C2 FUNDAMENTAL THEORY OF DYNAMICAL SYSTEMS
EXAMPLE SHEET 1
The questions on this sheet are quite open ended; do as much of each as you feel is useful.
1.
Pick two or three systems (e.g. pendulum, electronic oscillator etc.). In each case decide what
the state variables are, describe the phase space and if possible write down an equation for the
dynamics.
2.
Consider the map xn+1 = λ x n where x n ∈R, and λ is some constant. Write down the general
solution for xn in terms of x0. Investigate the dependence of the behaviour of the map on the
value of λ (hint: the values λ = ±1 are important).
3.
Redo Q2 with the two parameter family of maps x n+1 = λ x n + µ , where the parameters λ,µ∈R.
Hint: if λ ≠ 1, consider the change of variables x(y) = y - µ(1-λ)-1; if λ =1, what happens?
4.
Consider the two dimensional map
xn+1
=
λxn
y n+1
=
µy n
Write down the general solution for (xn,y n ) in terms of (x0,y 0). Investigate the dynamics, and
sketch the phase portraits for the cases:
5.
i)
λ>µ>1
ii)
λ < µ < -1
iii)
λ=µ>1
iv)
λ > 1, 0 < µ < 1
v)
λ > 1, -1 < µ < 0
vi)
λ > 1, µ < -1
vii)
0 < λ < µ< 1
viii)
0 < λ = µ< 1
ix)
-1 < λ < µ < 0
viii)
λ = 1, 0 < µ < 1
Consider the differential equation
dx
dt
=
λx
where x∈R. Write down the general solution for x(t) in terms of x(0) and investigate the
dependence of the dynamics on λ.
6.
Consider the two dimensional differential equation
dx
dt
dy
dt
=
λx
=
µy
Write down the general solution for (x(t),y(t)) in terms of (x(0),y(0)) and investigate the
dependence of the dynamics on λ,µ (in a similar fashion to Q4).
C2 Exercise Sheet 1
7.
2
Consider the differential equation
dx
dt
xα
=
for α ≥ 1. For what times is the solution of this equation defined (as a function of α).
8.
Consider the differential equation:
dx
dt
=
F(x)
where x∈Rn. Let ϕ(x,t) be the solution of this with initial condition x at t = 0 (so that ϕ(x,0) = x).
Show that (hint: uniqueness of solutions)
ϕ(ϕ(x,s),t)
9.
ϕ(x,t+s)
=
Define ϕ(x,t) as in Q8. Let ψ(x,t) be the solution of
dx
dt
=
-F(x)
with initial condition x at t = 0. Show that
ϕ(ψ(x,t),t)
=
x
ψ(ϕ(x,t),t)
=
Hence deduce that if the time T map f is defined by f(x) = ϕ(x,T) for some T > 0, then the inverse
f -1 satisifies f -1(x) =ψ(x,T).
10. Rewrite the following as first order equations
i)
..
.
x + α(x2 - 1)x + x
=
0
Van der Pol oscillator
ii)
..
.
x + δx - x + x3
=
0
Duffing oscillator
11. Rewrite the following as first order autonomous equations
i)
..
.
x + α(x2 - 1)x + x
=
β cos ωt
Forced Van der Pol
ii)
..
.
x + δx - x + x3
=
β cos ωt
Forced Duffing
12. Consider the differential equation
dx
dt
dy
dt
=
λx - ωy
=
ωx + λy
By writing this in polar co-ordinates, or otherwise, compute the general solution for (x(t),y(t))
in terms of (x(0),y(0)). Plot phase portraits for λ < 0, λ = 0 and λ > 0. LetΣ = { (x,y) : x > 0, y = 0 }. Is
this a valid Poincaré section? What about Σ' = { (x,y) : x ≥ 0, y = 0 } and Σ'' = { (x,y) : x ≠ 0, y = 0 }.
For those sets which are Poincaré section, compute the corresponding Poincaré (first return)
map.