References

[A-I] A.A. Andronov, E.A. Leontovich, I.I. Gordon and A.G. Maier, Qual­itative Theory of Second-Order Dynamical Systems, John Wiley and Sons, New York, 1973.

[A-II] A.A. Andronov, et. al., "Theory of Bifurcations of Dynamical Sys­tems on a Plane," Israel Program for Scientific Translations, Jerusalem, 1971.

[B] I. Bendixson, "Sur les courbes definies par des equations differentielles," Acta Math., 24 (1901), 1-88.

[C] H.S. Carslaw, Theory of Fourier Series and Integrals, Dover Publica­tions, Inc., New York, 1930.

[C/H] S.N. Chow and J.K. Hale, Methods of Bifurcation Theory, Springer­Verlag, New York, 1982.

[C/L] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw Hill, New York, 1955.

[Cu] C.W. Curtis, Linear Algebra, Allyn and Bacon Inc., Boston, 1974.

[D] H. Dulac, "Sur les cycles limites," Bull. Soc. Math. France, 51 (1923), 45-188.

[G] L.M. Graves, The Theory of Functions of Real Variables, McGraw Hill, New York, 1956.

[GIG] M. Golubitsky and V. Guillemin, Stable Mappings and their Singu­larities, Springer-Verlag, New York, 1973.

[G/H] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.

[H] P. Hartman, Ordinary Differential Equations, John Wiley and Sons, New York, 1964.

[HIS] M.W. Hirsch and S. Smale, Differential Equations, Dynamical Sys­tems and Linear Algebra, Academic Press, New York, 1974.

[L] S. Lefschetz, Differential Equations: Geometric Theory, Interscience, New York, 1962.

[Lo] F. Lowenthal, Linear Algebra with Linear Differential Equations, John Wiley and Sons, New York, 1975.

[N/S] V.V. Nemytskii and V.V. Stepanov, Qualitative Theory of Differen­tial Equations, Princeton University Press, Princeton, 1960.

396 References

[P] H. Poincare, "Memoire sur les courbes definies par une equation differentielle," J. Mathematiques, 7 (1881), 375-422; Oeuvre (1880-1890), Gauthier-Villar, Paris.

[R] W. Rudin, Principles of Mathematical Analysis, McGraw Hill, New York, 1964.

[Ru] D. Ruelle, Elements of Differentiable Dynamics and Bifurcation The­ory, Academic Press, New York, 1989.

[S] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, New York, 1982.

[W] P. Waltman, A Second Course in Elementary Differential Equations, Academic Press, New York, 1986.

[Wi] S. Wiggins, Global Bifurcations and Chaos, Springer-Verlag, New York, 1988.

Index

A a-limit cycle, 185, 186 a-limit point, 175 a-limit set, 175 Analytic function, 69 Analytic manifold, 106 Annular region, 270 Antipodal points, 250 Asymptotic stability, 128, 130 Asymptotically stable periodic or-

bits, 184 Atlas, 107, 117,226 Attracting set, 177, 179 Attractor, 177, 180 Autonomous system, 65

B Behavior at infinity, 248, 252 Bendixson sphere, 248, 269 Bendixson's criteria, 245 Bendixson's index theorem, 280 Bendixson's theorem, 139 Bifurcation

homoclinic, 337, 350, 363, 368, 380

Hopf, 314, 317, 343, 352, 357 period doubling, 333, 339 pitchfork, 308, 310, 330, 332,

339,342 saddle connection, 300, 304,

350 saddle node, 306, 310, 326,

331,341 transcritical, 307, 310, 329,

331 value, 272, 306

Bifurcation at a nonhyperbolic equi­librium point, 306

Bifurcation at a nonhyperbolic pe­riodic orbit, 324, 333

Bifurcation from a center, 385 Bifurcation from a multiple focus,

320 Bifurcation from a multiple limit

cycle, 333 Bifurcation from a multiple sepa­

ratrix cycle, 364 Bifurcation from a simple separa-

trix cycle, 364 Bifurcation set, 291 Bifurcation theory, 291 Bifurcation value, 104, 272, 306 Blowing up, 248, 265

