Stochastics Group Dylan O’Connell, Lance Ford, Derek Kielty Rajeshwari Majumdar, Heather McCain

Stochastics Group
Dylan O’Connell, Lance Ford, Derek Kielty
Rajeshwari Majumdar, Heather McCain
UCONN Math REU
June 4, 2015
ż = z n ODE
We begin our study of stochastic processes by considering the following
Ordinary Differential Equation (ODE)
dz
= zn
dt
∀n ≥ 2, z ∈ C
Stability
Definition
An ODE with initial condition z|t=0 = z0 explodes if the ODE solution
approaches infinity in finite time, that is
lim ||z(t)|| = ∞
t→t 0
for some t 0 > 0.
Stability of z 2
dz
= z 2 , z|t=0 = z0
dt Z
Z t
t
dz
→
=
dt
2
0 z
0
1
1
→
− =t
z0 z
1
→ z(t) = 1
z0 − t
Thus z(t) will approach infinity in finite time when z0 lies on the positive
real axis.
Phase Portraits
z 2 phase portrait
Im z
100
50
0
Re z
-50
-100
-100
-50
0
50
100
Phase Portraits
z 3 phase portrait
Im z
100
50
0
Re z
-50
-100
-100
-50
0
50
100
Phase Portraits
z 4 phase portrait
Im z
100
50
0
Re z
-50
-100
-100
-50
0
50
100
Why add noise?
Can push an unstable trajectory back to a stable region
What is Brownian Motion?
In 1827, botanist Robert Brown noticed pollen and other materials
exhibiting random motion
Physicists ran with the idea, and Einstein examined the path taken by
atoms
Helped prove the existence of atoms
Known in math as the Wiener Process
i.e., random walk
What is it as a DE term?
Brownian motion is considered a stochastic process which turns an ODE
into a Stochastic Differential Equation (SDE)
dz(t) = (z(t))n dt + k1 dW (1) (t) + k2 dW (2) (t)
”The infinitesimals k1 dW (1) (t) and k2 dW (2) (t) thus represent
independent “kicks” in the directions of k1 and k2 , respectively.”
–David P. Herzog
Kappa’s in a Nutshell
Kappa’s determine the “direction of the noise”
Linear independence means that there is some component of noise in
any direction
Tr a j e c t or y of ż = z 3
0.1
0
−0.1
Im(z )
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
0
5
10
15
20
25
30
35
40
45
50
Numerically solving the stochastic differential equations
We use Euler’s Method to numerically solve differential equations.
Numerically solving the stochastic differential equations
We break z n into its real and imaginary parts
dx
= f (x, y ) real
dt
dy
= g (x, y ) imaginary
dt
For z 2 we get
z 2 = (x + iy )2
z 2 = x 2 + 2xyi − y 2
z 2 = x 2 − y 2 + 2xyi
Which gives us that
dx
= x 2 − y 2 real
dt
dy
= 2xy
imaginary
dt
Numerically solving the stochastic differential equations
This is what our equations will look in MatLab
xk+1 = xk + f (xk , yk )dt + Re(K1 )db + Re(K2 )dw
yk+1 = xk + g (xk , yk )dt + Im(K1 )db + Im(K2 )dw
Where db and dw are defined as
√
db = ∆t ∗ (a random value from the standard normal distribution)
√
dw = ∆t ∗ (a random value from the standard normal distribution)
If we were looking at z 2 we would get
xk+1 = xk + (x 2 − y 2 ) dt + Re(K1 )db + Re(K2 )dw
yk+1 = xk + 2xy dt + Im(K1 )db + Im(K2 )dw
Numerically solving the stochastic differential equations
A choice of two Kappa’s will give a positive probability that the system
will stabilize if:
k1 and k2 are linearly independent, or
{k1n−1 , k2n−1 } contains a complex number.
Numerically solving the stochastic differential equations
Figure: Mathematica
Figure: MatLab
xk+1 = xk + (x 3 − 3xy 2 ) dt + Re(K1 )db + Re(K2 )dw
yk+1 = xk + (3x 2 y − y 3 ) dt + Im(K1 )db + Im(K2 )dw