Flashcards

B5.6

Ben Tidswell Part B uni Inglês 23 flashcards

Flashcards do conjunto (23)

1Suppose E⊂R^2 is an open subset of the plane and 𝐟∈C^1(E). State the Poincaré-Bendixson theorem
2Let E be an open subset of R^n containing 𝐱_0 and let φ_t:E->E be the the flow of ẋ= 𝐟(𝐱). State the Stable manifold theorem
3Let φ_t:E⊆R^n->E be the flow of ẋ= 𝐟(𝐱) with fixed point 𝐱_0 where 𝐟∈C^r(E). State the Centre manifold theorem
4Let (𝐱_0,𝐲_0) be close enough to the origin and (𝐱(t),𝐲(t)) be the solution of: ẋ=A𝐱+𝐟(𝐱,𝐡(𝐱)), ẏ=D𝐡(𝐱)ẋ=B𝐡(𝐱)+𝐠(𝐱,𝐡(𝐱)), starting at (𝐱_0,𝐲_0). State the Shadowing theorem
5For E an open set of R^m containing 𝐱_0 consider the iterative map 𝐱_(n+1)=𝐆(𝐱_n) with fixed point 𝐱_0, where 𝐆:E->E and 𝐆^-1 exists on E. State the Stable and unstable manifold theorem
6Let P:R^n->R^n be a diffeomorphism s.t. P has a hyperbolic fixed point of saddle type, 𝐩, and a transverse homoclinic point 𝐪∈W^s(𝐩)∩W^u(𝐩). State The Smale-Birkhoff Homoclinic Theorem.
7State Melnikov's Theorem ('63) including any assumptions
8State the Taylor expansion for a function f(x) about a point x=a
9For the system ẍ=-∇U(𝐱(t)), state what is meant by a first integral and give an example of one
10Describe the Poincaré-Lindstedt method to approximate behaviour of weakly nonlinear oscillators ẍ+x=εf(x)
11Let Λ be an invariant compact set for an invertible iterative map f:ℳ->ℳ. Define what it means for f to be chaotic.
12State the key 3 properties of the shift map
13Consider the 1D map: x_n+1 = f(x_n) State how to determine the stability of a p-periodic orbit {x_1,...,x_p} $Consider the 1D map: x_n+1 = f(x_n) State how to determine the stability of a p-periodic orbit {x_1,...,x_p}$
14For ẋ=f(x,μ) state the 4 main types of one-dimensional one-parameter bifurcations and provide a diagram(subcritical and supercritical should switch)
15State the two generic cases for the matrix D_(𝘅)𝗳 at the bifurcation point (𝘅(μ_0),μ_0) and name the bifurcations that occur
16Suppose the 2D system: ẋ=f(x,y;μ), ẏ=g(x,y;μ) has a fixed point (x_0,y_0) with a pair of complex eigenvalues that cross the imaginary axis Re(λ)=0 at the bifurcation point μ=μ_0. State the steps to convert the system into Normal form
17State all the standard and hyperbolic trig identities of the form (trig function)^2±(other trig function)^2=1
18Define what it means for a fixed point to be hyperbolic for a system ẋ=𝗳(𝘅)
19When does a bifurcation occur at a fixed point 𝘅_0 for the mapping 𝘅_(n+1)=𝗚(𝘅_n)
20Consider ẋ=𝗳(𝘅) with 𝘅∈E⊆ℝ^n, 𝗳∈C^1(E) with a fixed point 𝘅_0. State the theorem regarding a Lyapunov function V and its implications on the stability of 𝘅_0
21State the standard forms of conical sections
22Consider the general one-parameter system ż=𝗙(𝘇,μ̃) and assume that for μ̃=μ̃_c, there is a non-hyperbolic fixed point 𝘇_c such that the matrix M=D𝗙(𝘇_c) has at least one eigenvalue with zero real part. Describe the steps to find the solution on the extended centre manifold
23State equivalent expressions to cos(x±y) and sin(x±y)