TRACING THE SOLUTION SURFACE WITH

FOLDS OF A TWO PARAMETER SYSTEMS

S.-L. Chang∗

Center for General Education,

Southern Taiwan University of Technology, Tainan, Taiwan, 710

C.-S. Chien†‡

Department of Applied Mathematics,

National Chung-Hsing University, Taichung, Taiwan, 402

B.-W. Jeng†

Department of Applied Mathematics,

National Chung-Hsing University, Taichung, Taiwan, 402

We describe a special Gauss-Newton method for tracing solution manifolds with singularities ofmulti-parameter systems. First we choose one of the parameters as the continuation parameter,and fix the others. Then we trace one-dimensional solution curves by using continuation methods.Singularities such as folds, simple and multiple bifurcations on each solution curve can be easilydetected. Next, we choose an interval for the second continuation parameter, and trace one-dimensional solution curves for certain values in this interval. This constitutes a two-dimensionalsolution surface. The procedure can be generalized to trace a k-dimensional solution manifold.Numerical results in 1D, 2D and 3D second order semilinear elliptic eigenvalue problems given byP. L. Lions [1982] are reported.

Keywords: parameter-dependent nonlinear systems, solution manifold, singularities, continuationmethods, two-grid schemes.

∗Supported by the National Science Council of R.O.C.(TAIWAN) through Project NSC 92-2115-M-218-003.†Supported by the National Science Council of R.O.C.(TAIWAN) through Project NSC 92-2115-M-005-001.‡Author for correspondence.

1

1. Introduction

We consider parameter-dependent nonlinear systems of equations of the following form

F (x,Λ) = 0, (1)

where F : X1 × Rk → X2 is a smooth mapping with x ∈ X1, Λ ∈ Rk a parameter vector, X1 andX2 are appropriate Banach spaces. Equation (1) arises in many physical systems. For conveniencewe consider the two parameter case. For example, the fourth order von Karman equations

F (f,w, λ, ) =

∆2f + 1

2 [w,w] = 0∆2w − [w, f ] + λwxx = 0

in Ω = [0, ] × [0, 1] (2)

describe a long rectangular plate P in compression. Here f(x, y) is the Airy stress function producedin P , and w(x, y) represents the deformation of P from its flat state, and the bracket operator [·, ·]is defined by

[u, v] = uxxvyy − 2uxyvxy + uyyvxx.

We consider two sets of boundary conditions for f ,(a) homogeneous Dirichlet conditions f = ∆f = 0;(b) homogeneous Neumann conditions fn = (∆f)n = 0,

and two sets of boundary conditions for w,(i) simply supported on all four edges, w = ∆w = 0;(ii) simply supported on unloaded sides, clamped on loaded end.

See [Golubitsky & Schaeffer, 1985, pp.439–454] and the further references cited therein for details.

In [1982] P. L. Lions gave a survey concerning positive solutions of semilinear elliptic equationsof the following form

−∆u = f(u) in Ω, u ∈ C2(Ω),u > 0 in Ω, u = 0 on ∂Ω,

(3)

where Ω is a bounded regular domain in Rn, and f(u) is some nonlinear smooth function. Forf(u) = u(1 − sin u) + up, 1 < p < N∗, where N∗ = ∞ if n = 1, 2, and N∗ = n+2

n−2 , if n ≥ 3,it was claimed that the solution curve has exactly one fold on the (λ, ‖u‖∞)-plane. See [Lions,1982, Remark 1.7] for details. Further study concerning the number of positive solutions of certainsemilinear elliptic equations can be found, e.g., in Schaaf and Schmitt [1988, 1990]. Recently Wang[2004] studied the evolution and qualitative behaviors of the two-point boundary value problems

−u′′(x) = λ(u(1 − sinu) + up), −1 < x < 1,u(−1) = u(1) = 0.

(4)

It was shown there that the number of folds on each solution curve varies with respect to the valuesof p. Clearly Wang [2004] gives partial results of the solutions of Eq. (4).

