Nonlinear Differential Equations and Dynamical SystemsSpringer Science & Business Media, 6 dec 2012 - 277 pagina's On the subject of differential equations a great many elementary books have been written. This book bridges the gap between elementary courses and the research literature. The basic concepts necessary to study differential equations - critical points and equilibrium, periodic solutions, invariant sets and invariant manifolds - are discussed. Stability theory is developed starting with linearisation methods going back to Lyapunov and Poincaré. The global direct method is then discussed. To obtain more quantitative information the Poincaré-Lindstedt method is introduced to approximate periodic solutions while at the same time proving existence by the implicit function theorem. The method of averaging is introduced as a general approximation-normalisation method. The last four chapters introduce the reader to relaxation oscillations, bifurcation theory, centre manifolds, chaos in mappings and differential equations, Hamiltonian systems (recurrence, invariant tori, periodic solutions). The book presents the subject material from both the qualitative and the quantitative point of view. There are many examples to illustrate the theory and the reader should be able to start doing research after studying this book. |
Inhoudsopgave
1 | |
6 | |
7 | |
2 | 14 |
5 | 22 |
Periodic solutions | 39 |
Stability analysis by the direct method | 101 |
Introduction to perturbation theory | 117 |
The method of averaging | 145 |
Introduction to the theory of stability | 157 |
Relaxation oscillations | 177 |
Bifurcation theory | 183 |
Chaos | 204 |
Hamiltonian systems | 218 |
The Morse lemma | 237 |
References | 271 |
Overige edities - Alles bekijken
Nonlinear Differential Equations and Dynamical Systems Ferdinand Verhulst Gedeeltelijke weergave - 2012 |
Nonlinear Differential Equations and Dynamical Systems Ferdinand Verhulst Gedeeltelijke weergave - 2006 |
Nonlinear Differential Equations and Dynamical Systems: With 107 Figures Ferdinand Verhulst Geen voorbeeld beschikbaar - 1989 |
Veelvoorkomende woorden en zinsdelen
A₁ apply approximation asymptotically stable autonomous equation averaging behaviour bifurcation called centre manifold chapter characterise characteristic exponents closed orbit coefficients Consider equation Consider the equation Consider the system convergent corresponding cos(t cos² critical point Determine differential equations domain eigenvalues equation 6.9 equation ï equilibrium solution example exists expansion Figure fixed point follows fundamental matrix Hamiltonian systems harmonic oscillator initial value problem integral lemma Liénard equation lim x(t limit cycle linear analysis linearisation Lipschitz-continuous Lyapunov mapping n-matrix negative neighbourhood of 0,0 periodic solution perturbation phase phase-flow phase-plane phase-space Poincaré Poincaré-Bendixson theorem Poincaré-Lindstedt method Poincaré-mapping positive attractor produces resonance manifold result righthand side saddle saddle point sin(t sin² sint solution of equation solution x(t T-periodic t₁ theory time-scale 1/ɛ transformation transversal trivial solution unstable manifolds variable Volterra-Lotka equations w-limitset x(to zero