; μ + > μ w /w + there is an repulsive cone.

11 99 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () We consider as a function of μ (, ) for μ = μ + w + /w, the invariant cone degenerates to the axis for lim (μ ) = δe (μ+ λ + )π/w + μ lim μ μ + w + /w (μ ) = sgn(δ) lim μ μ + w + /w (μ ) = sgn(δ) lim (μ ) = δe (λ+ /w + μ + )π/w + μ ( ) Note that the invariant half plane is given by δ. 4.. Case II: δ = and α /= In the case δ = the system possesses an invariant plane = with t (ξ) = π/w.forα /=, and return time t depends on ξ and is determined as smallest positive root of = sξ + α[c s E ]. /= the (7) The map P(ξ) isgivenby ( c s αs P = F E + E )( ξ ). For = we have t = π/w, s =,c = and P reduces to ( )( ) ξ P = F E + E, for /= P depends on ξ in a nonlinear way through t (ξ). Invariant directions are determined by the equation P(ξ)=μξ, μ for ξ W< W < +, i.e. ξ >and>a ξ + a 3, hence w < ξ α[λ w μ ] w > ξ α[λ w μ ] if α[λ w μ ] > if α[λ w μ ] <. We distinguish two cases to obtain invariant cones: F(s + c) =. Then t = π/w, = and s =,c =, consequently λ + /w + + λ /w =. To see this note that the fixed point equation first yields (FE + E ) =. If = then s =, hence t = π/w and c =, corresponding to a flat cone for which of course ξ = W< W < +. If /= then FE E + =. The first part of the fixed point equation requires s =, hence c =, consequently c = /F < in contradiction to c = E > following from Eq. (7). In this case we obtain a flat cone contained within the invariant plane. The cone corresponding to λ + /w + + λ /w = is attractive if μ + /w + + μ /w <. F(s + c) /= and FE + E =, hence μ + t + + μ t =.

12 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () Attractive invariant cone μ + /w + +μ /w < a ξ ξ Repulsive invariant cone μ + /w + +μ /w > b ξ ξ Fig.. Attractive/repulsive invariant cone consists of periodic orbits λ + = λ =,μ =,w = w =,t + = t = π: (a) μ + =.5 and (b) μ + =.5. For ξ = W < W + <, t (ξ) is determined by (7), or equivalently by = s ξ α c s E =: α h (t ). In terms of function h the restriction ξ = W < W + < can be written as ξ > and < h () ξ α if h () α >, (8a)

13 99 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () Attractive fixed point λ + /w + +λ /w < a ξ ξ Unstable fixed point λ + /w + +λ /w > b x 7 3 x 7 5 ξ ξ 4 x 7 6 Fig. 3. Attractive/repulsive fixed points, μ =.,μ + =.5,w + = w =,λ =,t + = t = π, α =,δ=.5: (a) λ + =.9 and (b) λ + =.. < h () ξ α if h () α >, where h () = w /[λ w μ ]. Set h (t ):= c s E, then h () = and h () = w /h (). (8b) Geometrically the line = h ()ξ /α determines the boundary of the sliding motion area in the (ξ, )-plane. For t defined by (7) we can determine an invariant cone spanned by ξ =, = h (t )/α, (9) provided (8a) and (8b) and t + μ + + t μ = hold. Attractivity of the cone is determined by the quantity F(c + s)(t ) <. Since t must satisfy t = (μ + /μ )π/w we can check by considering the graph of h /α for which parameters (8a) and (8b) holds, that is for t = (μ + /μ )π/w, ξ = we have to check if = h (t ) satisfies (8a) and (8b).

14 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () a 4 case h ()/α> 8 Sliding motion area h (t ) 6 4 T * 4 Direct transition between (T *,T ) t b 5 4 Case h ()/α< 3 h (t ) Direct transition between (,T ) T =T * Sliding motion area t Fig. 4. Graph of h (t ) for parameter values α =.5,μ + =.75,w + = w =.,μ =.,λ + =.5: (a) λ =.5 and (b) λ =.5. Let (T, T ) (, ) denote the interval such that for t (T, T ), = h (t ) satisfies (8a) and (8b). Let T denote the smallest positive root of h (t ). In the case < h ()/α we obtain T = T and T as the smallest positive root of h (t ) h ()=. In the case > h ()/α we obtain T = and T = T. Note that both T and T depend on the parameters w, λ, μ. We obtain a cone of periodic orbits if t = (μ + /μ )π/w (T,T ). Attractivity of the cone is determined by the quantity F(s + c), evaluated at t. The graph of h is determined by zeros of h (t ). Because of h () = and h () = λ μ w we obtain h () > in case (8a) and h () < in case (8b). Since h (π/(4w )) < in case (8a) there is T (,π/(4w )) such that h ( T ) =. The typical graph of h in case (8a) is given in Fig. 4a. In case (8b) there is two possibilities; see Fig. 4b. h (t)<(t (, π)) h (T ) = for some minimal (T (, π)).

