Extending Slow Manifold Near an Asymmetric Cusp Point

XIE Feng(谢峰),ZHANG Wanjun(张婉军)

(Department of Mathematics,School of Science,Donghua University,Shanghai 201620,China)

Abstract: In this paper,we investigate the dynamics of singularly perturbed differential equations near an asymmetric cusp point,where the critical manifold loses its hyperbolicity.Our results are proved by means of blow-up transformations combined with standard tools in the theory of dynamical system.

Key words: Slow manifold;Asymmetric cusp point;Singular perturbation;Blow-up

1.Introduction

A large class of physical problems,such as travelling wave in semiconductors[1−2],nonlinear network[3],and so on,can be modeled by the system of singularly perturbed ordinary differential equations

wheref,gareC∞-functions,and the dot denotes the derivative ofxandywith respect to the timeτ.Sett:=τ/ε,one obtains the following equivalent system

where the prime denotes the derivative ofxandywith respect tot.Here the variableycan be seen as slow variable ‘parameter’.The set

corresponds to the equilibria of (1.2) withε=0.We callSthe critical manifold.From the geometric singular perturbation theory[4−5],we know that a normally hyperbolic submanifoldS0⊂Spersists as a nearby locally invariant slow manifoldSεof (1.2) for sufficiently smallε.An important issue is to study the dynamics of system (1.2),and in particular,to clarify the behavior of the slow manifoldSεin the vicinity of points where the normal hyperbolicity is lost.These are bifurcation points of the critical manifoldS,for instance,folds,self-intersection points ofSand so on.The corresponding phenomena in the context of system (1.2) are sometimes referred to dynamic bifurcation[6−7].Using the developed blow-up methods[8−9],Krupa and Szmolyan[10−11]investigated the dynamic behavior in some vicinity of folds,canard points,transcritical and pitchfork singularities.

In this paper we focus our attention on planar problems and study asymmetric cusp points.The corresponding situations in higher-dimensional systems(1.1)with one-dimensional critical manifolds can be reduced to the planar case by the center manifold reduction.Roughly speaking,an asymmetric cusp is a bifurcation point of the critical manifoldS.For more-detail definition,we will present it at the begin of Section 2.We remark that the blow-up method developed in [8-10]and [11]can not be used directly in this paper.Because there using one blow-up transformation the system becomes regular.In our case after one time blow-up transformation,it is also a singular perturbation system.But we must say that the main ideas of the proof follow that given in [10-11].

2.The Dynamics Near the Asymmetric Cusp

Suppose that the singularly perturbed ordinary differential equations (1.2),with (x,y)∈R2andε>0 sufficiently small,satisfies the following defining conditions:

where byvwe mean a directional derivative along the eigenvector associated with the eigenvalue zero of the Hessian off.Under the conditions (2.1)-(2.3),the origin is the so-calledasymmetric cuspfor the graph off(x,y,0)=0 where we considerxas a function ofy(see for example,in [12]).Furthermore,we assume that

Without loss of generality,we suppose that

The critical manifoldSconsists ofSaandSr(see Figure 2.1).Sais stable andSris unstable.The standard singular perturbation theory[4,5]implies that forε >0,outside of a small neighborhood of (0,0) there exist two branches of slow manifolds bifurcated fromSaandSr,respectively.The left one is stable and the other is unstable.We denote bySa,εandSr,εthese two branches,respectively.We are interested in the behavior ofSa,εas it is extended through a neighborhood of(0,0).The following lemma provides a canonical normal form of system (1.2).

Lemma 2.1Assume that the conditions (2.1)-(2.4) hold.Set

Then system (1.2) with (x,y)∈R2can be transformed into the following canonical form:

Fig.2.1 Critical manifolds,slow manifolds

Fig.2.2 Sections for r2=0 in K2: (a) σ >0，(b) σ <0

ProofIntroduce a new variablesatisfying

Using (2.1)-(2.3),system (1.2) is transformed into the following form

whereh1andh2satisfy the order estimates given in the statement of Lemma 2.1.By introducing new variables,andas follows:

and making some easy computations,we can get the proof of the lemma.

For smallρ>0 and a small open intervalJcentered at 0 in R,set

Letπ:∆in→∆outbe the transition map derived by the flow of (1.2).We have the following main results.

Theorem 2.1Assume that the conditions (2.1)-(2.4) hold,then there existsε0>0 such that the following assertions hold forε ∈(0,ε0].

