References Chen Yushu, Nonlinear Vibrations, Tian lin Science and Technology Press 1983 (in Chinese) 2 M Golubitsky and WF Langford, Classification and Unfolding of Degenerate Hopf Bifurcations, Journal Differential Equations, No 3, 1981 3 Chen Yushu and WF Langford, The Subharmonic Bifurcation Solution of Nonlinear Mathieu's Equation and Euler Dynamic Buckling Problems, Acta Mechanica Sinica, No 4, 1988 4 M Globitsky and DG Schaeffer, Singularities and Groups in Bifurcation Theory, 1 and 2, Springer-Verlag, New York, 1985, 1988 5 Zhu Zhaoxuan, Chaos in Nonlinear Dynamics, Advance of Mechanics, 14, No 2, 1984 (in Chinese) 6 Lin Rei and Chen Yushu, A New Method for Multi-degrees of Freedom of Nonlinear Vibration Systems and Application in Analysis of Snake Motion of Rail Vehicles, Acta Applied Mechanics, 3, No 4, 1986 (in Chinese) 7 NK Cooperrider, Nonlinear Behavior in Rail Vechicle Dynamics, Numerical Methods for Bifurcation Problems, Proc Conf of UD, August 1984 8 K Zeman, H Troger and R Scheidle, Bifurcation Theory and Vehicle Dynamics with Application to the Tractor Semitrailer, Proc 7th IAVSD Symposium in Cambridge UK 1981, Swets and Zeitlinger BV, Lisse, 1982 9 RK Mehra and lV Carroll, Bifurcation Analysis of Aircraft High Angle of Attack Flight Dynamics, New Approches to Nonlinear Problems in Dynamics SIAM, Philadelphia, 1980 10 MP Paidoussis and NT Issid, Dynamic Stability of Pipes Conveying Fluid, Journal of Sound Vibration, No 33, 1974 11 PR Sethna, Bifurcation Theory and Averaging in Mechanical Systems, Numerical Methodfor Biforcation Problems, Proc Conf ofUD, August 1983, Birkhauser Verlag, 1984 12 GR Garalas, Non-linear Differencial Equations of Chemically Reacting ystems, Springer-Verlag, New York, 1968 13 V Volterra, Variations and Fluetuations of Number ofIndividuals in Animal Species Living Together, Journal de Conseil International Pour l'Exploration de la Mer, 3, Nol,1928 14 EC Zeeman, Catastrophe Theory, Scientific American, April 1976 151M Greenberg, On the Equilibrium Configurations of Compressible Slender Bars, Scientific American, 27, 1967 16 WT Kaiter, Post-Buckling Analysis ofa Simple Two-Bar Frame, Recent Progress in Applied Mechanics, Almquist and Wiksell, Stockholm, 1967 17 S Alntman, Equilibrium States of Nonlinearly Elastic Rod, Recent Progress in Applied Mechanics, Almquist and Wiksell, Stockholm, 23, 1968
Transcript
References
Chen Yushu, Nonlinear Vibrations, Tian lin Science and Technology Press 1983 (in Chinese)
2 M Golubitsky and WF Langford, Classification and Unfolding of Degenerate Hopf Bifurcations, Journal Differential Equations, No 3, 1981
3 Chen Yushu and WF Langford, The Subharmonic Bifurcation Solution of Nonlinear Mathieu's Equation and Euler Dynamic Buckling Problems, Acta Mechanica Sinica, No 4, 1988
4 M Globitsky and DG Schaeffer, Singularities and Groups in Bifurcation Theory, 1 and 2, Springer-Verlag, New York, 1985, 1988
5 Zhu Zhaoxuan, Chaos in Nonlinear Dynamics, Advance of Mechanics, 14, No 2, 1984 (in Chinese)
6 Lin Rei and Chen Yushu, A New Method for Multi-degrees of Freedom of Nonlinear Vibration Systems and Application in Analysis of Snake Motion of Rail Vehicles, Acta Applied Mechanics, 3, No 4, 1986 (in Chinese)
7 NK Cooperrider, Nonlinear Behavior in Rail Vechicle Dynamics, Numerical Methods for Bifurcation Problems, Proc Conf of UD, August 1984
8 K Zeman, H Troger and R Scheidle, Bifurcation Theory and Vehicle Dynamics with Application to the Tractor Semitrailer, Proc 7th IAVSD Symposium in Cambridge UK 1981, Swets and Zeitlinger BV, Lisse, 1982
9 RK Mehra and lV Carroll, Bifurcation Analysis of Aircraft High Angle of Attack Flight Dynamics, New Approches to Nonlinear Problems in Dynamics
SIAM, Philadelphia, 1980
10 MP Paidoussis and NT Issid, Dynamic Stability of Pipes Conveying Fluid, Journal of Sound Vibration, No 33, 1974
11 PR Sethna, Bifurcation Theory and Averaging in Mechanical Systems, Numerical Methodfor Biforcation Problems, Proc Conf ofUD, August 1983, Birkhauser Verlag, 1984
12 GR Garalas, Non-linear Differencial Equations of Chemically Reacting ystems, Springer-Verlag, New York, 1968
13 V Volterra, Variations and Fluetuations of Number ofIndividuals in Animal Species Living Together, Journal de Conseil International Pour l'Exploration de la Mer, 3, Nol,1928
14 EC Zeeman, Catastrophe Theory, Scientific American, April 1976
151M Greenberg, On the Equilibrium Configurations of Compressible Slender Bars, Scientific American, 27, 1967
