In this section we repeat some well known results about exponential dichotomies for or-dinary differential equations which can be found in [42], [60], and some facts about the operator L defined in (1.46).
Definition A.5 (Exponential dichotomy) The linear differential operator
Lz=z0−M(·)z, x∈J ⊂R, M(·) :J →Rm,m (A.11) with solution operator S(x, ξ) has an exponential dichotomy (ED) in the interval J = [x−, x+], x± ∈ R∪ {±∞} with data (K, α, π) if there exist a bound K > 0, a rate α >0 and a function π:J 3x7→π(x),π(x) a projector, such that the following holds
S(x, ξ)π(ξ) =π(x)S(x, ξ) (A.12)
and the Green’s function
Gc(x, ξ) =
(S(x, ξ)π(ξ), x≥ξ
−S(x, ξ)(I−π(ξ)), x < ξ, satisfies an exponential estimate
kGc(x, ξ)k ≤Ke−α|x−ξ|, x, ξ∈J. (A.13)
140 Chapter A. Auxiliary results
IfJ = (−∞,0] then the kernel of the projectorπ(0) is given by N(π(0)) ={z0 ∈Rm : sup
x≤0kS(x,0)z0k <∞}
and for J = [0,∞) the image of π(0) is given by R(π(0)) ={z0 ∈Rm : sup
x≥0kS(x,0)z0k<∞},
(see [41], Section 2). If Lhas an exponential dichotomy on (−∞,0] and [0,∞) with data (K±, α±, π±), then the kernel ofL is given by
N(L) ={S(·,0)z0 : z0 ∈ N(π−(0))∩ R(π+(0))}.
Note that, ifLhas exponential dichotomies onR±with data (K±, α±, π±) then the adjoint operator
L∗:L2→ H2, z 7→z0+MT(·)z (A.14) also has exponential dichotomies on R± with projectorsI−π±T and
N(L) ={S(x,0)z0 : z0∈ R(π+)∩ N(π−)}. Thus for φ∈ N(L) we obtain the exponential estimate
kφ(x)k ≤Ke−α|x|, x∈R. (A.15) Note that,Gcbeing a Green’s function means that the solution of the linear inhomogeneous equation
Lz= ¯r, x∈J (A.16)
is given by z(x) =R
JG(x, ξ)r(ξ)dξ.
Thus if the operator L has exponential dichotomies onR± with data (K±, α±, π±), then solutions of (A.16) onJ =R± are given by
[¯s−(¯r)](x) = Z 0
−∞
G−c (x, ξ)¯r(ξ)dξ
= Z x
−∞
S(x, ξ)P−(ξ)¯r(ξ)dξ− Z 0
x
S(x, ξ)(I−P−(ξ))¯r(ξ)dξ [¯s+(¯r)](x) =
Z ∞
0
G+c (x, ξ)¯r(ξ)dξ
= Z x
0
S(x, ξ)P+(ξ)¯r(ξ)dξ− Z ∞
x
S(x, ξ)(I−P+(ξ))¯r(ξ)dξ.
(A.17)
Using the dichotomy estimates, these solutions can be estimated for ¯r∈ L2by (cf. Lemma 3.21 in [60])
ks¯±(¯r)kL2 +k[¯s±(¯r)](0)k ≤Ck¯rkL2. (A.18) In order to infer the existence of exponential dichotomies on R± for the operator L de-fined in (1.46) from the existence of exponential dichotomies for the constant coefficient operatorsL±= limx→∞ dxd −M(x), the following well known “Roughness Theorem” ([41], [3]) is used. It describes the persistence of exponential dichotomies under perturbations which decay forx→ ∞ to zero.
A.3. Exponential dichotomies for ordinary differential equations 141
Lemma A.6 Assume that the operatorLz=z0−M(·)zpossesses an exponential dichotomy on J = [x0,∞),x0 ∈R with data(K, α, π). Consider the perturbed operator
Lz˜ =z0−(M(·) + ∆(·))z
with∆∈ C(J,Rm,m)andk∆(x)k →0asx→ ∞. ThenL˜has an exponential dichotomy auf [x0,∞) with data ( ˜K,α,˜ π), and˜
k˜π(x)−π(x)k →0 as x→ ∞.
