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1.

図書
図書
1. Dynamical systems and nonlinear waves in plasmas. 1st ed (: hbk)
Asit Saha, Santo Banerjee
出版情報: Boca Raton : CRC Press, Taylor & Francis Group, 2021  x, 207 p. ; 24 cm
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Dynamical systems
Waves in plasmas
Bifurcation of small amplitude waves in plasmas
Bifurcation of arbitrary amplitude waves in plasmas
Bifurcation analysis of supernonlinear waves
Chaos, multistability and stable oscillation in plasmas
Dynamical systems
Waves in plasmas
Bifurcation of small amplitude waves in plasmas
2.

図書
図書
2. Lyapunov-Schmidt methods in nonlinear analysis and applications
by Nikolay Sidorov ... [et al.]
出版情報: Dordrecht ; London : Kluwer Academic Publishers, c2002  xx, 548 p. ; 25 cm
シリーズ名: Mathematics and its applications ; v. 550
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Preface
On Regularization of Linear Equations on the Basis of Perturbation Theory / 1.：
Generalised Jordan chains, sets and root numbers of linear operators
Regularization of linear equations with Fredholm operators / 2.：
Principal theorem on regularization of linear equations by the perturbation method / 3.：
Regularization of linear equations on the basis of perturbation theory in Hilbert spaces / 4.：
Computation of eigenvalues and eigenvectors of linear operators by pseudo-perturbation method / 5.：
Notes / 6.：
Investigation of Bifurcation Points of a Nonlinear Equations
Lyapunov-Schmidt BEq in the problem of a bifurcation point
General existence theorems for the bifurcation points
Construction of asymptotics in a neighborhood of a bifurcation point
Asymptotic bifurcation points
On perturbation of the branch points of nonlinear equations
Notes and generalization
Regularization of Computation of Solutions in a Neighborhood of the Branch Point
Construction of the regularizing equation in the problem at a branch point
Definition and properties of simple solutions
Regularization of calculations of simple solutions of nonlinear equations
Regularization of method for continuation along parameter in a neighborhood of a branch point
Notes and remarks
Iterations, Interlaced Equations and Lyapunov Convex Majorants in Nonlinear Analysis
Iterations and uniformization of branching solutions in nonlinear analysis
BEq and the selection of the initial approximation / 1.1：
On the role of supporting lines and Newton diagrams in the construction of the initial approximation / 1.2：
A one-step iteration method / 1.3：
An N-step iteration method / 1.4：
On regularization in the sense of Tikhonov, modifications and possible generalizations of an N-step method / 1.5：
Remarks / 1.6：
Interlaced and potential BEq
The property of (S, K)-interlacing of an equation and its inheritance by the BEq / 2.1：
(T, M)-interlaced and (T[superscript 2], M)-interlaced BEq / 2.2：
[alpha]-parametric interlaced BEq / 2.3：
Interlaced BEq of potential type / 2.4：
Surface bundle of a domain of free parameters / 2.5：
Parametrization of solutions and the method of successive approximations / 2.6：
On the role of Lyapunov convex majorants in the nonlocal existence theorems of implicit functions
Majorants independent of parameters / 3.1：
Majorants depending on a parameter / 3.2：
Investigation of the existence domain of the solution of equation F(u, [varepsilon]) = 0. / 3.3：
Methods of Representation Theory and Group Analysis in Bifurcation Theory
Nonlinear equations invariant under transformation groups
Lyapounov-Schmidt BEq and some methods of their reduction
Some applications
Hereditary symmetry of branching equations and resolving systems
Invariance properties of BEq
Resolving systems for differential equations with Fredholm operator at the derivative and their symmetry
On the Grobman-Hartman theorem for equations with degenerate operator at the derivative
Construction and investigation of the branching equation by group analysis methods