15 994 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () μ + +3 μ /4< μ + +3 μ /4= a.5 b ξ ξ ξ ξ μ + +3 μ /4> c ξ ξ Fig. 5. Attractive/periodic/repulsive invariant cone for case ii (), μ =.,w + = w =,λ =.783,t + = π, t = 3π/4,δ=.,α=.5: (a) λ + =.7768, μ + =.765, (b) λ + =.83, μ + =.75 and (c) λ + =.846, μ + =.783. attractive invaiant cones 3 5 a 5 a 5 4 ξ ξ 3 4 Fig. 6. Two attractive invariant cone, λ + =.5,μ + =.75,w + = w =,λ =.5,μ =.,t + = π, δ =.,α=.5 where t = π for a and t =.555 for a.

16 T. Küpper, H.A. Hosham / Mathematics and Computers in Simulation 8 () We consider a few special cases explicitly. () Since T <π/(4w ) we can always arrange that s + c = for (3π/4w ) = t = (μ + /μ )(π/w + ), hence if we set μ + = 3μ /4 and require λ μ w > we obtain an attractive cone given by ξ =, = /(Fαs) without any restriction on λ +, see Fig. 5. () If λ + /w + + λ /w = and μ + + μ <, then by (i) there exists an attractive invariant flat cone with t = π/w. We use this constellation to construct another invariant cone by using case (ii) for a different t = (μ + /μ )(π/w + ); that cone is attractive as well if F(s + c) < for this choice of t. For example this situation holds for the special choice of parameters used in Fig. 6. Both locally attractive cones are separated by a repulsive manifold on which solutions converge towards. An interpretation of this example in terms of generalized center manifolds provides a situation that there are (locally) two generalized center manifolds at the same time, hence there is a chance of bifurcation of two separate periodic orbits if nonlinear terms are added. 5. Conclusions In a previous paper the existence of invariant cones for nonsmooth piecewise linear systems has been established. This approach can be extended to include nonlinear perturbation of the basic piecewise linear systems. With regard to further bifurcation analysis the notion of a generalized center manifold can be formulated. Using a class of three-dimensional examples already developed in [8] we investigate various ways to generate invariant cones. As a typical situation we consider the a unfolding of a flat cone for a smooth system triggered by a nonsmooth perturbations; further we show that multiple invariant cones may exist and finally we study the mechanism to determine attractivity of the original systems. Acknowledgement This work was supported by Department of Mathematics, Faculty of Science, Al-Azhar University of Assiut, Egypt. References [] M. di Bernardo, C. Budd, A.R. Champneys, P. Kowalczyk, A.B. Nordmark, G. Olivar, P.T. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Review 5 (4) (8) [] M. di Bernardo, C. Budd, A.R. Champneys, P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, 8. [3] M. di Bernardo, C.J. Budd, A.R. Champneys, Corner collision implies border-collision bifurcation, Physics D 54 (3 4) () [4] V. Carmona, E. Freire, E. Ponce, F. Torres, Bifurcation of invariant cones in piecwise linear homogeneous system, Int. J. Bifur. Chaos 5 (8) (5) [5] V. Carmona, E. Freire, J. Ros, F. Torres, Limit cycle bifurcation in 3d continuous piecewise linear systems with two zones. Application to Chua s circuit, Int. J. Bifur. Chaos 5 () (5) [6] H. A. Hosham, T. Küpper, D. Weiß, Invariant manifold in nonsmooth systems, Physica D: Nonlinear Phenomena, submitted. [7] M. Kunze, Non-smooth Dynamical Systems, Springer-Verlag, Berlin, (Lecture Notes in Mathematics 744). [8] T. Küpper, Invariant cones for non-smooth systems, Mathematics and Computers in Simulation 79 (8) [9] T. Küpper, Concepts of non-smooth dynamical systems, in: Mathematics and st Century, World Scientific Publishing Co.,, pp [] T. Küpper, S. Moritz, Generalized Hopf bifurcation of nonsmooth planar systems., R. Soc. Lond. Philos, Trans. Ser. A: Math. Phys. Eng. Sci. 359 (789) () [] R.I. Leine, H. Nijmeijer, Dynamics and bifurcations of nonsmooth mechanical systems, vol. 8 of Lecture Notes in Applied and Computational Mechanics. Springer-Verlag, Berlin, 4. [] H.E. Nusse, E. Ott, J.A. Yorke, Border-collision bifurcations: an explanation for observed difurcation phenomena, Phys. Rev. E(3) 49 () (994) [3] Y. Zou, T. Küpper, Generalized Hopf bifurcation for nonsmooth planar systems: the coner case, Northeast. Math. J. 7 (4) () [4] Y. Zou, T. Küpper, Hopf bifurcation for nonsmooth planar dynamical systems, Northeast. Math. J. 7 (3) () [5] Y. Zou, T. Küpper, Generalized Hopf bifurcation emanated from a corner, Nonlin. Anal. TAM 6 () (5) 7. [6] Y. Zou, T. Küpper, W.J. Beyn, Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlin. Sci. 6 () (6)