It is well known that the solution curve of simple turning points are regular solutions of aparticular extended system of Eq. (1) with λ2 fixed. The extended system for numerical purposes

2

was first introduced by Seydel [1979], and then discussed in [Moore & Spence, 1980]. See also[Jepson & Spence, 1985] and the further references cited therein. In [1982] Rheinboldt proposednumerical methods for computing folds by following a path of turning points on the solution surfaceof Eq. (1). Later [Jepson & Spence, 1985] analyzed the theory of fold curves of Eq. (1), which is usedto design algorithms for computing fold curves, where cusp points, bifurcation points, and points ofisola formation are all simple turning points in this fold curve. In short, the computations of one-dimensional solution curves, fold curves and solution surfaces, etc., require solutions of borderedlinear systems. Research articles on this topic can be found, e.g., in [Chan 1984a, 1984b; Govaerts,1995, 1997, 2000]. Perhaps AUTO [Doedel, 1981] is one of the most well known numerical softwareswhich deals with solutions of bordered linear systems associated with Eq. (1). This software hasbeen updated regularly until recently [Doedel et al., 2000].

In Eqs. (2) and (3) we treat the parameter and p as the second continuation parameter,respectively. We will trace solution surface of Eq. (1) by varying two parameters, say λ1 and λ2

simultaneously. More precisely, for certain values of λ2 in some interval I, we wish to trace eachone-dimensional solution curve with λ1 as the continuation parameter and λ2 fixed. The folds oneach solution curve can be computed easily. See e.g., [Keller, 1987].

The numerical methods described above can be generalized to compute k-dimensional solutionmanifolds with singularities, k > 2. Research articles concerning the computations of k-dimensionalmanifolds can be found, e.g., in [Allgower & Schmidt, 1985; Allgower & Gnutzmann, 1987; Rhein-boldt, 1988; Henderson, 2002]. For simplicity we consider λ1, λ2 and λ3 as the continuationparameters. Suppose for a fixed value of λ3 we already obtain a solution surface with singularities.By varying the values of λ3 in some interval, we can compute the corresponding surfaces in a similarway.

Actually, the numerical method we describe above can be viewed as a special case of the Gauss-Newton method, where one uses Newton’s method to solve the discretization of Eq. (1), whichis an underdetermined nonlinear system. See [Allgower & Georg, 1996, Chapter 2]. To find theunique solution of the associated linear system, k additional constraint conditions must be given.For the physical problem with three parameters, we consider the Brusselator equations [Golubitsky& Schaeffer, 1985, pp.312–317]

∂u∂t = f(u, v, λ, α, ) = d1∆u − (λ + 1)u + u2v + α,∂v∂t = g(u, v, λ, α, ) = d2∆v + λu − u2v,

(x, y) ∈ Ω = (0, ) × (0, 1) (5)

with boundary conditions

u(x, y, t) = α, v(x, y, t) =λ

α, (x, y) ∈ ∂Ω.

Here we treat λ, α and the length of the domain Ω as continuation parameters. The numericalstudy of steady state solutions of Eq. (5) can be found, e.g., in [Chien & Chen, 1998].

The stationary axisymmetric flows of a viscous, incompressible Newtonian fluids between long,concentric cylinders is another example with multi-parameters. We assume that the temperature

3

of the fluids is constant with constant viscosity and density. Furthermore, the inner cylinder withradius R1, rotates at the angular speed Ω1R1, and the outer cylinder with radius R2 at the angularspeed Ω2R2. Let µ = Ω2/Ω1 be the rotation rate, η = R1/R2 the radius ratio and Re the Reynoldsnumber. Because of the geometry of the domain, the Navier-Stokes equations are described interms of cylindrical polar coordinates (r, θ, z). Since the problem we consider is axisymmetric, thevelocity components v = (u, v,w) depends only (r, z) and time. Discretizing the Navier-Stokesequations by using the finite differences or the finite elements, one obtain a nonlinear algebraicsystem

F (x,Re, k, η, µ) = 0, (6)

where η is a fixed constant, and Re, k, and µ may be viewed as continuation parameters. We referto Meyer-Spasche [1999] for details. The further reports concerning tracing solution curves of theNavier-Stokes equations using continuation method can be found, e.g., in Glowinski et al. [1985] andSanchez et al. [2002]. It would be interesting to study the solution manifolds of the Navier-Stokesequations. The numerical investigation is beyond the scope of this paper.