BEq of solutions invariant relative to subgroups of the original equation group symmetry
Potential BEq
Direct methods of BEq group invariance usage for its general form construction by admitted group symmetry
Applications of Lie--Ovsyannikov theorem about invariant manifolds for the construction of BEq general form by allowing group symmetry / 4.1：
Non-linearly perturbed Helmholtz equations
Domain symmetry and bifurcational solutions asymptotics / 5.1：
Periodic solutions / 5.2：
Capillary--gravity waves in fluid layers
Capillary--gravity waves in a floating fluid spatial layer / 6.1：
Capillary--gravity waves at the interface of two fluids flow / 6.2：
Capillary--gravity waves on a cylindrical surface / 6.3：
Ferrofluid layer in a magnetic field / 6.4：
Fluid phase state crystallization problem in statistical crystal theory / 7.：
The statement of the problem / 7.1：
Subspaces N(B[subscript s]). Their expansions on irreducible subspaces relative to O[subscript h] / 7.2：
The BEq construction / 7.3：
Asymptotics of small solutions families for n[subscript s] = 1, 3 / 7.4：
Solutions invariant relative to normal divisors O[subscript h] / 7.5：
Andronov--Hopf bifurcation under group symmetry conditions / 8.：
BEq derivation in non-stationary bifurcation / 8.1：
Symmetry inheritance theorem / 8.2：
BEq construction by group analysis methods / 8.3：
On the asymptotics of small solutions / 8.4：
Stability of the bifurcating solutions / 9.：
Singular Differential Equations in Banach Spaces
Fundamental operator functions
Generalized functions in Banach spaces
Fundamental operator functions of singular differential operators
Fundamental operator functions of singular integral and integro-differential operators
The initial value problem for a differential equation having a Noetherian operator at the derivative. Periodic solutions and the property of convergence
Auxiliary information on Jordan sets of Noetherian operators
The initial value problem for a linear differential equation
The initial value problem for a nonlinear differential equation
Integral pseudo-solutions
Non-stationary differential equations with singularities
The initial value problem for a non-stationary linear equation and systems of 1st kind integral Volterra equations
The initial value problem for a non-stationary linear differential equation and a system of integral Volterra equations with a singularity
Branching differential equations of the initial value problem with singularity
The initial value problem for a nonlinear differential equation and equations with a singular point / 3.4：
Partial differential equations with the Fredholm operator in the main part
The theory of semigroups and groups of operators with kernels
Relative resolvents. Relatively adjoint elements
Relatively spectrally bounded operators and analytical groups of operators with kernells
Relatively sectorial operators and analytical semigroups of operators with kernels / 5.3：
Steady-State Solutions of the Vlasov--Maxwell System
Introduction
Stationary solutions of a VM system
The reduction of VM system to the resolving elliptic system (2.28), (2.29)
The reduction of the resolving system to the unique resolving equation
Existence of solutions of the boundary value problem (2.40)-(2.42)
Applications of the reduction theorems
Normalized solutions for a one-component distribution function
Non-stationary problem
Bifurcation points and nontrivial solutions of the stationary VM system
Statement of the boundary value problem and the problem at a bifurcation point / 4.2：
Resolution of the bifurcation equation / 4.3：
The existence theorem for bifurcation points and the construction of asymptotic solutions / 4.4：
Appendices
Positive solutions of the nonlinear singular boundary value problem of magnetic insulation / A：
References
Preface
On Regularization of Linear Equations on the Basis of Perturbation Theory / 1.：
Generalised Jordan chains, sets and root numbers of linear operators
3.

