My title - Departamento de Matemática da Universidade do Minho

More documents

Recommendations

Info

6 FLOWS 50 Theorem (smooth dependence on parameters). Let v(t, x, λ) be a family of vector fields defined on some domain of the extended phase space D ⊂ R × X depending on a parameter λ ∈ Λ ⊂ R. If v is of class C k with k ≥ 1, then in some neighborhood of any (t 0 , x 0 , λ 0 ) ∈ D × Λ the local solutions of ẋ = v(t, x, λ) with initial condition x(t 0 ) = x 0 are differentiable (indeed C k ) functions of (t, x, λ). Proof. See [BN05] Warning. Continuous dependence does not exclude sensitive dependence on both initial conditions and parameters, even in the linear case! For example, the distance between solutions of ẋ = µx with different x(0) and/or µ may diverge for large time ...
7 OSCILLATIONS AND CYCLES 51 7 Oscillations and cycles 7.1 Harmonic oscillator Harmonic oscillator. equation The harmonic oscillator is the (phenomenon modeled by the) Newton ¨q = −ω 2 q . This is a quite universal equation, since it describes small oscillations around a “generic” stable equilibrium of any one-dimensional Newtonian system 16 (indeed, take a Newton equation mẍ = −dU ′ (x) of a particle in a potential field U. An equilibrium position is a zero of the force, i.e. a point x 0 where U ′ (x 0 ) = 0. It is “stable” if x 0 is a local minimum of the potential, so that the Taylor expansion of the potential around x 0 in powers of q = x − x 0 starts with U(x) = α + 1 2 βq2 + . . . , for some positive second derivative U ′′ (x 0 ) = β. If we are only interested in small displacements of x around x 0 , we can safely disregard high order terms and approximate the Newton equation as m¨q ≃ −βq, which is an harmonic oscillator with resonant frequency ω = √ β/m). Call p = ˙q the momentum. The Newton equation ¨q = −ω 2 q is equivalent to Hamilton’s first order equations ˙q = p ṗ = −ω 2 q . If we define the complex variable z = ωq + i ˙q, Newton equation then takes the form of a first order linear equation in the complex line, namely ż = −iωz, whose solution is z (t) = e −iωt z (0). In terms of the original (physical) variables, the solutions read periodicity of exp q (t) = q 0 cos (ωt) + v 0 ω sin (ωt) = A sin(ωt + φ) where the amplitude A and the initial phase φ depend on the initial conditions q(0) = q 0 and ˙q(0) = v 0 . So, all trajectories are periodic with common period 2π/ω, and orbits are ellipses in the q- ˙q plane, determined by the conserved energy E = 1 2 ( ˙q 2 + ω 2 q 2) = ω 2 A 2 . Harmonic oscillator, phase curves and time series. 16 “The harmonic oscillator, which we are about to study, has close analogs in many other fields; although we start with a mechanical example of a weight on a spring, or a pendulum with a small swing, or certain other mechanical devices, we are really studying a certain differential equation. This equation appears again and again in physics and other sciences, and in fact is a part of so many phenomena that its close study is well worth our while. Some of the phenomena involving this equation are the oscillations os a mass on a spring; the oscillations of charge flowing back and forth in an electrical circuit; the vibrations of a tuning fork which is generating sound waves; the analogous vibrations of the electrons in an atom, which generate light waves; the equations for the operation of a servosystem, such as a thermostat trying to adjust a temperature; complicated interactions in chemical reactions; the growth of a colony of bacteria in interaction with the food supply and the poison the bacteria produce; foxes eating rabbits eating grass, and so on; ...” Richard P. Feynman [Fe63]
Page 1 and 2: 1405R1 - Tópicos de Sistemas Dinâ
Page 3 and 4: CONTEÚDO 3 9 Transversalidade e bi
Page 5 and 6: 1 INTRODUÇÃO 5 Hidrodinâmica. O
Page 7 and 8: 2 ITERATION/RECURSION 7 2 Iteration
Page 9 and 10: 2 ITERATION/RECURSION 9 do primeiro
Page 11 and 12: 2 ITERATION/RECURSION 11 If a is th
Page 13 and 14: 2 ITERATION/RECURSION 13 Nice pictu
Page 15 and 16: 2 ITERATION/RECURSION 15 Experiênc
Page 17 and 18: 3 SISTEMAS DINÂMICOS TOPOLÓGICOS,
Page 19 and 20: 3 SISTEMAS DINÂMICOS TOPOLÓGICOS,
Page 21 and 22: 4 NÚMEROS E DINÂMICA 21 4 Número
Page 23 and 24: 4 NÚMEROS E DINÂMICA 23 4.2 Deslo
Page 25 and 26: 4 NÚMEROS E DINÂMICA 25 Rotaçõe
Page 27 and 28: 4 NÚMEROS E DINÂMICA 27 Observe q
Page 29 and 30: 4 NÚMEROS E DINÂMICA 29 Frações
Page 31 and 32: 5 ÓRBITAS REGULARES E PERTURBAÇÕ
Page 39 and 40: 6 FLOWS 39 6 Flows “... forse sti
Page 41 and 42: 6 FLOWS 41 6.2 Integration of one-d
Page 43 and 44: 6 FLOWS 43 Counter-example. Both th
Page 45 and 46: 6 FLOWS 45 which undergoes a catast
Page 47 and 48: 6 FLOWS 47 Uniqueness of solutons.
Page 49: 6 FLOWS 49 Dependence on initial da
Page 53 and 54: 7 OSCILLATIONS AND CYCLES 53 Phase
Page 55 and 56: 7 OSCILLATIONS AND CYCLES 55 The ge
Page 57 and 58: 7 OSCILLATIONS AND CYCLES 57 para a
Page 59 and 60: 8 LINEARIZAÇÃO 59 8 Linearizaçã
Page 61 and 62: 9 TRANSVERSALIDADE E BIFURCAÇÕES
Page 63 and 64: 10 STATISTICAL DESCRIPTION OF ORBIT
Page 75 and 76: 11 RECORRÊNCIAS 75 Exercícios.
Page 77 and 78: 11 RECORRÊNCIAS 77 O conjunto não
Page 79 and 80: 12 TRANSITIVIDADE E ÓRBITAS DENSAS
Page 81 and 82: 12 TRANSITIVIDADE E ÓRBITAS DENSAS
Page 83 and 84: 13 HOMEOMORFISMOS DO CÍRCULO 83 13
Page 85 and 86: 13 HOMEOMORFISMOS DO CÍRCULO 85 de
Page 87 and 88: 14 PERDA DE MEMÓRIA E INDEPENDÊNC
Page 97 and 98: 15 DIMENSIONS, FRACTALS AND ENTROPY
Page 99 and 100: 15 DIMENSIONS, FRACTALS AND ENTROPY
Page 101 and 102:
15 DIMENSIONS, FRACTALS AND ENTROPY
Page 103 and 104:
16 ERGODICITY AND CONVERGENCE OF TI
Page 105 and 106:
16 ERGODICITY AND CONVERGENCE OF TI
Page 107 and 108:
REFERÊNCIAS 107 Referências [AA67
show all

My title - Departamento de Matemática da Universidade do Minho

Create successful ePaper yourself

Delete template?

Save as template?