C C(E),68 Cl(E), 68, 292 C l diffeomorphism, 126, 172, 195,

370 C l function, 68 C l norm, 292, 294 C l vector field, 261 C k (E),68 C k conjugate vector fields, 173 Ck equivalent vector fields, 172 Ck function, 68 C k manifold, 106 Ck norm, 319 Canonical region, 270 Cauchy sequence, 73 Center, 23, 24, 138, 142 Center focus, 138, 142 Center manifold of an equilibrium

point, 115 Center manifold of a periodic or­

bit, 210 Center manifold theorem, 115 Center manifold theorem for pe­

riodic orbits, 210 Center subspace, 5, 9, 51, 55

398

Center subspace of a map, 370 Center subspace of a periodic or-

bit, 208 Central projection, 249 Characteristic exponent, 204, 205 Characteristic multiplier, 204, 205 Chart, 107 Cherkas' theorem, 246 Chillingworth's theorem, 171 Circle at infinity, 250 Closed orbit, 184 Competing species, 273 Complete family of rotated vector

fields, 347 Complete normed linear space, 73,

292 Complex eigenvalues, 28, 36 Compound separatrix cycle, 190,

226 Conservation of energy, 153 Continuation of solutions, 90 Continuity with respect to initial

conditions, 10, 20, 79 Continuous function, 68 Continuously differentiable func-

tion, 68 Contraction mapping principle, 77 Convergence of operators, 11 Critical point, 101 Critical point of multiplicity m,

309 Critical points at infinity, 252, 258 Cusp, 149, 150, 161 Cycle, 184 Cyclic family of periodic orbits,

361 Cylindrical coordinates, 94

D Dr, 67 Dkr, 69 Deficiency indices, 42 Degenerate critical point, 23, 154,

289

Index

Degenerate equilibrium point, 23, 154,289

Derivative, 67, 69 Derivative of the Poincare map,

196, 198, 203, 205, 207, 324

Derivative of the Poincare map with respect to a param­eter, 332

Diagonal matrix, 6 Diagonalization, 6 Diffeomorphism, 126, 164, 195 Differentiability with respect to ini-

tial conditions, 79 Differentiability with respect to pa-

rameters, 83 Differentiable, 67 Differentiable manifold, 106, 117 Discrete dynamical system, 173 Displacement function, 197, 328,

360 Dulling's equation, 386 Dulac's criteria, 246 Dulac's theorem, 188, 199 Dynamical system, 2, 163, 164,

169, 173 Dynamical system defined by a

differential equation, 165, 169, 170

E Eigenvalues

complex, 28, 36 distinct, 6 pure imaginary, 23 repeated, 33

Elementary Jordan blocks, 40, 49 Elliptic domain, 147, 150 Elliptic region, 271 Elliptic sector, 146 Equilibrium point, 2, 65, 101 Euler-Poincare characteristic of a

surface, 274, 281 Existence uniqueness theorem, 73

Index

Exponential of an operator, 12, 13,15,17

F fa, 261 Fixed point, 102, 370 Floquet's theorem, 203 flow

of a differential equation, 95 of a linear system, 54 of a vector field, 95 on a manifold, 261 on 82, 251, 255, 302 on a torus, 182, 220, 286, 289,

300 Focus, 22, 24, 25, 13~ 142 Fundamental existence uniqueness

theorem, 73 Fundamental matrix solution, 60,

77, 83, 85, 203 Fundamental theorem for linear

systems, 17

G Gauss' model, 273 General solution, 1 Generalized eigenvector, 33, 51 Generalized Poincare Bendixson

theorem, 227 Generic property, 301, 303 Genus, 281, 284 Global behavior of limit cycles and

periodic orbits, 352, 353, 358

Global existence theorem, 166, 169, 170,171

Global Lipschitz condition, 170 Global phase portrait, 255, 263,

266,269 Global stability, 184 Global stable and unstable man-

ifolds, 113, 185, 370 Gradient system, 157, 160 Graphic, 190, 226, 305, 350 Gronwall's inequality, 79

399

H Hamiltonian system, 153, 159, 192,

216 Harmonic oscillator, 153 Hartman Grobman theorem, 119 Hartman's theorem, 126 Heteroclinic orbit, 189 Hilbert's 16th problem, 243 Homeomorphism, 106 Homoclinic bifurcation, 338, 350,

363, 368 Homoclinic explosion, 339 Homoclinic orbit, 188, 337 Homoclinic tangle, 372 Hopf bifurcation, 272, 314, 317,

338, 343, 352, 364 Horseshoe map, 372, 375 Hyperbolic equilibrium point, 101 Hyperbolic fixed point of a map,

370 Hyperbolic flow, 54 Hyperbolic periodic orbit, 208 Hyperbolic region, 271 Hyperbolic sector, 146

I Ir(C),274 Ir(xo), 276, 281 Implicit function theorem, 194,

228,324 Improper node, 21 Index

of a critical point, 276, 281 of a Jordan curve, 274 of a saddle, node, focus or

center, 280 of a separatrix cycle, 279 of a surface, 274, 281, 282

Index theory, 273 Initial conditions, 1, 70, 79 Initial value problem, 16, 29, 70,