図書
図書
3. From equilibrium to chaos : practical bifurcation and stability analysis
Rüdiger Seydel
出版情報: New York : Elsevier, c1988  xv, 367 p. ; 24 cm
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4.

図書
図書
4. Nonlinear analysis
editor, Th.M. Rassias
出版情報: Singapore : World Scientific, c1987  xi, 557 p. ; 23 cm
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5.

図書
図書
5. Applications of centre manifold theory (: U.S : [softcover] ; : Germany)
Jack Carr
出版情報: New York : Springer-Verlag, c1981  142 p. ; 25 cm
シリーズ名: Applied mathematical sciences ; v. 35
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6.

図書
図書
6. Singularities and groups in bifurcation theory (v. 1 : us - v. 2 : gw)
Martin Golubitsky, David G. Schaeffer
出版情報: New York ; Tokyo : Springer-Verlag, c1985-c1988  2 v. ; 25 cm
シリーズ名: Applied mathematical sciences ; v. 51, 69
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Preface
of Vol. I
Introduction
Group Theoretic Preliminaries
Symmetry-Breaking in Steady-State Bifurcation
Case Study 4: The Planar BFnard Problem
Equivariant Normal Forms
Equivariant Unfolding Theory
Case Study 5: The Traction Problem for Mooney-Rivlin Material
Symmetry-Breaking in Hopf Bifurcation
Hopf Bifurcation with 0(2) Symmetry
Further Examples of Hopf Bifurcation with Symmetry
Mode Interactions
Mode Interactions with 0(2) Symmetry
Case Study 6: The Taylor-Couette System
Bibliography
Index
Preface
of Vol. I
Introduction
7.

図書
図書
7. Elementary stability and bifurcation theory (: New York ; : Berlin)
Gérard Iooss, Daniel D. Joseph
出版情報: New York : Springer-Verlag, c1980  xv, 286 p. ; 24 cm
シリーズ名: Undergraduate texts in mathematics
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Asymptotic Solutions of Evolution Problems
Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension
Imperfection Theory and Isolated Solutions Which Perturb Bifurcation
Stability of Steady Solutions of Evolution Equations in Two Dimensions and n Dimensions. Appendices
Bifurcation of Steady Solution in Two Dimensions and the Stability of the Bifurcating Solutions. Appendix
Methods of Projection for General Problems of Bifurcation into Steady Solutions
Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions
Bifurcation of Periodic Solutions in the General Case
Subharmonic Bifurcation of Forced T-Periodic Solutions
Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions. Appendices
Secondary Subharmonic and Symptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf's Type) in the Autonomous Case
Stability and Bifurcation in Conservative Systems
Asymptotic Solutions of Evolution Problems
Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension
Imperfection Theory and Isolated Solutions Which Perturb Bifurcation
8.

図書
図書
8. The Hamiltonian Hopf bifurcation (: gw ; : us)
Jan-Cees van der Meer
出版情報: Berlin ; New York : Springer-Verlag, c1985  vi, 115 p. ; 25 cm
シリーズ名: Lecture notes in mathematics ; 1160
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9.

図書
図書
9. Numerical methods for bifurcation problems and large-scale dynamical systems
Eusebius Doedel, Laurette S. Tuckerman, editors
出版情報: New York : Springer, c2000  x, 471 p. ; 25 cm
シリーズ名: The IMA volumes in mathematics and its applications ; v. 119
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10.

