Ordinary Differential Equations with Applications

Introduction to Ordinary Differential Equations * Linear Systems and Stability of Nonlinear Systems * Applications * Hyperbolic Theory * Continuation of Periodic Solutions * Homoclinic Orbits, Melnikov?s Method, and Chaos * Averaging * Local Bifurcation * References * Index.

[1]  Laura Hamilton Waxman.  The Solar System , 1885, Nature.

[2]  A. Liapounoff,et al.  Problème général de la stabilité du mouvement , 1907 .

[3]  Balth van der Pol Jun Docts. Sc.,et al.  LXXII. The heartbeat considered as a relaxation oscillation, and an electrical model of the heart , 1928 .

[4]  G. A. Watson.  A treatise on the theory of Bessel functions , 1944 .

[5]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[6]  N. N. Bautin,et al.  On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type , 1954 .

[7]  S. Ulam,et al.  Studies of nonlinear problems i , 1955 .

[8]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[9]  H. Trotter.  On the product of semi-groups of operators , 1959 .

[10]  P. Hartman.  A lemma in the theory of structural stability of differential equations , 1960 .

[11]  W. A. Coppel.  On a differential equation of boundary-layer theory , 1960, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[12]  J. Breakwell,et al.  Satellite orbits about a planet with rotational symmetry , 1960 .

[13]  H. Flanders.  Differential Forms with Applications to the Physical Sciences , 1964 .

[14]  S. Smale.  Diffeomorphisms with Many Periodic Points , 1965 .

[15]  Jorge Sotomayor,et al.  Generic one-parameter families of vector fields on two-dimensional manifolds , 1968 .

[16]  J. Moser,et al.  On a theorem of Anosov , 1969 .

[17]  M. Hirsch,et al.  Stable manifolds for hyperbolic sets , 1969 .

[18]  E. Velasco,et al.  Generic properties of polynomial vector fields at infinity , 1969 .

[19]  H. Weitzner,et al.  Perturbation Methods in Applied Mathematics , 1969 .

[20]  J. E. Drummond.  An existence theorem for differential equations , 1970, Bulletin of the Australian Mathematical Society.

[21]  Paul F. Byrd,et al.  Handbook of elliptic integrals for engineers and scientists , 1971 .

[22]  A. Andronov.  Theory of bifurcations of dynamic systems on a plane = Teoriya bifurkatsii dinamicheskikh sistem na ploskosti , 1971 .

[23]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[24]  J. Robbin,et al.  Topological conjugacy and structural stability for discrete dynamical systems , 1972 .

[25]  A. Andronov,et al.  Qualitative Theory of Second-order Dynamic Systems , 1973 .

[26]  V. Arnold,et al.  Ordinary Differential Equations , 1973 .

[27]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[28]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[29]  Martin Goland,et al.  Principles of aeroelasticity , 1975 .

[30]  G. Sell,et al.  The Hopf Bifurcation and Its Applications , 1976 .

[31]  William S. Massey,et al.  Algebraic Topology: An Introduction , 1977 .

[32]  Freddy Dumortier,et al.  Singularities of vector fields on the plane , 1977 .

[33]  John Milnor,et al.  Analytic Proofs of the “Hairy Ball Theorem” and the Brouwer Fixed Point Theorem , 1978 .

[34]  Vimal Singh,et al.  Perturbation methods , 1991 .

[35]  Frank C. Hoppensteadt,et al.  Dynamics of the Josephson junction , 1978 .

[36]  D. Aronson,et al.  Multidimensional nonlinear di u-sion arising in population genetics , 1978 .

[37]  D. Jordan,et al.  Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers , 1979 .

[38]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[39]  J. Tinsley Oden,et al.  Applied functional analysis : a first course for students of mechanics and engineering science , 1979 .

[40]  Stephen Smale.  The Mathematics of Time: Essays on Dynamical Systems, Economic Processes, and Related Topics , 1980 .

[41]  C. Robinson,et al.  Qualitative dynamics from asymptotic expansions: Local theory , 1980 .

[42]  J. Roblin.  Algebraic Kupka-Smale theory , 1981 .

[43]  Jerrold E. Marsden,et al.  A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam , 1981 .

[44]  A. Lichtenberg,et al.  Regular and Stochastic Motion , 1982 .

[45]  James F. Geer,et al.  Power Series Expansions for the Frequency and Period of the Limit Cycle of the Van Der Pol Equation , 1982 .

[46]  Carmen Chicone,et al.  On General Properties of Quadratic Systems , 1982 .

[47]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[48]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[49]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[50]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[51]  W. M. Oliva,et al.  An Introduction to Infinite Dimensional Dynamical Systems - Geometric Theory , 1983 .

[52]  Philip Holmes,et al.  Repeated Resonance and Homoclinic Bifurcation in a Periodically Forced Family of Oscillators , 1984 .

[53]  Glen R. Hall,et al.  Resonance Zones in Two-Parameter Families of Circle Homeomorphisms , 1984 .

[54]  J. Palis,et al.  Geometric theory of dynamical systems : an introduction , 1984 .

[55]  James F. Geer,et al.  Perturbation analysis of the limit cycle of the free van der Pol equation , 1984 .

[56]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[57]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[58]  Carmen Chicone,et al.  On a class of complete polynomial vector fields in the plane , 1986 .

[59]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[60]  Carmen Chicone,et al.  Limit cycles of a class of polynomial vector fields in the plane , 1986 .

[61]  V. Arnold,et al.  Dynamical Systems III , 1987 .

[62]  S. Childress,et al.  Topics in geophysical fluid dynamics. Atmospheric dynamics, dynamo theory, and climate dynamics. , 1987 .

[63]  J. Murdock.  Perturbations: Theory and Methods , 1987 .