(i) The manifoldSa,εpasses through ∆outat a point (ρ,h(ε)),where

forσ/=0,and

forσ=0,while−Ω0is the smallest positive zero of the modified Bessel function of the second kind BesselK.

(ii) The transition mapπis a contraction with contraction rateO(e−ε/c),where c is a positive constant.

In the following proof of the theorem,forσ/=0 orσ=0 we need to use different blow-up transformations.Hence we distinguish these two cases.

Case (a)σ/=0.

Consider the extended system

whereh1,h2satisfy the order estimates given in the statement of Lemma 2.1.

For this system we apply the following blow-up transformation:

denoted byΦ,which mapsB0=S2×[0,r0],r0>0 into R3.The blow-up vector fieldis of the form

As can be easily checked,there are four equilibria on the invariant circle==0,i.e.on the equator,two corresponding to the two branches of the critical manifold,one corresponding to the incoming critical fibre and one denoted byqoutcorresponding to the outgoing critical fibre.We need three charts to describe all the dynamics,namely the usual rescaling chartK2defined by

the chartK1defined by

and the chartK3defined by

Obviously,we have the following changes of coordinates between the charts:

κ12:K1→K2given by

κ21:K2→K1given by

κ23:K2→K3given by

κ32:K3→K2given by

We introducedenotes an object in the blow-up which corresponds to an objectPin the original problem.Ifis described in the blown-up manifold and belongs to some charts then we usePito denote the object in the corresponding chartsKi,i=1,2,3.

The dynamics of the blown-up vector fieldin a neighborhood of the upper half-sphere is studied in the chartK2.Applying the change (2.7) to desingularize the origin of system(2.6),we obtain

where the prime denotes the derivative with respect tot2.We note that (2.14) is also a singular perturbation system for small positiver2.

We need to describe the transition map for (2.14) within a bounded domainD2with the origin as an inner point.In such a domain we can deduce properties of the flow of (2.14)from the extending geometric singular perturbation theory described in [10].The dynamics outersideD2will be investigated in the chartsK1andK3.We denote byγ0the union of the

Proposition 2.1A small neighborhood ofq0is mapped diffeomorphically onto a neighborhood ofΠ2(q0) and

ProofBy transformation ˆy2=y2−σ1/3,system (2.14) becomes

whose critical manifold contains a unique fold.Then the result follows from Theorem 2.1 in[10].

where the prime denotes the derivative with respect to a rescaled time variablet1.

Remark 2.1Forr1>0,(2.15) has the same orbits as the blown-up vector fieldwith the corresponding solutions having a different time parametrization.Since we deal with transition maps between sections,time parametrization of solutions has no significance to our analysis.

System (2.15) has two equilibriapa=(−1,0,0) andpr=(1,0,0) on the liner1=ε1=0,see Fig.2.3.The pointpahas a one-dimensional stable eigenspace and a twodimensional centre eigenspace.The pointprhas a one-dimensional unstable eigenspace and a two-dimensional centre eigenspace.The dynamics in the invariant planeε1=0 is governed by

Fig.2.3 Geometry and dynamics in chart K1

It follows from the Implicit Function Theorem that this system has a normally hyperbolic curveSa,1of equilibria containingpaand a curveSr,1of equilibria containingprfor smallr1.Actually,Sa,1andSr,1are precisely the branches of the critical manifoldSdescribed in Section 2.1.The dynamics in the invariant planer1=0 is governed by

It has a one-dimensional centre manifold at each ofpaandpr,denoted byNa,1andNr,1.

We restrict our attention to the setD1:={(x1,r1,ε1):x1∈R,0≤r1≤ρ,0≤ε1≤δ},whereρ >0 andδ >0 are the constants defined in Section 2.1.For the system (2.15) we have the following results:

Proposition 2.2System (2.15) has the following properties forρ >0 andδ >0 sufficiently small:

1) The linearization of system (2.15) atpa(respectively,pr) has the following real eigenvalues:λ1=−2(respectively,λ1=2) with the eigenvector (1,0,0),λ2=0 with an eigenvector tangent toSa,1(respectively,Sr,1),andλ3=0 with the eigenvector (σ,0,2)(respectively,(σ,0,−2)) tangent to the centre direction in the invariant planer1=0;

2)There exists an attracting two-dimensionalCk-center manifoldMa,1atpathat contains the line of equilibriaSa,1and the center manifoldNa,1.InD1the manifoldMa,1is given as a graph of the functionx1=ha(r1,ε1).The branch ofNa,1inr1=0 andε1>0 is unique;

3) There exists a repelling two-dimensionalCk-center manifoldMr,1atprthat contains the line of equilibriaSr,1and the center manifoldNr,1.InD1the manifoldMr,1is given as a graph of the functionx1=hr(r1,ε1).The branch ofNr,1inr1=0 andε1>0 is unique;

4) There exists a stable invariant foliationFswith the baseMa,1and one-dimensional fibers.For any−2

5)There exists an unstable invariant foliationFuwith the baseMr,1and one-dimensional fibers.For any 0

ProofThe first assertion follows from simple computations.Assertions 2)-5) follow from the first assertion and the standard theory of center manifold[5].