図書
図書
10. Numerical bifurcation analysis for reaction-diffusion equations
Zhen Mei
出版情報: Berlin : Springer, c2000  xiv, 414 p. ; 24 cm
シリーズ名: Springer series in computational mathematics ; 28
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Reaction-Diffusion Equations / 1：
Introduction / 1.1：
Bifurcations and Pattern Formations / 1.2：
Boundary Conditions / 1.3：
Continuation Methods / 2：
Parameterization of Solution Curves / 2.1：
Natural parameterization / 2.1.1：
Parameterization with arclength / 2.1.2：
Parameterization with pseudo-arclength / 2.1.3：
Local Parameterization of Solution Manifolds / 2.2：
Predictor-Corrector Methods / 2.3：
Euler-Newton method / 2.3.1：
A continuation-Lanczos algorithm / 2.3.2：
A continuation-Arnoldi algorithm / 2.3.3：
Computation of Multi-Dimensional Solution Manifolds / 2.4：
Detecting and Computing Bifurcation Points / 3：
Generic Bifurcation Points / 3.1：
One-parameter problems / 3.1.1：
Two-parameter problems / 3.1.2：
Test Functions / 3.2：
Test functions for turning points / 3.2.1：
Test functions for simple bifurcation point / 3.2.2：
Test functions for Hopf bifurcations / 3.2.3：
Minimally extended systems / 3.2.4：
Computing Simple Bifurcation Points / 3.3：
Simple bifurcation points / 3.3.1：
Extended systems / 3.3.2：
Newton-like methods / 3.3.3：
Rank-1 corrections for sparse problems / 3.3.4：
A numerical example / 3.3.5：
Computing Hopf Bifurcation Points / 3.4：
Hopf points / 3.4.1：
Newton method for extended systems / 3.4.2：
Branch Switching at Simple Bifurcation Points / 4：
Structure of Bifurcating Solution Branches / 4.1：
Behavior of the Linearized Operator / 4.2：
Euler-Newton Continuation / 4.3：
Branch Switching via Regularized Systems / 4.4：
Other Branch Switching Techniques / 4.5：
Bifurcation Problems with Symmetry / 5：
Basic Group Concepts / 5.1：
Equivariant Bifurcation Problems / 5.2：
Equivariant Branching Lemma / 5.3：
A Semi-linear Elliptic PDE on the Unite Square / 5.4：
Liapunov-Schmidt Method / 6：
Liapunov-Schmidt Reduction / 6.1：
Equivariance of the Reduced Bifurcation Equations / 6.2：
Derivatives and Taylor Expansion / 6.3：
Equivalence, Determinacy and Stability / 6.4：
Simple Bifurcation Points / 6.5：
Truncated Liapunov-Schmidt Method / 6.6：
Branch Switching at Multiple Bifurcation Points / 6.7：
Branch switching with prescribed tangents / 6.7.1：
Branch switching with scaling techniques / 6.7.2：
Corank-2 Problems with Dm-symmetry / 6.8：
Semilinear elliptic PDEs on a square / 6.8.1：
A semilinear elliptic PDE on a hexagon / 6.8.2：
Center Manifold Theory / 7：
Center Manifolds and Their Properties / 7.1：
Approximation of Center Manifolds / 7.2：
Symmetry and Normal Form / 7.3：
Hopf bifurcations / 7.4.1：
Waves in Reaction-Diffusion Equations / 7.5：
Oscillating waves / 7.5.1：
Long waves / 7.5.2：
Long time and large spatial behavior / 7.5.3：
A Bifurcation Function for Homoclinic Orbits / 8：
A Bifurcation Function / 8.1：
Approximation of Homoclinic Orbits / 8.2：
Solving the Adjoint Variational Problem / 8.3：
Preserving the inner product / 8.3.1：
Systems with continuous symmetries / 8.3.2：
The Approximate Bifurcation Function / 8.4：
Examples / 8.5：
Freire et al.'s circuit / 8.5.1：
Kuramoto-Sivashinsky equation / 8.5.2：
One-Dimensional Reaction-Diffusion Equations / 9：
Linear Stability Analysis / 9.1：
The general system / 9.2.1：
The Brusselator equations / 9.2.2：
Solution Branches at Double Bifurcations / 9.3：
The reflection symmetry and its induced action / 9.3.1：
(k,m) = (odd, odd) or (odd, even) / 9.3.2：
(k,m) = (even, even) / 9.3.3：
Central Difference Approximations / 9.3.4：
General systems / 9.4.1：
Numerical Results for the Brusselator Equations / 9.4.2：
The length <$>\ell = 1<$>, diffusion rates d1 = 1, d2 = 2 / 9.5.1：
The length <$>\ell = 10<$>, diffusion rates d1 = 1, d2 = 2 / 9.5.2：
Reaction-Diffusion Equations on a Square / 10：