16 WT Kaiter, Post-Buckling Analysis ofa Simple Two-Bar Frame, Recent Progress in Applied Mechanics, Almquist and Wiksell, Stockholm, 1967
17 S Alntman, Equilibrium States of Nonlinearly Elastic Rod, Recent Progress in Applied Mechanics, Almquist and Wiksell, Stockholm, 23, 1968
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INDEX
A a and ~ limit sets 17 E Andronov-Pontriagin bifurcation theorem 88 elliptic point 290 Andronov-Pontriagin stability theorem 76 equivalence of static bifurcation 93 Arnold diffusion 303 asymptotially stable motion 26 F attracting set 22 Feigenbaum constant 275 autonomous systems 6 fmite determination 107 averaging method fmite generation 104
bifurcation theory 230 fixed point 17 Duffmg equation 248 Floquet theorem 29 Hamiltonian and global bifurcation 261 Flow 3 local bifurcation 255 flow box theorem 55 Poincare map 237 fractal dimensions 303
frequency spectrum 305 B
Banach space 7 G Bellman--Gronwelliemma 12 generalized eigen space 205 Bendixson theorem 37 generalized implicit function theorem 92 bifurcation 84 generic map 71 bifurcation diagram 85 generality 47 bifurcation parameter 86 geometric description of averaging method 241 bifurcation of double zero eigenvalues 203 geometric multiplicity 45 bifurcation set 87,103 germ 103 bifurcations of closed orbit 90 global bifurcation 87
C H calculation of centre manifolds 157 Hartman--Grobman (HG) theorem 58
central manifold 154 Hopfbifurcation of Lorenz's systems 159 central manifold theorem 155 centre subspace 46,154
Hopfbifurcation solutions 146,192
chaos 265 Hopf bifurcation theorem 144,176
chaotic sea 295 hyperbolic fixed point of map G 68
classification of codimensions 118 hyperbolicity 47
coefficients of Hopf bifurcation solution 198 hysteresis set 116
complex and fine structure 277 heteroclinic orbit 19
complex normal form of Hopfbifurcation 179 I compress map theorem 7 ideal 103 connected set 21 identical operator 170 construction of chaotic regions 11,319,324 continuous dynamical systems 3,66
diffeomorphic map 76 invariant subspace of linear flow 45
differential dynamical systems 3 J dimensions of manifold 60 Japanese attractor 271 divergence of flow 35 double limits set 116
Jordan's curves theorem 32
452 Bifurcation and Chaos in Engineering
K R KAMtheorem 301 real normal form of Hopf bifurcation 182 KAM torus 307 recognition 102 KBM transformation 237 response diagram 133
restrict tangent space lOS L
Liapunov eigen exponent 304 S Liapunov-Schmidt reduction 93 saddle·node bifurcation 166 linear flow 37 secondary Hopf bifurcation 307 linear map 67 self·adjoint operator 125 Lipschitz condition 8 semi·dynamical systems 4 local bifurcation 88 semi·simple case 44 local transverse section 30 Shacovsky sequence 276 Lorenz attractor 271 shift phase operator 129
simple bifurcation 119 M simple bifurcation of Duffing's equation 123
map 2 singularity theory 102 maximum ideal 103 Smale horseshoe 277 maximum solution 10 special orthogonal group 129 Melnikov's function 286 stability characteristic 318 multi· harmonic excitation 320 stability of Hopf bifurcation solution 149,194
stability of map 74 N stability of motion 23
network of chaotic river 296 stable manifold theorem 60 Newtonian algorithm 317 stable manifold theorem of map 69 non·autonomous systems 6 stable motion 23 non·degenerate conditions 133 stable motion of orbit 30 non· linear map 68 stable subspace 46,155 non· linear Mathieu operator 129 state variables 85 non· wandering point 22 static bifurcation 93 normal form of vector field 169 static bifurcation point 93 normed space 6 strong equivalence lOS number of harmonic terms 318 structural stability 76
structural stability of planar systems 81 0 subharmonic resonant bifurcation solutions 136
Orbit 15 supercritical bifurcation 169 order of bifurcation in planar vector field 91 symmetry of non· linear Mathieu operator 129 orthogonal group 129 orthogonality 233 T
Taken's form 210 P time reverse operator 129
Peixoto's structural stability theorem 81 topological entropy 304 periodic orbit of map 70 transition set 116,319 periodic window 275 transversity 30 phase portraits 2,63 two·dimensional equilibrium points 87 phase-time course 16 physical parametric space 319 U pitchfork bifurcation 86,169 Unfolding 112 Poincare eigenvalue lemma 171 universal 267 Poincare grid 295 universal constant 307 Poincare map 73 universal unfolding 112,138 Poincare -Birkhoffnormal form 172 unstable motion 23 Poincare resonant lemma 171 Poincare-Bendix son theorem 29 V power ideal of k orders 104 vector field 4 perturbation set 105