73,76,78 Invariant manifolds, 107, 111, 114,

207, 223, 370 Invariant subset, 98, 176

400

Invariant subspace, 16, 20, 54

J Jacobian matrix, 67 Jordan block, 40, 42, 49 Jordan canonical form, 39, 47 Jordan curve, 187, 228, 274 Jordan curve theorem, 187

K Kernel of a linear operator, 42 Klein bottle, 282, 288

L L(Rn),lO Left maximal interval, 90 Level curves, 158 Liapunov function, 129 Liapunov number, 200, 317, 352 Liapunov theorem, 130 Lienard equation, 135, 234, 385 Lienard system, 234 Lienard's theorem, 234 Limit cycle, 178, lS6 Limit cycle of multiplicity k, 198 Limit orbit, 177 Limit set, 175 Linear

approximation, 101 flow, 54 subspace, 51 system, 1, 20 transformation, 7, 20

Linearization about a periodic or­bit, 203

Linearization of a differential equa-tion, 101, 203

Liouville's theorem, 85, 214 Lipschitz condition, 71 Local limit cycle, 240 Local stable and unstable mani­

folds, 113, 185 Locally Lipschitz, 71 Lorenz system, 104, 180, 184, 335

M Manifold, 106

center, 115

Index

global stable and unstable, 114, 185, 370

invariant, 107, 111, 114, 185, 207, 223, 370

local stable and unstable, 113, 185,370

Maps, 342, 344, 377 Markus' theorem, 271 Maximal family of periodic orbits,

362 Maximal interval of existence, 65,

67, 86, S9, 93 Melnikov function, 379, 382, 384 Melnikov's method, 292, 375, 378 Morse-Smale system, 302 Multiple eigenvalues, 33 Multiple focus, 200, 320 Multiple limit cycle, 19S, 333 Multiple separatrix cycle, 364 Multiplicity of a critical point,

309 Multiplicity of a focus, 200 Multiplicity of a limit cycle, 198

N Negative half-trajectory, 175 Negatively invariant set, 98 Neighborhood of a set, 176 Newtonian system, 155, 161 Nilpotent matrix, 33, 50 Node, 22, 24, 25, 13S, 142 Nonautonomous linear system,

77,85 Nonautonomous system, 62, 65,

76 Nondegenerate critical point, 154 Nonhomogeneous linear system,

60 Nonhyperbolic equilibrium point,

101,145 Nonlinear systems, 65 Nonwandering point, 300

Index

Nonwandering set, 300, 305 Norm

C1-norm, 292, 294 Ck-norm, 319 Euclidean, 11 matrix, 10, 15 operator, 10, 15 uniform, 73

Number of limit cycles, 230, 234, 239, 241

o w-limit cycle, 185, 186 w-limit point, 175 w-limit set, 175 Operator norm, 10 Orbit, 174, 177, 183 Orbit of a map, 370 Ordinary differential equation, 1 Orient able manifold, 107, 117 Orthogonal systems of differential

equations, 159

p Parabolic region, 271 Parabolic sector, 146 Peixoto's theorem, 301 Pendulum, 155 Period, 184 Period doubling bifurcation, 333,

339 Period doubling cascade, 341 Periodic orbit, 184 Periodic orbit of saddle type, 186 Periodic solution, 184 Perko's planar termination prin-

ciple, 362 Phase plane, 2 Phase portrait, 2, 9, 20 Picard's method of successive ap­

proximations, 72 Pitchfork bifurcation, 308, 310, 312,

330, 332, 336, 339 Poincare-Bendixson theorem, 227

401

Poincare-Bendixson theorem for two-dimensional manifolds, 231

Poincare index theorem, 282 Poincare map, 193, 195,200,324,

332,334 Poincare map for a focus, 200 Poincare sphere, 248, 255 Polar coordinates, 28, 136, 144,

344 Positive half trajectory, 174 Positively invariant set, 98 Predator prey problem, 273 Projective geometry, 217, 248 Projective plane, 250, 282, 285,

286 Proper node, 21, 138 Pure imaginary eigenvalues, 23 Putzer algorithm, 39

R Real distinct eigenvalues, 6 Regular point, 227 Rest point or equilibrium point,