D4-Symmetry / 10.1：
Eigenpairs of the Laplacian / 10.2：
Bifurcation Points / 10.3：
Steady state bifurcation points / 10.4.1：
Hopf bifurcation points / 10.4.2：
Mode Interactions / 10.5：
Steady/steady state mode interactions / 10.5.1：
Hopf/steady state mode interactions / 10.5.2：
Hopf/Hopf mode interactions / 10.5.3：
Kernels of Du G0 and <$>(D_u G_0)^{\ast}<$> / 10.6：
Simple and Double Bifurcations / 10.7：
Simple bifurcations / 10.8.1：
Double bifurcations induced by the D4 symmetries / 10.8.2：
Normal Forms for Hopf Bifurcations / 11：
Domain Symmetries and Their Extensions / 11.1：
Actions of D4 on the Center Eigenspace / 11.3：
The Normal Form / 11.4：
Analysis of the Normal Form / 11.5：
Odd parity / 11.5.1：
Even parity / 11.5.2：
Brusselator Equations / 11.6：
Linear stability analysis / 11.6.1：
Bifurcation scenario / 11.6.2：
Nonlinear degeneracy / 11.6.3：
Steady/Steady State Mode Interactions / 12：
Induced Actions / 12.1：
Interaction of Two D4-Modes / 12.2：
Interaction of two even modes / 12.2.1：
Interaction of an even mode with an odd mode / 12.2.2：
Interaction of two odd modes / 12.2.3：
Mode Interactions of Three Modes / 12.3：
Induced actions / 12.3.1：
Interactions of the modes (m,n,k) =(even, odd, odd) / 12.3.2：
Interactions of the modes (m,n,k) =(even, odd, even) / 12.3.3：
Interactions of Four Modes / 12.4：
Interactions of the modes (m, n, k, l) = (even, odd, even, odd) / 12.4.1：
Interactions of the modes (m, n, k, l) = (even, even, even, odd) / 12.4.2：
Reactions with Z2-Symmetry / 12.5：
Hopf/Steady State Mode Interactions / 13：
Normal Forms / 13.1：
Bifurcation Scenario / 13.4：
Calculations of the Normal Form / 13.5：
Homotopy of Boundary Conditions / 14：
Homotopy of boundary conditions / 14.1：
Boundary conditions for different components / 14.1.2：
Mixed boundary conditions along the sides / 14.1.3：
Dynamical boundary conditions / 14.1.4：
A Brief Review of Sturm-Liouville Theory / 14.2：
Laplacian with Robin Boundary Conditions / 14.3：
Variational Form / 14.4：
Continuity of Solutions along the Homotopy / 14.5：
Neumann and Dirichlet Problems / 14.6：
Properties of Eigenvalues / 14.7：
One-dimensional problems / 14.7.1：
Two-dimensional problems / 14.7.2：
Bifurcations along a Homotopy of BCs / 15：
Stability and Symmetries / 15.1：
Variations of Bifurcations along the Homotopy / 15.3：
(κ1, κ2) = (odd, even) or (even, odd) / 15.4.1：
(κ1, κ2) = (odd, odd) / 15.4.2：
(κ1, κ2) = (even, even) / 15.4.3：
A Numerical Example / 15.5：
Discretization with finite difference methods / 15.5.1：
Homotopy of (κ1(μ), κ2(μ)) from (1,2) to (2,3) / 15.5.2：
Homotopy of (κl(μ), κ2(μ)) from (1,3) to (2,4) / 15.5.3：
Homotopy of (κ1(μ), κ2(μ)) from (2,4) to (3,5) / 15.5.4：
Forced Symmetry-Breaking in BCs / 15.6：
Bifurcation points / 15.6.1：
Bifurcation scenarios / 15.6.2：
A Mode Interaction on a Homotopy of BCs / 16：
Symmetries and Normal Forms / 16.1：
Generic Bifurcation Behavior / 16.3：
Solutions with the modes φ1, φ2 / 16.3.1：
Pure φ3-mode solutions / 16.3.2：
Interactions of three modes / 16.3.3：
Scales of Solution Branches / 16.4：
Secondary Bifurcations / 16.5：
Secondary Hopf bifurcations / 16.5.1：
Truncated Bifurcation Equations / 16.6：
Derivatives with respect to homotopy parameter / 16.6.1：
Reduced Stability / 16.7：
Stability of solution branches at (0, λ1(μ),μ) / 16.7.1：
Stability of solution branches at (0, λ2(μ), μ) / 16.7.2：
Stability of solution branches at mode interaction / 16.7.3：
Solution branches along (0; λ1(μ),μ) / 16.8：
Solution branches along (0, λ2(μ),μ) / 16.8.2：
Mode interaction / 16.8.3：
Switching and continuation of solution branches / 16.8.4：
List of Figures
List of Tables
Bibliography
Index
Reaction-Diffusion Equations / 1：
Introduction / 1.1：
Bifurcations and Pattern Formations / 1.2：
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