The numerical method described in Chien et al. [2000] shows how the solution surface of Eq. (2)can be obtained, and the figures given therein would show how it looks like. Our aim here is togive a detailed investigation concerning the solution surface of Eq. (4) and the associated two-and three-dimensional problems. This paper is organized as follows. Section 2 briefly reviews theresults of P. L. Lions [1982] and Wang [2004]. In Sec. 3 we describe a special Gauss-Newton method[Allgower & Georg, 1996] to trace solution manifolds of Eq. (1). The fold curve on the solutionsurface can be easily computed. We give some centered difference formulae for the coefficient matrixassociated with the higher order Laplace operator in Sec. 4. Our numerical results concerning thesolution surfaces of Eq. (4) and the two- and three-dimensional problems are reported in Sec. 5,where the one- and three-dimensional problems are discretized by centered differences, and thetwo-dimensional problem is discretized by the six-node triangular elements. A summary of ournumerical results is given in Sec. 6.

2. A Brief Survey of Well Known Results

In [1982] P. L. Lions gave a survey concerning the problem of the existence of positive solutions forsemilinear elliptic equations of the following form

−∆u = f(u) in Ω, u ∈ C2(Ω),u > 0 in Ω, u = 0 on ∂Ω,

(7)

where Ω is a bounded regular domain in Rn, and f(t) is some given nonlinearity. For simplicitywe assume f(0) = 0, f(t) > 0 for all t > 0, lim

t→+∞f(t)t

= 0, with = n+2n−2 if n ≥ 3, < ∞ if n = 1, 2.

Moreover, f is superlinear, that is, limt→+∞

f(t)t > λ1, where λ1 is the first eigenvalue of −∆ with

Dirichlet boundary conditions. In particular, if f ′(0) = 1, f(t) < t for t > 0, t small, Lions gave the

4

bifurcation diagram for f(t) = t(1− sin t)+ tp, 1 < p < n+2n−2 . It was claimed that the solution curve

has exactly one fold on the (λ, ‖u‖∞)-plane. See [Lions, 1982, Remark 1.7]. Recently, Wang [2004]studied the evolution and qualitative behaviors of positive solutions of the two-point boundaryvalue problem

−u′′(x) = λ(u(1 − sinu) + up), −1 < x < 1,u(−1) = u(1) = 0,

(8)

where λ > 0 is the bifurcation parameter and p ≥ 1 is the evolution parameter. The followingresults for Eq. (8) are given in [Wang, 2004].

(i) For p > 2, the bifurcation curve has exactly one fold on the (λ, ‖u‖∞)-plane.

(ii) For p = 2, the bifurcation curve is a monotone curve on the (λ, ‖u‖∞)-plane.

(iii) For 1 < p ≤ p = (1+2 cos 1−sin 1)(1−2 cos 1+sin 1) ≈ 1.6286, the bifurcation curve has at least two folds on the

(λ, ‖u‖∞)-plane.

(iv) For p = 1, the bifurcation curve has infinitely many folds on the (λ, ‖u‖∞)-plane. Moreoverfor any integer n ≥ 1, there exist δ > 0 such that Eq. (8) has at least n distinct positivesolutions for λ ∈ (π2

8 − δ, π2

8 + δ), and infinitely many positive solutions for λ = π2

8 .

However, there are still some facts which remain to be clarified. For example, how does themonotone curve at p = 2 evolve to the bifurcation curve with one fold? How does the solutioncurve evolve from p = 3 to the monotone curve at p = 2? In next section we will describe thenumerical methods to trace solution curves of Eq. (7) for various values of p. The folds on eachsolution curve can be easily detected.

3. Numerical Methods for Tracing Solution Surfaces

Discretizing the physical problems described in Sec. 1 by using finite difference method or finiteelement method, we obtain

H(x,Λ) = 0, (9)

where H : RN × Rk → RN is a smooth mapping with x ∈ RN , Λ ∈ Rk, k ≥ 1. Equation (9)is an underdetermined nonlinear system of equations. By [Rheinboldt, 1986, Theorem 4.2] thesolution of Eq. (9) is a k-dimensional manifold without boundary. We denote the solution manifoldof Eq. (9) by M = (x, λ) | H(x, λ) = 0. Assume that the parametrization via pseudo-arclength isavailable on the solution manifold. Of special interest here is the detection of folds on the solutionsurface. Well-known numerical methods for tracing folds can be found, e.g., in [Rheinboldt, 1982;Jepson & Spence, 1985].