[64]  Carmen Chicone,et al.  The monotonicity of the period function for planar Hamiltonian vector fields , 1987 .

[65]  C. Meunier,et al.  Multiphase Averaging for Classical Systems , 1988 .

[66]  Vladimir I. Arnold,et al.  Dynamical Systems I , 1988 .

[67]  James H. Davenport,et al.  Computer Algebra: Systems and Algorithms for Algebraic Computation , 1988 .

[68]  Mark Levi,et al.  Stability of the Inverted Pendulum—A Topological Explanation , 1988 .

[69]  S. Hastings,et al.  Oscillating solutions of the Falkner-Skan equation for positive β , 1988 .

[70]  R. Abraham,et al.  Manifolds, tensor analysis, and applications: 2nd edition , 1988 .

[71]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[72]  Stephen Wiggins,et al.  Global Bifurcations and Chaos , 1988 .

[73]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[74]  H. Friedrich,et al.  The hydrogen atom in a uniform magnetic field — An example of chaos , 1989 .

[75]  Carmen Chicone,et al.  Bifurcation of critical periods for plane vector fields , 1989 .

[76]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[77]  Simultaneous equilibrium and heteroclinic bifurcation of planar vector fields via the Melnikov integral , 1990 .

[78]  Jorge V. José,et al.  Chaos in classical and quantum mechanics , 1990 .

[79]  J. Vickers.  MANIFOLDS, TENSOR ANALYSIS AND APPLICATIONS (2nd edition) (Applied Mathematical Sciences 75) , 1990 .

[80]  V. M. Tikhomirov,et al.  A Study of the Diffusion Equation with Increase in the Amount of Substance, and its Application to a Biological Problem , 1991 .

[81]  Carmen Chicone,et al.  The twisting tennis racket , 1991 .

[82]  Carmen Chicone,et al.  Bifurcation of Limit Cycles from Quadratic Isochrones , 1991 .

[83]  J. R. E. O’Malley.  Singular perturbation methods for ordinary differential equations , 1991 .

[84]  Jack K. Hale,et al.  Structural stability for time-periodic one-dimensional parabolic equations , 1992 .

[85]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[86]  P. Colwell.  Bessel functions and Kepler's equation , 1992 .

[87]  Carmen Chicone,et al.  Bifurcations of nonlinear oscillations and frequency entrainment near resonance , 1992 .

[88]  O. K. Yoon,et al.  Introduction to differentiable manifolds , 1993 .

[89]  S. Friedlander,et al.  Hydrodynamic instability for certain ABC flows , 1993 .

[90]  Remarks on periods of planar Hamiltonian systems , 1993 .

[91]  C. Robinson.  Dynamical Systems: Stability, Symbolic Dynamics, and Chaos , 1994 .

[92]  K. Yagasaki.  Chaos in a pendulum with feedback control , 1994 .

[93]  Miklós Farkas,et al.  Periodic Motions , 1994 .

[94]  H. Zoladek,et al.  Quadratic Systems with Center and Their Perturbations , 1994 .

[95]  S. Chow,et al.  Normal Forms and Bifurcation of Planar Vector Fields , 1994 .

[96]  Carmen Chicone,et al.  Lyapunov-Schmidt Reduction and Melnikov Integrals for Bifurcation of Periodic Solutions in Coupled Oscillators , 1994 .

[97]  C. Chicone.  A geometric approach to regular perturbation theory with an application to hydrodynamics , 1995 .

[98]  Carlos Gutierrez,et al.  A solution to the bidimensional Global Asymptotic Stability Conjecture , 1995 .

[99]  C. Chicone,et al.  Gravitational ionization : periodic orbits of binary systems perturbed by gravitational radiation , 1995, gr-qc/9508065.

[100]  On the ionization of a Keplerian binary system by periodic gravitational radiation , 1996, gr-qc/9605031.

[101]  J. Barrow-Green.  Poincare and the Three Body Problem , 1996 .

[102]  R. Benguria,et al.  Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation , 1994, patt-sol/9408001.

[103]  A. J. Roberts.  Low-dimensional modelling of dynamical systems , 1997 .

[104]  Christiane Rousseau,et al.  Local Bifurcations of Critical Periods in the Reduced Kukles System , 1997, Canadian Journal of Mathematics.

[105]  Addendum: On the ionization of a Keplerian binary system by periodic gravitational radiation [J. Math. Phys. 37, 3997–4016 (1996)] , 1997 .

[106]  The geodesic flow generates a fast dynamo: an elementary proof , 1997 .

[107]  Gravitational ionization: a chaotic net in the Kepler system , 1996, gr-qc/9608054.

[108]  Richard S. Palais.  The symmetries of solitons , 1997 .

[109]  T. Weissert.  The genesis of simulation in dynamics , 1997 .

[110]  Carmen Chicone.  Invariant tori for periodically perturbed oscillators , 1997 .

[111]  Santiago Ibáñez,et al.  Singularities of vector fields on , 1998 .

[112]  J. Zukas.  Introduction to the Modern Theory of Dynamical Systems , 1998 .

[113]  S. Smale,et al.  Finding a horseshoe on the beaches of Rio , 1998 .

[114]  G. Iooss,et al.  Topics in bifurcation theory and applications , 1999 .

[115]  Yuri Latushkin,et al.  Evolution Semigroups in Dynamical Systems and Differential Equations , 1999 .

[116]  Hans F. Weinberger.  An Example of Blowup Produced by Equal Diffusions , 1999 .

[117]  Carmen Chicone,et al.  On the continuation of periodic orbits , 2000 .

[118]  Weishi Liu,et al.  On the Continuation of an Invariant Torus in a Family with Rapid Oscillations , 1999, SIAM J. Math. Anal..