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Mishchenko E.F., Rozov N.Kh. Ч Differential Equations with Small Parameters and Relaxation Oscillations
Mishchenko E.F., Rozov N.Kh. Ч Differential Equations with Small Parameters and Relaxation Oscillations



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Ќазвание: Differential Equations with Small Parameters and Relaxation Oscillations

јвторы: Mishchenko E.F., Rozov N.Kh.

јннотаци€:

A large amount of work has been done on ordinary differential equations with small parameters multiplying derivatives. This book investigates questions related to the asymptotic calculation of relaxation oscillations, which are periodic solutions formed of sections of both slow-and fast-motion parts of phase trajectories. A detailed discussion of solutions of differential equations involving small parameters is given for regions near singular points.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1980

 оличество страниц: 228

ƒобавлена в каталог: 13.12.2009

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
AiryТs equation      69
Almost-discontinuous periodic solutions      139Ч170
Almost-discontinuous periodic solutions, asymptotic calculation of trajectory for      203Ч209
Almost-discontinuous periodic solutions, existence of      139Ч142 203Ч209
Almost-discontinuous periodic solutions, in systems of arbitrary order      199Ч216
Almost-discontinuous periodic solutions, uniqueness of      139Ч142
Approximate solution, defined      30
Approximations, asymptotic      see Asymptotic approximations
Approximations, zeroth      see Zeroth approximation
Arbitrary-order asymptotic solutions      171Ч198
Arbitrary-order systems      16Ч18
Arbitrary-order systems, almost-discontinuous periodic solutions in      199Ч216
Arbitrary-order systems, asymptotic calculation of solutions for      171Ч198
Asymptotic approximations      31Ч32
Asymptotic approximations, for trajectory in neighborhood of junction point      187Ч192
Asymptotic approximations, for trajectory of periodic solution      143Ч144
Asymptotic approximations, of trajectory at beginning of junction system      179Ч187
Asymptotic approximations, of trajectory at end of junction section      193Ч196
Asymptotic approximations, of trajectory for initial slow-motion and drop parts      132Ч137
Asymptotic approximations, of trajectory in fast-motion part      107Ч111
Asymptotic approximations, partial sum of series from      51Ч52
Asymptotic approximations, vs. actual trajectories at end of junction section      99Ч103
Asymptotic approximations, vs. actual trajectories in immediate vicinity of junction point      77Ч82
Asymptotic expansion, defined      31Ч32
Asymptotic expansion, for end of junction part of trajectory      96Ч99
Asymptotic formula for relaxation-oscillation period      164Ч168
Asymptotic formula for trajectory approximations      34
Asymptotic methods, application of      30Ч31
Asymptotic methods, general theory of      29Ч30
Asymptotic representations for drop part, proof of      125Ч132
Asymptotic representations for fast-motion part, derivation of      111Ч114
Asymptotic representations for junction part, proof of      103Ч107
Asymptotic representations, of initial drop part      123
Asymptotic representations, proof of      52Ч56
Asymptotic series, for coefficients of expansion near junction point      82Ч88
Asymptotic solutions, arbitrary-order      174Ч198
Asymptotic solutions, second-order      39Ч137
Auto-oscillations, relaxation-oscillations and      7
BendixonТs criteria      21Ч22
Bessel equation      69
Bessel function, modified      69
Bifurcation value      16
Closed trajectory, isolated and stable properties of      200
Coefficients of expansion, asymptotic series for      82Ч88
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      24Ч29 199
Degenerate system, trajectories of solutions in      10Ч11 29
Dependence, continuous      2
Dependence, discontinuity in      4
Dependence, smooth      1Ч3
Dependence, types of      1Ч37
Discontinuity, in dependence      4
Discontinuous periodic solution, defined      140
Discontinuous solution, defined      29
Discontinuous solution, phase trajectories and      45
Discontinuous solution, zeroth order in      173Ч176
Discontinuous system, trajectory of      29
Displacement vector      197Ч198
Dividing solution, for RiccatiТs equation      71
DorodnitsynТs formula      168Ч170
Drop part of trajectory, asymptotic approximations of      119Ч125
Drop part of trajectory, defined      48
Drop part of trajectory, initial      48 123
Drop part of trajectory, proof of asymptotic representations of      125Ч132
Drop part of trajectory, special variables for      114Ч119
Drop point following junction point      45
Drop point, defined      173
Drop point, junction point and      29
Drop time, calculation of      157Ч164
Drop-off, defined      13
Equilibrium position, solutions as      26
Expansion, asymptotic      31Ч32
Expansion, coefficients of      82Ч88
Fast motion, defined      16Ч20
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      107Ч111
Fast-motion part of traj ectory, defined      45 48 174
Fast-motion time, calculation of      156Ч157
