MATHEMATICAL PRELIMINARIES |
Banach spaces and examples |
Linear transformations |
Fixed point theorems |
GENERAL PROPERTIES OF DIFFERENTIAL EQUATIONS |
Existence |
Continuation of solutions |
Uniqueness and continuity properties |
Continuous dependence and stability |
Extension of the concept of a differential equation |
Differential inequalities |
Autonomous systems-generalities |
Autonomous systems-limit sets, invariant sets |
Remarks and suggestions for further study |
TWO DIMENSIONAL SYSTEMS |
Planar two dimensional systems-the Poincaré-Bendixson theory |
Differential systems on a torus |
Remarks and suggestions for further study |
LINEAR SYSTEMS AND LINEARIZATION |
General linear systems |
Stability of linear and perturbed linear systems |
nth Order scalar equations |
Linear systems with constant coefficients |
Two dimensional linear autonomous systems |
The saddle point property |
Linear periodic systems |
Hill’s equation |
Reciprocal systems |
Canonical systems |
Remarks and suggestion for further study |
PERTURBATION OF NONCRITICAL LINEAR SYSTEMS |
Nonhomogeneous linear systems |
Weakly nonlinear equations-noncritical case |
The general saddle point property |
More general systems |
The Duffing equation with large damping and large forcing |
Remarks and extensions |
SIMPLE OSCILLATORY PHENOMENA AND THE METHOD OF AVERAGING |
Conservative systems |
Nonconservative second order equations-limit cycles |
Averaging |
The forced van der Pol equation |
Duffing’s equation with small damping and small harmonic forcing |
The subharmonic of order 3 for Duffing’s equation |
Damped excited pendulum with oscillating support |
Exercises |
Remarks and suggestions for further study |
BEHAVIOR NEAR A PERIODIC ORBIT |
Stability of a periodic orbit |
Sufficient conditions for orbital stability in two dimensions |
Autonomous perturbations |
Remarks and suggestions for further study |
INTEGRAL MANIFOLDS OF EQUATIONS WITH A SMALL PARAMETER |
Methods of determining integral manifolds |
Statement of results |
A “nonhomgeneous linear” system |
The mapping principle |
Proof of Theorem 2.1 |
Stability of the perturbed manifold |
Applications |
Exercises |
Remarks and suggestions for further study |
PERIODIC SYSTEMS WITH A SMALL PARAMETER |
A special system of equations |
Almost linear systems |
Periodic solutions of perturbed autonomous equa |
Remarks and suggestions for further study |
ALTERNATIVE PROBLEMS FOR THE SOLUTION OF FUNCTIONAL EQUATIONS |
Equivalent equations |
A generalization |
Alternative problems |
Alternative problems for periodic solutions |
The Perron-Lettenmeyer theorem |
Remarks and suggestions for further study |
THE DIRECT METHOD OF LIAPUNOV |
Sufficient conditions for stability and instability in autonomous systems |
Circuits containing Esaki diodes |
Sufficient conditions for stability in nonautonomous systems |
The converse theorems for asymptotic stability |
Implications of asymptotic stability |
Wazewski’s principle |
Remarks and suggestions for further study |
APPENDIX |
ALMOST PERIODIC FUNCTIONS |
REFERENCES |
INDEX |