For convenience we rewrite Eq. (9) as

H(y) = 0, y = (x, λ) (10)

5

where H : RN+k → RN is a smooth mapping. Assume 0 is a regular value of H. We briefly discusshow the Gauss-Newton method can be used to solve Eq. (10). For details we refer to [Allgower& Gerog, 1996]. Let ui−1 be an approximation solution of Eq. (9) such that H(ui−1) ≈ 0, andvi ∈ RN+k be a predicted point such that H(vi) ≈ 0 and ‖ui−vi−1‖ ≈ hi, where hi is the steplengthin a predictor-corrector continuation algorithm. To determine the next approximate solution ui,we consider the Taylor expansion about vi:

H(ui) = H(vi) + H ′(vi)(ui − vi) + O(‖ui − vi‖2), (11)

where H ′(vi) ∈ RN×(N+k). As in Newton’s method, the linearization of Eq. (10) consists ofneglecting the higher order term O(‖ui − vi‖2). This defines a Gauss-Newton map v → N (v) suchthat N (vi) is an approximation to the solution ui. That is, the Newton point N (vi) satisfies thefollowing equation

H(vi) + H ′(vi)(N (vi) − vi) = 0. (12)

Thus the solution of Eq. (12) can be expressed as

N (vi) := vi − H ′(vi)+H(vi) = 0, (13)

where H ′(vi)+ ∈ R(N+k)×N is the Moore-Penrose of H ′(vi). See, e.g., [Golub & Van Loan, 1996,Chapter 5] for details. Note that Eq. (12) is an underdetermined nonlinear system of equations.To obtain a unique solution of this system, k additional constraint conditions must be imposed onthis system. For k = 1, we can choose the following constraint condition

t(H ′(vi))T (N(vi) − vi) = 0, (14)

where t(A) denotes the tangent vector induced by the matrix A. Further constraint conditions forthis case can be found, e.g., in Allgower and Georg [1996, 2003] and Keller [1987]. For the casek > 2, besides the condition given in Eq. (14), we have to determine k − 1 additional constraintconditions. The whole process amounts to solving bordered linear systems of the following form[

A B

CT D

] [x

λ

]=

[f

g

], (15)

where A ∈ RN×N , B, C ∈ RN×k, D ∈ Rk×k and f ∈ RN , g ∈ Rk.

The numerical method we propose here for tracing solution manifold of Eq. (9) can be viewedas a special case of the Gauss-Newton method. Let λ = [λ1, · · · , λk]T . Tracing one-dimensionalcurves of Eq. (9) amounts to solving a minimum bordering system of Eq. (12). To trace the solutionsurface of Eq. (9), we have to vary two parameters, say, λ1 and λ2 simultaneously. This may beaccomplished by choosing a proper interval I for λ2 and some positive integers p1, · · · , pm ∈ I.For each fixed pi ∈ I, we trace the one-dimensional curve by solving the minimum bordered linearsystems of Eq. (12). It is straightforward to generalize the above-mentioned procedure to computea k-dimensional manifold of Eq. (9) with singularities. Finally, theoretical results for the Gauss-Newton map can be found in Allgower and Georg [1996, Chapter 2].

6

4. Centered Difference Approximations

Consider the following linear eigenvalue problem

−∆u = λu in Ω = (0, 1)n,

u = 0 on ∂Ω.(16)

We discretize Eq. (16) by the centered difference approximations with uniform meshsize h = 1N+1 on

each axis for some positive integer N . Let A(n) ∈ RNn×Nnbe the discretization matrix associated

with the Laplacian −∆. For n = 1, the discretization matrix A(1) ∈ RN×N is tridiagonal andsymmetric positive definite. Note that A(n) can be generated from A(1) and the identity matrixIN ∈ RN×N via matrix tensor products. For completeness we recall the following definition.

Let R = (rij) ∈ Rm×n and S = (sij) ∈ Rp×q. The matrix tensor product of R and S, denotedby R ⊗ S, is defined by

R ⊗ S =

⎛⎜⎜⎜⎜⎝r11S r12S · · · r1nS

r21S r22S · · · r2nS...

.... . .

...rm1S rm2S · · · rmnS

⎞⎟⎟⎟⎟⎠ .

One can easily verify that

A(n) = A(1) ⊗ IN ⊗ · · · ⊗ IN + IN ⊗ A(1) ⊗ · · · ⊗ IN + · · · + IN ⊗ · · · ⊗ IN ⊗ A(1), (17)

where each term on the right hand side of Eq. (17) is the tensor products of n − 1 identity matrixIN and one A(1).