We now define the following sections:

Proposition 2.3For sufficiently smallρ,δandβ1,the transition mapΠ1has the following properties

ProofFrom the second and third equations of (2.15),we have

Integrating the above equation,we obtain

whileρandδare the same constants as before,andβ3>0 is sufficiently small (see Fig.2.4).

Fig.2.4 Geometry and dynamics of system (2.19) near the equilibrium qout

To obtain a formula for the mapΠ3,we divide (2.19) by the factorF(r3,y3,ε3) and obtain

whereG(r3,y3,ε3) is aCk-function.System (2.20) has the same phase portrait as that of system (2.19) in a neighborhood of the origin.

Proposition 2.4The transition mapΠ3defined by the flow of system (2.19) has the form

This completes the proof of the proposition.

In the following proof we combine the results acquired in the individual charts.

Proof of Statement (i) in Theorem 2.1We define the mapΠ:by the composition

Consequently,the corresponding expansion for the functionh(ε) in Theorem 2.1 is

We complete the proof of the statement.

Case (b)σ=0.Consider the extended system

whereh1,h2satisfy the order estimates given in the statement of Lemma 2.1.

For this system,using the following blow-up transformation:

denoted byΦ,which mapsB0=S2×[0,r0],r0>0 into R3,we obtain the following blow-up vector fieldof the form

They have four equilibria on the invariant circle==0,two corresponding to the two branches of the critical manifold,one corresponding to the incoming critical fibre and one corresponding to the outgoing critical fibre.We need three charts to describe all the dynamics,namely the usual rescaling chartK2,defined by

the chartK1defined by

and the chartK3defined by

Obviously,we have the following changes of coordinates between the charts

κ12:K1→K2given by

κ21:K2→K1given by

κ23:K2→K3given by

κ32:K3→K2given by

The dynamics of the blown-up vector fieldin a neighborhood of the upper half-sphere is studied in chartK2.Using (2.22) to desingularize (2.21) in a neighborhood of the origin,we obtain

where the prime denotes derivative with respect tot2.

Whenr2=0,the system (2.29) becomes

This is a Riccati equation whose solutions can be expressed in terms of special functions.We have the following results.

Proposition 2.5System (2.30) has the following properties:

1) There exists a unique orbitγ2which can be parametrized as (x2,y2(x2)),x2∈R,where

2) Every orbit has a horizontal asymptotey=yr,whereyrdepends on the orbit such thaty2approaches the liney2=yrfrom above asx2→+∞.

ProofEquation dx2/dy2=is a special Riccati equation.Its general solution can be expressed as follows

wherecis an arbitrary constant,and BesselI and BesselK are the modified Bessel functions of the first and second kinds,respectively (see for instance [13]).The results follow from the properties of the Bessel functions.

Forδ >0 we define the transition mapderived by the flow of system(2.29),where

Let.From Proposition 2.5 and the regular perturbation theory,we get

Proposition 2.6A small neighborhood ofq0is mapped diffeomorphically onto a neighborhood ofΠ2(q0) and

Applying transformations (2.23) and (2.24) to (2.21) respectively,we obtain inK1

For system (2.31),we can get similar results to Proposition 2.2 and Proposition 2.3.To obtain a formula for the mapΠ3(defined in Proposition 2.4),we divide (2.32) by the factorF(r3,y3,ε3),and then obtain

whereG(r3,y3,ε3) is aCk-function.System (2.33) can be linearized by a near identity transformation of the form

Proposition 2.7The transition mapΠ3formed by the flow of system (2.32) has the form

This completes the proof of the proposition.

Using the analysis in the caseσ/=0 and working in a similar way to the proof of the caseσ/=0,we can prove the theorem except the estimation ofh(ε).

Consequently,the expansion of the functionh(ε) in Theorem 2.1 is of the form

This completes the proof of the theorem.