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raffoul y advanced differential equations 2022

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Textbook in PDF format Preliminaries and Banach spaces Preliminaries Escape velocity Applications to epidemics Metrics and Banach spaces Variation of parameters RC circuit Special differential equations Exercises Existence and uniqueness Existence and uniqueness of solutions Existence on Banach spaces Existence theorem for linear equations Continuation of solutions Dependence on initial conditions Exercises Systems of ordinary differential equations Existence and uniqueness x ′ = A(t)x Fundamental matrix x ′ = Ax Exponential matrix e^At x ′ = A(t)x + g(t) Discussion Exercises Stability of linear systems Deﬁnitions and examples x ′ = A(t)x Floquet theory Mathieu’s equation Applications to Mathieu’s equation Exercises Qualitative analysis of linear systems Preliminary theorems Near-constant systems Perturbed linear systems Autonomous systems in the plane Hamiltonian and gradient systems Exercises Nonlinear systems Bifurcations in scalar systems Stability of systems by linearization An SIR epidemic model Limit cycle Lotka–Volterra competition model Bifurcation in planar systems Manifolds and Hartman–Grobman theorem The stable manifold theorem Global manifolds Center manifold Center manifold and reduced systems Hartman–Grobman theorem Exercises Lyapunov functions Lyapunov method Stability of autonomous systems Time-varying systems; non-autonomous Global asymptotic stability Instability ω-limit set Connection between eigenvalues and Lyapunov functions Exponential stability Exercises Delay differential equations Introduction Method of steps Existence and uniqueness Stability using Lyapunov functions Stability using ﬁxed point theory Neutral differential equations Neutral Volterra integro-differential equations Exponential stability Existence of positive periodic solutions Exercises New variation of parameters Applications to ordinary differential equations Periodic solutions Applications to delay differential equations The main inversion Variable time delay Exercises Index

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raffoul y advanced differential equations 2022

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