It has been shown in [3], Lemma 2.1, the existence of exponential dichotomies for L on R± follows from the hyperbolicity of the matrices M±= limx→±∞M(x).
Corollary A.7 Assume that the matrix M ∈ C(R,Rm,m) has limits M±= lim
x→±∞M(x),
which are hyperbolic. Let X±s and X±u be the stable and unstable invariant subspace of M±, respectively.
Then L has an exponential dichotomy on R− = (−∞,0] and R+ = [0,∞) with data (K±, α±, π±). The projectors π− and π+ satisfy
x→−∞lim (I−π−(x)) =E−u, lim
x→+∞π+(x) =E+s, where E−u denotes the projector onto X−u and E+s the projector onto X+s. If the number of stable and unstable eigenvalues of M± is m, then we have
dimN(π−(0)) = dimR(E−u) =m, dimR(π+(0)) = dimR(E+s) =m.
Moreover, the operatorL is a Fredholm operator of index k+s −k−s =k−u −k+u wherek±u resp. k±s denotes the number of unstable resp. stable eigenvalues of M±.
Instead of a single matrix function M(·) we often consider families of matrix functions M(·, s), in general the hyperbolicity of the matricesM±(s) = limx→∞M(x, s) is related to the characteristic equations (1.6) as follows. A solution (Y,Λ) ∈ Rm,p ×Rp,p of the quadratic eigenvalue problem
AYΛ2+BYΛ + (C−sI)Y = 0, A, B, C∈Rm,m. is related via linearization to the eigenvalue problem
M(s)W −WΛ for the matrix
M(s) =
0 I
−A−1(C−sI) −A−1B
∈R2m,2m (A.19)
via
W = Y
YΛ
.
142 Chapter A. Auxiliary results
Thus the spectral condition (SC) implies that the matricesM±(s) are hyperbolic for alls with Re (s)>−β withm stable andm unstable eigenvalues (cf. Lemma 3.30 in [60]).
It has been shown in [3], [60] that this implies that the operators
L(s)z=z0−M(·, s)z, with (A.20) M(x, s) =
0 I A−1(sI−C(x)) −A−1B(x)
possesses exponential dichotomies on both half linesR± if Re (s)>−β.
Note that Λ−sI and L(s) are strongly related. As has been proven in [50], the Jordan-block structures of Λ−sI and L(s) are the same, as well as the Fredholm properties.
In the following we fix the notation for the corresponding invariant subspaces and its projectors.
Definition A.8 We denote the (orthogonal) projector onto the stable subspace of M−(s) by Es−(s), i.e. R(E−s(s)) =R(W−s(s)), for
W−s(s) =
Y−s(s) Y−s(s)Λs−(s)
∈R2m,m where Y−s(s),Λs−(s) solve the quadratic eigenvalue problem
AYΛ2+B−YΛ + (C−−sI)Y = 0 and Reσ(Λs−(s))<0.
Similarly, we denote the projector onto the unstable subspace of M+(s) by E+u(s), i.e.
R(E+u(s)) =R(W+u(s)), for
W+u(s) =
Y+u(s) Y+u(s)Λu+(s)
where Y+u(s),Λu+(s) solve the quadratic eigenvalue problem AYΛ2+B+YΛ + (C+−sI)Y = 0 and Reσ(Λu+(s))>0.
In case s= 0, we omit the s dependency, e.g. write justY+u,Λs−.
143
Appendix B
Notation
D(P) domain of definition of the operatorP. N(P) nullspace or kernel ofP.
R(P) image or range ofP.
kPkX→Y norm of a bounded operator P :X→Y: kPk = sup
x∈D(P) x6=0
nkP xkY
kxkX
o.