In particular, for n = 3 we have A(3) ∈ RN3×N3with

A(3) = A(1) ⊗ IN ⊗ IN + IN ⊗ A(1) ⊗ IN + IN ⊗ IN ⊗ A(1)

=

⎛⎜⎜⎜⎜⎜⎝

1 2 · · · n

1 BN2 −IN2 02 −IN2 BN2

. . ....

. . . . . . −IN2

n 0 −IN2 BN2

⎞⎟⎟⎟⎟⎟⎠,

where

BN2 =

⎛⎜⎜⎜⎜⎜⎝

1 2 · · · n

1 AN −IN 02 −IN AN

. . ....

. . . . . . −IN

n 0 −IN AN

⎞⎟⎟⎟⎟⎟⎠

7

with

AN = (aij) =

⎧⎪⎨⎪⎩6 if i = j,−1 if i = j + 1 or j = i + 1,0 otherwise.

It is well known that the eigenpairs of A(1) are

µp = 2(N + 1)2(1 − cos pπN+1), 1 ≤ p ≤ N,

Up(xi) = sin ipπN+1 , 1 ≤ i, p ≤ N.

(18)

The eigenpairs of A(n) can be expressed in terms of Eq. (18), i.e.,

µp1,p2,···,pn = µp1 + µp2 + · · · + µpn

= 2(N + 1)2(n − cos p1πN+1 − · · · − cos pnπ

N+1), 1 ≤ p1, · · · , pn ≤ N,

Up1,p2,···,pn(x(1)i1

, · · · , x(n)in

) = sin p1i1πN+1 · · · sin pninπ

N+1

= Up1(x(1)i1

) · · ·Upn(x(n)in

), 1 ≤ p1, i1, · · · , pn, in ≤ N.

(19)

Equation (19) generalizes the formula of Kuttler and Sigillito [1984]. See also Chien [1987] fordetails.

5. Numerical Results

Example 1. One dimensional problem. We consider the bifurcation behaviors of positivesolutions for the two point boundary value problem

−u′′(x) = λ(u(1 − sin(u)) + up), 0 < x < 1u(0) = u(1) = 0,

(20)

where the bifurcation parameter λ > 0 and the evolution parameter p ≥ 1. Equation (20) wasdiscretized by the centered difference approximations with uniform mesh h = 0.0005 on [0, 1]. Thecontinuation-Lanczos algorithm [Chang & Chien, 2003] was used to trace the solution curves ofEq. (20). Table 1 lists the first five fold points of Eq. (20) for various values of p. One can useAUTO to get more accurate locations of folds on the solution curves.

Figure 1 shows the numerical solution of bifurcation curves with respect to 1 ≤ p ≤ 3 onthe (λ, ‖u‖∞)-plane. The solution surface on the (λ, p)-plane with 1 ≤ p ≤ 3 is shown Figure 2.Figure 3 depicts the contours of five positive solutions of Eq. (20) for λ = 4.0 at p = 1.15. Wesummarize the numerical results as follows.

1. All of the solution curves intersect at a point on the (λ, ‖u‖∞)-plane, and the point of inter-section is located at (λ, ‖u‖∞) ≈ (8.42, 1.33). See Fig. 1.

8

2. For 1 ≤ p ≤ 1.85, the bifurcation curve has at least two folds on the (λ, ‖u‖∞)-plane.

3. For 1.15 ≤ p ≤ 1.65, the bifurcation curve has at least one fold with ‖u‖∞ < ‖u‖∞, and boththe λ value and the ‖u‖∞ of the first fold increase with respect to p.

4. For 1.7 ≤ p ≤ 1.85, the bifurcation curve has no folds when ‖u‖∞ < ‖u‖∞, while for1.8 ≤ p ≤ 1.85, the bifurcation curve has exactly two folds on the (λ, ‖u‖∞)-plane with‖u‖∞ > ‖u‖∞.

5. For 1.9 ≤ p ≤ 2.1, the bifurcation curve is a monotone curve on the (λ, ‖u‖∞)-plane.

6. For p ≥ 2.2, the solution curve has exactly one fold (λ∗, ‖u∗‖∞) on the (λ, ‖u‖∞)-plane, andboth the λ value and the ‖u‖∞ increases with respect to p.

Example 2. Two dimensional problem. We consider the two-dimensional eigenvalue problem

−∆u = λ(u(1 − sin(u)) + up) in Ω = (0, 1)2,u = 0 on ∂Ω.

