Flashcards
B5.6
Cards in this deck (23)
- 1Suppose EβR^2 is an open subset of the plane and πβC^1(E). State the PoincarΓ©-Bendixson theorem
- 2Let E be an open subset of R^n containing π±_0 and let Ο_t:E->E be the the flow of αΊ= π(π±). State the Stable manifold theorem
- 3Let Ο_t:EβR^n->E be the flow of αΊ= π(π±) with fixed point π±_0 where πβC^r(E). State the Centre manifold theorem
- 4Let (π±_0,π²_0) be close enough to the origin and (π±(t),π²(t)) be the solution of: αΊ=Aπ±+π(π±,π‘(π±)), αΊ=Dπ‘(π±)αΊ=Bπ‘(π±)+π (π±,π‘(π±)), starting at (π±_0,π²_0). State the Shadowing theorem
- 5For E an open set of R^m containing π±_0 consider the iterative map π±_(n+1)=π(π±_n) with fixed point π±_0, where π:E->E and π^-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, π©, and a transverse homoclinic point πͺβW^s(π©)β©W^u(π©). 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 αΊ=-βU(π±(t)), state what is meant by a first integral and give an example of one
- 10Describe the PoincarΓ©-Lindstedt method to approximate behaviour of weakly nonlinear oscillators αΊ+x=Ξ΅f(x)
- 11Let Ξ be an invariant compact set for an invertible iterative map f:β³->β³. 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}
- 14For αΊ=f(x,ΞΌ) 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_(π
)π³ at the bifurcation point (π
(ΞΌ_0),ΞΌ_0) and name the bifurcations that occur
- 16Suppose the 2D system: αΊ=f(x,y;ΞΌ), αΊ=g(x,y;ΞΌ) has a fixed point (x_0,y_0) with a pair of complex eigenvalues that cross the imaginary axis Re(Ξ»)=0 at the bifurcation point ΞΌ=ΞΌ_0. State the steps to convert the system into Normal form
- 17State all the standard and hyperbolic trig identities of the form (trig function)^2Β±(other trig function)^2=1
- 18Define what it means for a fixed point to be hyperbolic for a system αΊ=π³(π
)
- 19When does a bifurcation occur at a fixed point π
_0 for the mapping π
_(n+1)=π(π
_n)
- 20Consider αΊ=π³(π
) with π
βEββ^n, π³βC^1(E) with a fixed point π
_0. State the theorem regarding a Lyapunov function V and its implications on the stability of π
_0
- 21State the standard forms of conical sections
- 22Consider the general one-parameter system ΕΌ=π(π,ΞΌΜ) and assume that for ΞΌΜ=ΞΌΜ_c, there is a non-hyperbolic fixed point π_c such that the matrix M=Dπ(π_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(xΒ±y) and sin(xΒ±y)
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- uni Part B Flashcards on B5.6, created by Ben Tidswell on 28/03/2023.
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- Ben Tidswell
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