σ(L), %(L) spectrum and resolvent of an operatorL
C(X, Y) bounded continuous operators from X toY with sup norm.
Ck(X, Y) k-times continuous differentiable operators fromX toY. LetK∈ {R,C}
Ck(R,Kn) k-times continuous differentiable functions fromR toKn. Cbk(R,Kn) functions, which possess continuous, bounded derivatives
f(j)= dxdjjf up to order kequipped with the norm kfkk,∞ =
k
X
j=0
kf(j)k∞=
k
X
j=0
sup
x∈Rkf(j)(x)k. Lp(R,Kn) Lebesgue integrable functions from RtoKn, with norm
kfkLp :=
Z
Rkf(x)kp dx 1
p
, 1≤p <∞.
hu, vi L2 inner product, hu, vi=R
Ru(x)Hv(x)dx
Hk(R,Kn) Sobolev space of functions f ∈ L2(R,Kn), which possess L2-integrable derivatives up to order kwith norm
kfkHk :=
k
X
j=0
kf(j)k2L2
1 2
=
Z
R k
X
j=0
kf(j)(x)k2 dx
1 2
.
144 Chapter B. Notation
u0 derivative of a functionu(x)
ux, ut partial derivatives of a functionu(x, t).
Bρ(x) closed ball of radiusρ aroundx∈X: Bρ(x) ={y∈X :kx−yk ≤ρ} J, Je, discrete intervals: J = [n−, n+], Je= [n−−1, n++ 1],
Jr, Jl Jr= [n−, n++ 1], Jl= [n−−1, n+]
GJ,h,x0 equidistant finite gridGJ,h,x0 ={xn: xn=x0+nh, n∈J}. SJ(Km) Banach space of sequences{zn}n∈J,zn∈Km with
kzk∞= supn∈Jkznk
δ+, δ−, δ0 finite difference operators: δ+ :SJr → SJ, δ− :SJl → SJ, δ0 :SJe → SJ
(δ+u)n= 1h(un+1−un), (δ−u)n= 1h(un−un−1), (δ0u)n= 2h1 (un+1−un−1)
Forz∈SJ:
k·k1,∞ kzk1,∞=kzk∞+kδ+zk∞ k·k2,∞ kzk2,∞=kzk1,∞+kδ+δ−zk∞
k·kL2,h discreteL2-norm forz∈SJ: kzkL2,h = (Pn+
n=n−hkznk2)12 k·kH1
h,k·k1,L2,h discreteH1-normkzk1,L2,h =kzkH1
h = (kzk2L2,h+kδ+zk2L2,h)12, k·kH2h,k·k2,L2,h discreteH2-normkzk2,L2,h =kzkHh2 = (kzk2H1h+kδ+δ−zk2L2,h)12 hu, vir,s hu, vir,s=Ps
n∈rh uHnvn
hu, vih L2,h inner product inSJ,J = [n−, n+]: hu, vih=hu, vin−,n+
Eρ functions which decay with its derivative, i.e. g∈ Eρ(J,Rm) if kg(x)(k)k ≤Ke−%|x|,k= 0,1 for some K >0
vec(u) u∈SJ(Rm) : vec(u) = (uTn
−, . . . , uTn+)∈Rm(n+−n−+1)
O, o Landau symbols
145
List of Figures
1.1 The sector Sω,ζ contained in the resolvent set . . . 10
1.2 Regions for resolvent estimates . . . 19
1.3 Path of integration for definition (1.32). . . 25
2.1 Overview over dichotomy estimates . . . 48
3.1 Regions for resolvent estimates . . . 70
4.1 Path of integration . . . 109
5.1 Nagumo, traveling front . . . 122
5.2 Nagumo wave, evolution of u(t) . . . 122
5.3 Frozen wave, evolution of the parameterµ(t) . . . 123
5.4 Nagumo, time evolution of|µ(t)−µ¯|and ku(t)−u¯|Jk∞ . . . 123
5.5 Approximation error, Dirichlet b.c. . . 124
5.6 Approximation error, Neumann b.c. . . 125
5.7 Eigenvalue near 0, approximation error, Dirichlet b.c. . . 126
5.8 Nagumo, spectra . . . 127
5.9 Nagumo, spectra . . . 127
5.10 QCGL, stable and unstable pulse . . . 128
5.11 QCGL, front . . . 129
5.12 QCGL, rotating vs. frozen pulse . . . 130
5.13 QCGL, rotating vs. frozen front . . . 130
5.14 QCGL, time evolution ofµr, µt . . . 130
5.15 QCGL, time evolution of|µ(t)−µ˜|and ku(t)−u˜k∞ . . . 131
5.16 QCGL, approximation error for the unstable pulse . . . 132
5.17 QCGL, approximation error for the double zero eigenvalue . . . 133
146 LIST OF FIGURES
5.18 QCGL, spectra . . . 135 5.19 QCGL, bordered system, zoom in, spectra . . . 136 5.20 QCGL, front, spectrum, zoom in with border . . . 136
147
Bibliography
[1] H. Airault. ´Equations asymptotiques pour des cas sp´eciaux de l’´equation de Nagumo.