101 Right maximal interval, 90 Rotated vector field (mod G =

0),354 Rotated vector fields, 346

S Saddle, 21, 24, 25, 102, 138, 140 Saddle connection, 300, 304, 350 Saddle-node, 148, 149 Saddle-node bifurcation, 306, 310,

326,331 Schwartz's theorem, 231 Sector, 146, 258 Semicomplete family of rotated

vector fields, 374 Semisimple matrix, 50 Semi-stable limit cycle, 186, 198,

202,349 Separatrix, 21, 28, 139, 270

402

Separatrix configuration, 256, 271

Separatrix cycle, 188, 189, 226, 350,367

Shift map, 375, 377 Simple limit cycle, 198 Simple separatrix cycle, 364 Singular point, 102 Sink, 26, 56, 102, 129 Smale-Birkhoff homoclinic theo-

rem, 375 Smale horseshoe map, 343, 372,

375 Smale's theorem, 375 Smooth curve, 260 Solution curve, 2, 95, 174 Solution of a differential equation,

70 Solution of an initial value prob-

lem, 70 Sotomayor's theorem, 310 Source, 26, 56, 102 Spherical pendulum, 153, 219 Spiral region, 270 Stability theory, 51, 128 Stable equilibrium point, 128 Stable focus, 22, 138 Stable limit cycle, 186, 198 Stable manifold of an equilibrium

point, 113 Stable manifold of a periodic or­

bit, 185, 207 Stable manifold theorem, 107 Stable manifold theorem for maps,

370 Stable manifold theorem for peri-

odic orbits, 207 Stable node, 22, 138 Stable periodic orbit, 184 Stable separatrix cycle, 364 Stable subspace, 5, 9, 51, 55, 58 Stable subspace of a map, 370 Stable subspace of a periodic or-

bit, 208 Stereographic projection, 217

Strange attractor, 182, 338 Strip region, 270

Index

Structural stability, 291, 292, 294, 301,302

Structural stable dynamical sys­tern, 293

Structurally stable vector field, 292, 294, 301

Sub critical Hopf bifurcation, 317 Subharmonic Melnikov function,

384 Subharmonic periodic orbit, 384 Subspaces, 5, 9, 51, 59 Successive approximations, 72, 76,

111, 121, 124 Supercritical Hopf bifurcation,

317 Surface, 272, 281 Symmetric system, 144 System of differential equations,

1, 65, 163

T Tp S2, 260 TpM, 260 Tangent bundle, 260 Tangent plane, 260 Tangent space, 260 Tangent vector, 260 Tangential homo clinic bifurcation,

364, 375, 380, 382 Topological saddle, 138, 140, 149 Topologically conjugate, 118, 166 Topologically equivalent, 106, 118,

165, 166, 169, 270, 293 Trajectory, 95, 174, 177, 183 Transcritical bifurcation, 307, 310,

312, 329, 331 Transversal, 194, 227 Transversal intersection of mani­

folds, 194, 302, 369 Transverse homo clinic orbit, 292,

324, 363, 370, 371 Transverse homo clinic point, 371,

375, 379, 382

Index

Thiangulation of a surface, 281, 288

Two-dimensional surface, 281

U Uncoupled linear systems, 17 Uniform continuity, 78 Uniform convergence, 73, 91 Uniform norm, 73 Uniqueness of limit cycles, 234,

238 Uniqueness of solutions, 66, 73 Unstable equilibrium point, 128 Unstable focus, 22, 138 Unstable limit cycle, 186, 198 Unstable manifold of an equilib-

rium point, 113 Unstable manifold of a periodic

orbit, 185, 207 Unstable node, 22, 138 Unstable periodic orbit, 184 Unstable separatrix cycle, 364 Unstable subspace, 4, 9, 51, 55,

58 Unstable subspace of a map, 370 Unstable subspace of a periodic

orbit, 208 Upper Jordan canonical form, 40,

48

403

V Van der Pol equation, 135, 237,

244 Variation of parameters, 62 Variational equation or lineariza­

tion of a differential equa­tion, 101,203

Vector field, 3, 95, 102, 165, 172, 260, 281, 292

Vector field on a manifold, 260, 262, 265, 281, 302

w W C (O),115 W 8 (O), 113, 371 WU(O), 113, 371 WC(r),210 W8(r), 185,207 WU(r), 185, 207 Weak focus or multiple focus, 200,

320 Wedge product, 332, 346 Whitney's theorem, 260 Wintner's principle of natural ter-

mination, 362

Z Zero eigenvalues, 148 Zero of a vector field, 101 Zhang's theorem, 238, 239

Italic page numbers indicate where a term is defined.