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      89Ч95
Infinite time interval, solution dependence on      3Ч4
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      62Ч67
Initial slow-motion and drop parts, asymptotic approximations of      132Ч137
Integral, generalized      90 95
Junction part of trajectory, asymptotic expansions for end of      96Ч99
Junction part of trajectory, defined      48
Junction part of trajectory, proof of asymptotic representations of      103Ч107
Junction point, asymptotic approximations for trajectory in neighborhood of      187Ч193
Junction point, asymptotic approximations vs. actual trajectories in immediate vicinity of      77Ч82
Junction point, asymptotic series for coefficients of expansion near      82Ч88
Junction point, defined      29 42 173
Junction point, local coordinates in neighborhood of      56Ч59 176Ч179
Junction point, trajectory in neighborhood of      72Ч76
Junction section, asymptotic approximat ions vs. actual trajectories at end of      99Ч103
Junction section, asymptotic approximation of trajectory at end of      193Ч196
Junction section, asymptotic approximations of trajectory at beginning of      179Ч187
Junction section, special variables for      67Ч68
Junction time, calculation of      145Ч156
Junction, trajectory on initial part of      60Ч62
Limit cycle, for Van der PolТs equation      35
Lyapunov function      205
Mapping, of trajectories      11Ч12
Multidimensional system, periodic solutions for      36
Multivibrators, symmetric      8
Mutual inductance, in Van der PolТs equation      6Ч8
Newton Ч 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      29Ч34
Periodic solution trajectory, asymptotic approximations for      143Ч144
Periodic solutions, DorodnitsynТs formula and      168Ч170
Periodic solutions, drop time and      157Ч164
Periodic solutions, exponentially stable      36
Periodic solutions, fast-inotion time and      156Ч157
Periodic solutions, junction time and      145Ч156
Periodic solutions, slow-motion time and      144Ч145
Periodic solutions, Van der PolТs equation and      168Ч170
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 PolТs equation      13
Phase trajectories, assumptions and definitions in      39Ч40
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$\acute{e}$Тs theorem      1Ч3
Rapid-motion equation system, defined      18
RayleighTs equation      6Ч7
Regular parts, defined      40
Regular point, defined      40
Relaxation motion, defined      20
Relaxation oscillations, asymptotic formula for period of      209Ч216
Relaxation oscillations, asymptotic theory and      35
Relaxation oscillations, auto-oscillations and      7
Relaxation oscillations, defined      9 13 16
Relaxation-oscillation period, asymptotic formula for      164Ч168
RiccatiТs equation      68Ч72
RiccatiТs equation, dividing solution for      71
Second-order systems, almost-discontinuous periodic solutions of      139Ч170
Second-order systems, asymptotic calculations and      34 39Ч137
Second-order systems, fast and slow motions in      9Ч16
Second-order systems, phase portrait of      13
Slow motion part, asymptotic approximations on      48Ч52
Slow motion part, defined      50 174
Slow motion part, in phase-velocity systems      48
Slow motion part, proof of asymptotic representations of      52Ч56
Slow motion, in arbitrary-order systems      16Ч20
Slow variable, defined      16Ч20
Slow-motion time, calculation of      144Ч145
Small parameters, dependence of solutions on      9
Small parameters, equations with      4Ч9
Smooth dependence      1Ч3 (see also Dependence)
Solutions, asymptotic calculation of      171Ч198
Solutions, asymptotic expansions of with respect to parameter      29Ч34
Solutions, differences between      10
Solutions, differentiability of with respect to parameter      3
Stable equilibrium, defined      13Ч14
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      179Ч187
Trajectory at end of junction section, asymptotic approximations for      193Ч196
Trajectory in neighborhood of junction point, asymptotic approximations for      187Ч193
Trajectory, asymptotic approximations for trajectory in neighborhood of junction point      72Ч76
Trajectory, asymptotic approximations on initial part of junction      60Ч62
Trajectory, asymptotic approximations on slow-motion parts of      48Ч52
Trajectory, asymptotic representation of, vs. actual      62Ч67
Trajectory, drop part of      see Drop part of trajectory
Trajectory, mapping of      11
Trajectory, of discontinuous system      29
Trajectory, on initial part of junction      60Ч62
Unstable part, defined      13 42
Vacuum tube oscillator, equation for      7
Van der PolТs equation      5Ч7 10 14
Van der PolТs equation, limit cycle for      35
Van der PolТs equation, periodic solutions and      168Ч170
Van der PolТs equation, phase portrait of      13
Zeroth approximation, defined      31Ч32
Zeroth approximation, discontinuous solution and      173Ч176
Zeroth approximation, of trajectory in neighborhood of junction point      75
Zeroth approximation, RiccatiТs equation and      68
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