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| Ïîèñê ïî óêàçàòåëÿì |
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| Mishchenko E.F., Rozov N.Kh. — Differential Equations with Small Parameters and Relaxation Oscillations |
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| Ïðåäìåòíûé óêàçàòåëü |
Airy’s equation 69
Almost-discontinuous periodic solutions 139—170
Almost-discontinuous periodic solutions, asymptotic calculation of trajectory for 203—209
Almost-discontinuous periodic solutions, existence of 139—142 203—209
Almost-discontinuous periodic solutions, in systems of arbitrary order 199—216
Almost-discontinuous periodic solutions, uniqueness of 139—142
Approximate solution, defined 30
Approximations, asymptotic see Asymptotic approximations
Approximations, zeroth see Zeroth approximation
Arbitrary-order asymptotic solutions 171—198
Arbitrary-order systems 16—18
Arbitrary-order systems, almost-discontinuous periodic solutions in 199—216
Arbitrary-order systems, asymptotic calculation of solutions for 171—198
Asymptotic approximations 31—32
Asymptotic approximations, for trajectory in neighborhood of junction point 187—192
Asymptotic approximations, for trajectory of periodic solution 143—144
Asymptotic approximations, of trajectory at beginning of junction system 179—187
Asymptotic approximations, of trajectory at end of junction section 193—196
Asymptotic approximations, of trajectory for initial slow-motion and drop parts 132—137
Asymptotic approximations, of trajectory in fast-motion part 107—111
Asymptotic approximations, partial sum of series from 51—52
Asymptotic approximations, vs. actual trajectories at end of junction section 99—103
Asymptotic approximations, vs. actual trajectories in immediate vicinity of junction point 77—82
Asymptotic expansion, defined 31—32
Asymptotic expansion, for end of junction part of trajectory 96—99
Asymptotic formula for relaxation-oscillation period 164—168
Asymptotic formula for trajectory approximations 34
Asymptotic methods, application of 30—31
Asymptotic methods, general theory of 29—30
Asymptotic representations for drop part, proof of 125—132
Asymptotic representations for fast-motion part, derivation of 111—114
Asymptotic representations for junction part, proof of 103—107
Asymptotic representations, of initial drop part 123
Asymptotic representations, proof of 52—56
Asymptotic series, for coefficients of expansion near junction point 82—88
Asymptotic solutions, arbitrary-order 174—198
Asymptotic solutions, second-order 39—137
Auto-oscillations, relaxation-oscillations and 7
Bendixon’s criteria 21—22
Bessel equation 69
Bessel function, modified 69
Bifurcation value 16
Closed trajectory, isolated and stable properties of 200
Coefficients of expansion, asymptotic series for 82—88
Degenerate system, arbitrary-order system and 17
Degenerate system, closed phase trajectories of 37 139
Degenerate system, defined 9
Degenerate system, function system as solution of 27
Degenerate system, nondegenerate system and 39
Degenerate system, periodic solution for 199
Degenerate system, phase trajectory of 173
Degenerate system, solutions of 24—29 199
Degenerate system, trajectories of solutions in 10—11 29
Dependence, continuous 2
Dependence, discontinuity in 4
Dependence, smooth 1—3
Dependence, types of 1—37
Discontinuity, in dependence 4
Discontinuous periodic solution, defined 140
Discontinuous solution, defined 29
Discontinuous solution, phase trajectories and 45
Discontinuous solution, zeroth order in 173—176
Discontinuous system, trajectory of 29
Displacement vector 197—198
Dividing solution, for Riccati’s equation 71
Dorodnitsyn’s formula 168—170
Drop part of trajectory, asymptotic approximations of 119—125
Drop part of trajectory, defined 48
Drop part of trajectory, initial 48 123
Drop part of trajectory, proof of asymptotic representations of 125—132
Drop part of trajectory, special variables for 114—119
Drop point following junction point 45
Drop point, defined 173
Drop point, junction point and 29
Drop time, calculation of 157—164
Drop-off, defined 13
Equilibrium position, solutions as 26
Expansion, asymptotic 31—32
Expansion, coefficients of 82—88
Fast motion, defined 16—20
Fast variable, defined 16 18
Fast-motion equation system, stationary solutions of 35
Fast-motion equations, defined 16 44
Fast-motion equations, exponentially stable period solution for 36
Fast-motion equations, Hamiltonian and 37
Fast-motion equations, scalar form of 26
Fast-motion part of traj ectory, asymptotic approximations of trajectory on 107—111
Fast-motion part of traj ectory, defined 45 48 174
Fast-motion time, calculation of 156—157
Fast-motion time, fast-motion equation and 44
Finite time interval, solution dependence in 4
Frugauer generators 8
Function system, as solution of degenerate system 27
Generalized integral 90 95
Improper integrals, regularization of 89—95
Infinite time interval, solution dependence on 3—4
Initial drop part, asymptotic representations of 123
Initial drop part, defined 48
Initial fast-motion part, asymptotic approximation for 134
Initial fast-motion part, defined 48
Initial junction section, asymptotic representations vs. actual trajectories in 62—67
Initial slow-motion and drop parts, asymptotic approximations of 132—137
Integral, generalized 90 95