(21)

Equation (21) was discretized by the finite element approximations with uniform mesh h = 1/32on the x- and y-axis, respectively. The continuation-Lanczos algorithm was used to trace thebifurcation curves of Eq. (21) for each fixed p. The bifurcation behaviors of this example is similarto those of Example 1. Table 2 lists the first five fold points of Eq. (21) for various values of p.Figure 4 shows the bifurcation curves of Eq. (21) on the (λ, ‖u‖∞)-plane. The solution surface onthe (λ, p)-plane is shown in Fig. 5. From the above, we have the following numerical results.

1. All of the solution curves intersect at a point on the (λ, ‖u‖∞)-plane, and the point of theintersection is located at (λ, ‖u‖∞) ≈ (16.75, 1.38). See Fig. 4.

2. For 1 ≤ p ≤ 1.7, the bifurcation curve has at least two folds on the (λ, ‖u‖∞)-plane.

3. For 1.2 ≤ p ≤ 1.65, the bifurcation curve has at least one fold with ‖u‖∞ < ‖u‖∞, and boththe λ value and the ‖u‖∞ of the first fold increase with respect to p.

4. For p = 1.7, the bifurcation curve has no folds when ‖u‖∞ < ‖u‖∞ and the bifurcation curvehas exactly two folds on the (λ, ‖u‖∞)-plane with ‖u‖∞ > ‖u‖∞.

5. For 1.75 ≤ p ≤ 2.0, the bifurcation curve is a monotone curve on the (λ, ‖u‖∞)-plane.

6. For p ≥ 2.05, the solution curve has exactly one fold (λ∗, ‖u∗‖∞) on the (λ, ‖u‖∞)-plane, andboth the λ∗ value and the ‖u∗‖∞ increase with respect to p.

9

Example 3. Three dimensional problem. We consider the following three-dimensional eigen-value problem

−∆u = λ(u(1 − sin(u)) + up) ≡ λfp(u) in Ω = (0, 1)3,u = 0 on ∂Ω.

(22)

Equation (22) was discretized by the seven-point centered difference approximations with uniformmesh h = 0.03125 on the x-, y- and z-axis, respectively. The associated discrete nonlinear systemcan be expressed as

H(U, λ) = A(3)U − λh2fp(U) = 0, (23)

where A(3) ∈ RN3×N3is defined as in Sec. 4 with N = 31. The Jacobian DH = [DUH,DλH] of H

is given byDUH = A(3) − λh2diag(f ′

p(U1), · · · , f ′p(UN3)),

andDλH = −h2[fp(U1), · · · , fp(UN3)]T .

The continuation-Lanczos algorithm was used to trace the bifurcation curves for each fixed p. Thebifurcation behaviors of this example is similar to those of Example 1. Table 3 lists the first fivefold points of Eq. (22) for various values of p. Figure 6 shows the bifurcation curves of Eq. (22) onthe (λ, ‖u‖∞)-plane. The solution surface on the (λ, p)-plane is displayed in Fig. 7. We have thefollowing numerical results.

1. All of the solution curves intersect at a point on the (λ, ‖u‖∞)-plane, and the point of inter-section is located at (λ, ‖u‖) ≈ (24.80, 1.70). See Fig. 6.

2. For 1 ≤ p ≤ 1.6, the bifurcation curve has at least two folds on the (λ, ‖u‖∞)-plane.

3. For 1.1 ≤ p ≤ 1.6, the λ value and the ‖u‖∞ of the first fold increases with respect to p.

4. For 1.7 ≤ p ≤ 2.0, the bifurcation curve is a monotone curve on the (λ, ‖u‖∞)-plane.

5. For p ≥ 2.1, the solution curve has exactly one fold (λ∗, ‖u∗‖∞) on the (λ, ‖u‖∞)-plane, andboth the λ∗ value and the ‖u∗‖∞ increases with respect to p.

6. Conclusions

We describe a special Gauss-Newton method for tracing the solution surface of a two parametersystem with folds. The numerical solution of a semilinear elliptic equation given by P. L. Lions[1982] is presented. It is obvious that Lions only gave partial result of this problem. It would beinteresting to study the bifurcation behavior of this problem theoretically.

AcknowledgmentsThe authors would like to thank two referees for many suggestions that improved the paper.

