Ordinary Differential Equations

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 equations 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