C. R. Acad. Sci. Paris S´er. I Math., 301(6):295–298, 1985.
[2] J. Alexander, R. Gardner, and C. Jones. A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math., 410:167–212, 1990.
[3] W.-J. Beyn. The numerical computation of connecting orbits in dynamical systems.
IMA Journal of Numerical Analysis, 9:379–405, 1989.
[4] W.-J. Beyn and J.-M. Kleinkauf. The numerical computation of homoclinic orbits for maps. SIAM J. Numer. Anal., 34(3):1207–1236, 1997.
[5] W.-J. Beyn, W. Kleß, and V. Th¨ummler. Continuation of low-dimensional invariant subspaces in dynamical systems of large dimension. InErgodic theory, analysis, and efficient simulation of dynamical systems, pages 47–72. Springer, Berlin, 2001.
[6] W.-J. Beyn and J. Lorenz. Stability of traveling waves: dichotomies and eigenvalue conditions on finite intervals. Numer. Funct. Anal. Optim., 20(3-4):201–244, 1999.
[7] W.-J. Beyn and V. Th¨ummler. Freezing solutions of equivariant evolution equations.
SIAM Journal on Applied Dynamical Systems, 3(2):85–116, 2004.
[8] Z. X. Chen and B. Y. Guo. Analytic solutions of the Nagumo equation. IMA J. Appl.
Math., 48(2):107–115, 1992.
[9] P. Chossat and R. Lauterbach. Methods in equivariant bifurcations and dynamical systems, volume 15 of Advanced Series in Nonlinear Dynamics. World Scientific Publishing Co. Inc., River Edge, NJ, 2000.
[10] S.-N. Chow and W. X. Shen. Stability and bifurcation of traveling wave solutions in coupled map lattices. Dynam. Systems Appl., 4(1):1–25, 1995.
[11] E. C. M. Crooks. Stability of travelling-wave solutions for reaction-diffusion-convection systems. Topol. Methods Nonlinear Anal., 16(1):37–63, 2000.
[12] E. C. M. Crooks and J. F. Toland. Travelling waves for reaction-diffusion-convection systems. Topol. Methods Nonlinear Anal., 11(1):19–43, 1998.
[13] S. S. D. Horva´ath, V. Petrov and K. Showalter. Instabilities in propagating reaction–
diffusion fronts. J. Chem. Phys., 98(8):6332–6343, 1993.
148 BIBLIOGRAPHY
[14] P. de Mottoni and M. Schatzman. Asymptotics of the Thual-Fauve pulse. In Asymp-totic and numerical methods for partial differential equations with critical parameters (Beaune, 1992), volume 384 ofNATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 225–239. Kluwer Acad. Publ., Dordrecht, 1993.
[15] E. Doedel. AUTO: a program for the automatic bifurcation analysis of autonomous systems. InProceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. I (Winnipeg, Man., 1980), volume 30, pages 265–284, 1981.
