Noise-driven dynamical systems - Noise-induced Phenomena in the Environmental Sciences - page 7

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Noise-driven dynamical systems

2.1 Introduction
We consider dynamical systems that can be represented through a stochastic differ-
ential equation in the form

d d t =

f ( φ ) + g ( φ ) ξ ( t ) , (2.1)

where ( t )

is a noise term accounting for the random external fluctuations forcing the dynamics
of

φ is the state variable, f ( φ )and g ( φ ) are deterministic functions of φ ,and ξ

.
The solution of Eq. ( 2.1 ) requires that the noise term

φ

( t ) be suitably specified.
The scope of this chapter is to describe the main features of the noise term and of
the resulting dynamics of

ξ

φ

in four cases, which are particularly interesting in the
environmental sciences: We model

ξ

( t ) as (i) dichotomous Markov noise (DMN),
(ii) white shot noise (WSN), (iii) white Gaussian noise, and (iv) Markovian colored
Gaussian noise. These representations of ( t ) are very well suited for investigating the

role of the random drivers typically found in the biogeosciences, and they are simple
enough to allow for the analytical (probabilistic) solution of Eq. ( 2.1 ).
We first consider (in Section 2.2 ) the case of dichotomous noise because it is more
general in that both WSN and white Gaussian noise can be obtained as limit cases of
the dichotomous noise. For this reason, these two white noises are described in detail
right after the case of DMN noise (i.e., in Sections 2.3 and 2.4 ); colored Gaussian
noise is presented in Section 2.5 .

ξ

2.2 Dichotomous noise
2.2.1 Definition and properties
The dichotomous Markov process is a stochastic process described by a state variable
ξ dn ( t ) that can take only two values, namely ξ = 1 and ξ = 2 , with transition dn dn

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