10

Table 1. The first five fold points of Eq. (20). The symbol “ ” means no fold.

1st 2nd 3rd 4th 5thp

λ ‖u‖∞ λ ‖u‖∞ λ ‖u‖∞ λ ‖u‖∞ λ ‖u‖∞1.00 9.446294161 1.74487112 3.668775057 5.26807055 7.094770372 8.33653923 4.056822022 11.7523731

1.05 9.279203835 1.72213485 3.568395214 5.28134552 6.579366566 8.33606127 3.865496636 11.7575036

1.10 9.123287610 1.68629530 3.465889630 5.30079970 6.094801099 8.33428416 3.671433117 11.7719752

1.15 6.166230215 0.17503935 8.980624436 1.63772235 3.369281288 5.45138362 5.634761863 8.24200780 3.477461104 11.8413886

1.20 6.441199617 0.16850090 8.853387580 1.58230230 3.257491387 5.39161938 5.205037988 8.20405443 3.280458878 11.7937472

1.25 6.714440051 0.19248420 8.740124161 1.53049861 3.149785532 5.32428862 4.802204615 8.15735976 3.086790595 11.7372341

1.30 6.975748686 0.23821676 8.643275385 1.47819382 3.045547270 5.54838257 4.427418706 8.46569111 2.893702493 11.9578919

1.35 7.218174110 0.23831143 8.564397558 1.40663358 2.929575800 5.48475645 4.086935845 8.41458364 2.701313640 11.9045235

1.40 7.453141626 0.26585771 8.519807317 1.33575428 2.818737827 5.42097796 3.767946845 8.36432153 2.515912849 11.8497520

1.45 7.678997833 0.28694687 8.482896660 1.25819712 2.708658015 5.65483317 3.470106906 8.31911387 2.334842369 12.0803232

1.50 7.896636566 0.31780115 8.470044152 1.17923977 2.592415645 5.58494625 3.194407639 8.25994888 2.158308841 12.0199712

1.55 8.106885740 0.34752990 8.484176238 1.07678465 2.481715091 5.53386782 2.937603646 8.22334920 1.990864314 12.2603419

1.60 8.364945220 0.40952577 8.528626630 0.97327848 2.365636450 5.76726841 2.701114143 8.17302837 1.827391176 12.2093975

1.65 8.556111605 0.45968612 8.608074230 0.83565818 2.252881335 5.71559576 2.486779351 7.80385109 1.673535179 12.4524502

1.70 2.137796387 5.96916578 2.290034587 7.77375567 1.526082103 12.4205126

1.75 2.024671407 5.92889556 2.108611320 7.74441806 1.386343777 12.6795519

1.80 1.908657595 6.18564819 1.944062167 7.39538293

1.85 1.792358599 6.45535972 1.796978266 7.06193355

1.90

1.95

2.00

2.10

2.20 10.55599464 0.32469102

2.30 10.85611945 0.37038343

2.40 11.15072219 0.40434560

2.50 11.43830539 0.42671288

2.60 11.71835168 0.44075714

2.70 11.99109647 0.45926680

2.80 12.25675375 0.47596390

2.90 12.51561522 0.49089837

3.00 12.76800616 0.50410746

11

Table 2. The first five fold points of Eq. (21). The symbol “ ” means no fold.

1st 2nd 3rd 4th 5thp

λ ‖u‖∞ λ ‖u‖∞ λ ‖u‖∞ λ ‖u‖∞ λ ‖u‖∞1.00 18.1900916 1.9542060 8.1455597 5.9709832 12.1541488 8.7354648 9.0828399 12.5766136

1.20 12.9113355 0.1659857 17.2694472 1.7746919 7.1989496 6.1143717 9.3388442 8.7843243 7.2914314 12.7126448

1.25 13.4607519 0.2372221 17.1011520 1.7033517 6.9531743 6.1524768 8.7235552 8.7460358 6.8492996 12.7702000

1.30 13.9818394 0.2319472 16.9624053 1.6349057 6.7044731 6.1976286 8.1425296 8.7427238 6.4138140 12.8213538

1.35 14.4728807 0.2939139 16.8563250 1.5642339 6.4531684 6.2649026 7.5942654 8.7287928 5.9868534 12.9203961

1.40 14.9450263 0.2968560 16.7861466 1.4859080 6.1995721 6.3092024 7.0781738 8.6647813 5.5701102 13.0304253