[16] C. Engstler. Matlab implementation of the Radau IIA method of order 5, 1998.
[17] A. Favini and A. Yagi. Degenerate differential equations in Banach spaces, volume 215 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York, 1999.
[18] B. Fiedler, B. Sandstede, A. Scheel, and C. Wulff. Bifurcation from relative equilibria of noncompact group actions: skew products, meanders, and drifts.Doc. Math., 1:No.
20, 479–505 (electronic), 1996.
[19] R. FitzHugh. Impulses and physiological states in theoretical models of nerve mem-brane. Biophysical Journal, 1:445–466, 1961.
[20] G. H. Golub and C. F. Van Loan. Matrix computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, third edition, 1996.
[21] E. Griepentrog and R. M¨arz. Differential-algebraic equations and their numerical treatment, volume 88 ofTeubner-Texte zur Mathematik [Teubner Texts in Mathemat-ics]. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1986.
[22] E. Hairer and G. Wanner. Solving ordinary differential equations. II, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1991. Stiff and differential-algebraic problems.
[23] D. Henry. Geometric theory of semilinear parabolic equations, volume 840 of Lecture notes in mathematics. Springer-Verlag, Berlin, Heidelberg, New York, 1981.
[24] A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Pysiology, 117:500–
544, 1952.
[25] S. A. J. Nagumo and S. Yoshizawa. An active pulse transmission line simulating nerve axon. InProceedings of the IRE, volume 50, pages 2061–2070, 1962.
[26] J. M. Kleinkauf. Numerische Analyse tangentialer homokliner Orbits. PhD thesis, Bielefeld, 1997.
[27] W. Kleß. Numerische Approximation lokal invarianter Mannigfaltigkeiten in großen dynamischen Systemen. PhD thesis, University of Bielefeld, 1997.
[28] G. Kreiss, H.-O. Kreiss, and A. Petersson. On the convergence to steady state of solu-tions of nonlinear hyperbolic-parabolic systems. SIAM Journal Numerical Analysis, 31:1577–1604, 1994.
BIBLIOGRAPHY 149
[29] H.-O. Kreiss and J. Lorenz. Initial boundary value problems and the Navier-Stokes equations. Academic press, 1989.
[30] J. Krishnan, K. Engelborghs, M. B¨ar, K. Lust, D. Roose, and I. G. Kevrekidis. A computer-assisted study of pulse dynamics in anisotropic media. Phys. D, 154(1-2):85–110, 2001.
[31] R. B. Lehoucq, D. C. Sorensen, and C. Yang. ARPACK users’ guide. Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods.
[32] P. Marcq, H. Chat´e, and R. Conte. Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation. Phys. D, 73(4):305–317, 1994.
[33] W. S. Martinson and P. I. Barton. A differentiation index for partial differential-algebraic equations. SIAM J. Sci. Comput., 21(6):2295–2315 (electronic), 2000.
[34] R. M¨arz. Differential algebraic systems anew. Appl. Numer. Math., 42(1-3):315–335, 2002. Ninth Seminar on Numerical Solution of Differential and Differential-Algebraic Equations (Halle, 2000).
[35] M. Miklavˇciˇc. Stability for semilinear parabolic equations with noninvertible linear operator. Pacific J. Math., 118(1):199–214, 1985.
[36] M. Miklavˇciˇc. Applied functional analysis and partial differential equations. World Scientific Publishing Co. Inc., River Edge, NJ, 1998.
[37] A. R. Mitchell. Numerical studies of travelling waves in nonlinear diffusion equations.
Bull. Inst. Math. Appl., 17(1):14–20, 1981.
[38] R. M. Miura. Accurate computation of the stable solitary waves for the FitzHugh-Nagumo equations. Journal of Mathematical Biology, 13:247–269, 1982.
[39] J. D. Murray. Mathematical biology, volume 19 ofBiomathematics. Springer-Verlag, Berlin, second edition, 1993.