Junction part of trajectory, asymptotic expansions for end of 96—99
Junction part of trajectory, defined 48
| Junction part of trajectory, proof of asymptotic representations of 103—107
Junction point, asymptotic approximations for trajectory in neighborhood of 187—193
Junction point, asymptotic approximations vs. actual trajectories in immediate vicinity of 77—82
Junction point, asymptotic series for coefficients of expansion near 82—88
Junction point, defined 29 42 173
Junction point, local coordinates in neighborhood of 56—59 176—179
Junction point, trajectory in neighborhood of 72—76
Junction section, asymptotic approximat ions vs. actual trajectories at end of 99—103
Junction section, asymptotic approximation of trajectory at end of 193—196
Junction section, asymptotic approximations of trajectory at beginning of 179—187
Junction section, special variables for 67—68
Junction time, calculation of 145—156
Junction, trajectory on initial part of 60—62
Limit cycle, for Van der Pol’s equation 35
Lyapunov function 205
Mapping, of trajectories 11—12
Multidimensional system, periodic solutions for 36
Multivibrators, symmetric 8
Mutual inductance, in Van der Pol’s equation 6—8
Newton — Leibnitz formula 90
Nondegenerate points, defined 40
Nondegenerate system, closed trajectory of 141
Nondegenerate system, defined 39
Nondegenerate system, phase velocity of 11
Nonregular point, defined 40 172
Nonregular point, nondegeneracy of 41
Numerical methods, approximate solution in 30
Ordinary points, defined 40
Parameter, asymptotic expansions of solutions with respect to 29—34
Periodic solution trajectory, asymptotic approximations for 143—144
Periodic solutions, Dorodnitsyn’s formula and 168—170
Periodic solutions, drop time and 157—164
Periodic solutions, exponentially stable 36
Periodic solutions, fast-inotion time and 156—157
Periodic solutions, junction time and 145—156
Periodic solutions, slow-motion time and 144—145
Periodic solutions, Van der Pol’s equation and 168—170
Phase plane, trajectory and 47
Phase point, stable equilibrium and 16 24
Phase portrait for fast and slow systems 21
Phase portrait of second-order system 13
Phase portrait of Van den Pol’s equation 13
Phase trajectories, assumptions and definitions in 39—40
Phase trajectories, discontinuous solution and 4 5
Phase trajectories, zeroth approximation of 45
Phase-velocity vector, infinite first component of 15
Phase-velocity vector, second component of 18
Poincar ’s theorem 1—3
Rapid-motion equation system, defined 18
RayleighTs equation 6—7
Regular parts, defined 40
Regular point, defined 40
Relaxation motion, defined 20
Relaxation oscillations, asymptotic formula for period of 209—216
Relaxation oscillations, asymptotic theory and 35
Relaxation oscillations, auto-oscillations and 7
Relaxation oscillations, defined 9 13 16
Relaxation-oscillation period, asymptotic formula for 164—168
Riccati’s equation 68—72
Riccati’s equation, dividing solution for 71
Second-order systems, almost-discontinuous periodic solutions of 139—170
Second-order systems, asymptotic calculations and 34 39—137
Second-order systems, fast and slow motions in 9—16
Second-order systems, phase portrait of 13
Slow motion part, asymptotic approximations on 48—52
Slow motion part, defined 50 174
Slow motion part, in phase-velocity systems 48
Slow motion part, proof of asymptotic representations of 52—56
Slow motion, in arbitrary-order systems 16—20
Slow variable, defined 16—20
Slow-motion time, calculation of 144—145
Small parameters, dependence of solutions on 9
Small parameters, equations with 4—9
Smooth dependence 1—3 (see also Dependence)
Solutions, asymptotic calculation of 171—198
Solutions, asymptotic expansions of with respect to parameter 29—34
Solutions, differences between 10
Solutions, differentiability of with respect to parameter 3
Stable equilibrium, defined 13—14
Stable equilibrium, phase point and 24
Stable part, defined 13 16 42
Stable region, defined 26
Stationary solutions, of fast-motion equation system 35
Surge, defined 13
Trajectory approximations, asymptotic formulas for 34
Trajectory at beginning of junction system, asymptotic approximations for 179—187
Trajectory at end of junction section, asymptotic approximations for 193—196
Trajectory in neighborhood of junction point, asymptotic approximations for 187—193
Trajectory, asymptotic approximations for trajectory in neighborhood of junction point 72—76
Trajectory, asymptotic approximations on initial part of junction 60—62
Trajectory, asymptotic approximations on slow-motion parts of 48—52
Trajectory, asymptotic representation of, vs. actual 62—67
Trajectory, drop part of see Drop part of trajectory
Trajectory, mapping of 11
Trajectory, of discontinuous system 29
Trajectory, on initial part of junction 60—62
Unstable part, defined 13 42
Vacuum tube oscillator, equation for 7
Van der Pol’s equation 5—7 10 14
Van der Pol’s equation, limit cycle for 35
Van der Pol’s equation, periodic solutions and 168—170
Van der Pol’s equation, phase portrait of 13
Zeroth approximation, defined 31—32
Zeroth approximation, discontinuous solution and 173—176
Zeroth approximation, of trajectory in neighborhood of junction point 75
Zeroth approximation, Riccati’s equation and 68
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