1.45 15.3960413 0.3418069 16.7557362 1.4051257 5.9437756 6.3942275 6.5938392 8.6143542 5.1648593 13.1522432

1.50 15.8317441 0.3806229 16.7698504 1.3073493 5.6861698 6.4817945 6.1405231 8.5505845 4.7719227 13.3496353

1.55 16.2504045 0.3983408 16.8337071 1.1934747 5.4266584 6.6117465 5.7177517 8.4545814 4.3910853 13.6251370

1.60 16.6540509 0.4415259 16.9535279 1.0764851 5.1650781 6.7576995 5.3252520 8.2981019

1.65 17.0412764 0.4882722 17.1376060 0.9246192 4.9004867 6.9470521 4.9636182 8.1251299

1.70 4.6299075 7.2668270 4.6350643 7.8027777

1.75

1.80

1.85

1.90

1.95

2.00

2.05 20.0073494 0.2900076

2.10 20.3091848 0.3442830

2.20 20.9133608 0.4048887

2.30 21.5020426 0.4417664

2.40 22.0717587 0.4682898

2.50 22.6224550 0.4907567

2.60 23.1548520 0.5126598

2.70 23.6698753 0.5277744

2.80 24.1683924 0.5458470

2.90 24.6513605 0.5573007

3.00 25.1194152 0.5686860

12

Table 3. The first five fold points of Eq. (22). The symbol “ ” means no fold.

1st 2nd 3rd 4th 5thp

λ ‖u‖∞ λ ‖u‖∞ λ ‖u‖∞ λ ‖u‖∞ λ ‖u‖∞1.0 26.34225439 2.16345920 13.26852968 6.71990312 16.59894207 9.18858228 14.24462710 13.2254108

1.1 17.38792981 0.105368194 25.75655627 2.04479064 12.49751673 6.80959692 14.81165568 9.23544480 12.87167965 13.3489922

1.2 19.17136906 0.178354104 25.30386237 1.94234699 11.70288989 6.96364123 13.18226912 9.25399558 11.50152022 13.5787858

1.3 21.00442162 0.271701158 24.99572658 1.82623200 10.87971610 7.07707236 11.70803881 9.28643809 10.15130043 13.8940639

1.4 22.46473897 0.325144383 24.86571059 1.65073492 10.03406523 7.33347971 10.38243283 9.04166599

1.5 23.79234972 0.478222225 24.96053880 1.43811824 9.16027034 7.67129928 9.21115027 8.69294194

1.6 25.01008248 0.519814929 25.33878301 1.17022160

1.7

1.8

1.9

2.0

2.1 30.41716328 0.390624117

2.2 31.29108667 0.458150632

2.3 32.13908678 0.499718116

2.4 32.95577543 0.531502468

2.5 33.73692200 0.556621089

2.6 34.49241231 0.578521454

2.7 35.21940989 0.598439309

2.8 35.91954710 0.614613791

2.9 36.58528786 0.608446128

3.0 37.22579584 0.672825737

13

0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

•

λ

||u|| ∞

p=1.0 p=1.5 p=3.0

Fig. 1: The solution curves of Eq. (20).

05

1015

11.5

22.5

30

2

4

6

8

10

12

14

λ

•••••••••••

••••

••••

•

p

||u|| ∞

the 1st fold

the 2nd fold

the 3rd fold

Fig. 2: The solution surface of Example 1.

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

x

u

Fig. 3: The contours u of Eq. (20) at p = 1.15 and λ = 4.0.

0 5 10 15 20 250

2

4

6

8

10

12

14

λ

||u|| ∞

p=2.05~3.0p=1.75~2.0p=1.0~1.70

Fig. 4: The solution curves of Eq. (21).

15

05

1015

2025

11.5

22.5

30

2

4

6

8

10

12

14

the 3rd foldthe 2nd foldthe 1st fold

λp

||u|| ∞

Fig. 5: The solution surface of Example 2.

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

λ

||u|| ∞

p=1.0 p=1.4 p=3.0

Fig. 6: The solution curves of Eq. (22).

16

0 5 10 15 20 25 30 35

11.5

22.5

30

2

4

6

8

10

12

14

λ

••••

••

••••

••••

••

p

||u|| ∞

the 1st fold

the 2nd fold

the 3rd fold

Fig. 7: The solution surface of Example 3.

17

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