[40] K. Palmer. Shadowing in dynamical systems, volume 501 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 2000. Theory and applications.
[41] K. J. Palmer. Exponential dichotomies and transversal homoclinic points. Journal of Differential Equations, 55:225–256, 1984.
[42] K. J. Palmer. Exponential dichotomies, the shadowing lemma and transversal ho-moclinic points. In Dynamics Reported, volume 1, pages 265–306. B. G. Teubner, Stuttgart, 1988.
[43] A. Pazy. Semigroups of linear operators and applications to partial differential equa-tions, volume 44 ofApplied Mathematical Sciences. Springer-Verlag, New York, 1983.
[44] S. Popp, O. Stiller, I. Aronson, A. Weber, and L. Kramer. Localized hole solutions and spatio-temporal chaos in the 1d complex Ginzburg-Landau equation. Physical review letters, 70, 1993.
150 BIBLIOGRAPHY
[45] P. J. Rabier and W. C. Rheinboldt. Theoretical and numerical analysis of differential-algebraic equations. In Handbook of numerical analysis, Vol. VIII, Handb. Numer.
Anal., VIII, pages 183–540. North-Holland, Amsterdam, 2002.
[46] M. Renardy and R. C. Rogers. An introduction to partial differential equations.
Springer-Verlag, Berlin, Heidelberg, New York, 1993.
[47] C. W. Rowley, I. G. Kevrekidis, J. E. Marsden, and K. Lust. Reduction and recon-struction for self-similar dynamical systems. Nonlinearity, 16:1257–1275, 2003.
[48] B. Sandstede. Convergence estimates for the numerical approximation of homoclinic solutions. IMA J. Numer. Anal., 17(3):437–462, 1997.
[49] B. Sandstede. Stability of travelling waves. In Handbook of dynamical systems, Vol.
2, pages 983–1055. North-Holland, Amsterdam, 2002.
[50] B. Sandstede and A. Scheel. Absolute and convective instabilities of waves on un-bounded and large un-bounded domains. Phys. D, 145(3-4):233–277, 2000.
[51] B. Sandstede and A. Scheel. Absolute and convective stability in large domains. In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pages 797–800. World Sci. Publishing, River Edge, NJ, 2000.
[52] B. Sandstede, A. Scheel, and C. Wulff. Dynamics of spiral waves on unbounded domains using center-manifold reductions. J. Differential Equations, 141(1):122–149, 1997.
[53] B. Sandstede, A. Scheel, and C. Wulff. Dynamical behavior of patterns with Euclidean symmetry. In Pattern formation in continuous and coupled systems (Minneapolis, MN, 1998), volume 115 of IMA Vol. Math. Appl., pages 249–264. Springer, New York, 1999.
[54] L. Shampine, M. W. Reichelt, and J. A. Kierzenka. Solving index-1 DAEs in MAT-LAB and Simulink. SIAM Rev., 41(3):538–552 (electronic), 1999.
[55] L. F. Shampine and M. W. Reichelt. The MATLAB ODE suite. SIAM J. Sci.
Comput., 18(1):1–22, 1997. Dedicated to C. William Gear on the occasion of his 60th birthday.
[56] T. Stykel. Analysis and numerical solution of generalized Lyapunov equations. PhD thesis, Technische Universit¨at Berlin, 2002.
[57] J. Swinton. The stability of homoclinic pulses: a generalisation of Evans’s method.
Phys. Lett. A, 163(1-2):57–62, 1992.
[58] O. Thual and S. Fauve. Localized structures generated by subcritical instabilities. J.
Phys. France, 49:1829–1833, 1988.
[59] O. Thual and S. Fauve. Solitary waves generated by subcritical instabilities in dissi-pative systems. Phys. Rev. Lett., 64(3):282–284, 1990.
[60] V. Th¨ummler. Numerische Stabilit¨atskriterien f¨ur wandernde Wellen. Master’s thesis, University of Bielefeld, 1998.