Elements of Lyapunov Theory - Nonlinear Oligopolies: Stability and Bifurcations - page 272

Appendix A
Elements of Lyapunov Theory

Consider a time-invariant nonlinear dynamical system

x .t C 1/ D g . x .t// (A.1)

or

x .t/ D g . x .t//; (A.2)

n with n . It is also assumed that g is continuous

where g W D 7! R D being a set in R

on D , and starting with arbitrary initial state x .0/ 2 D , the unique solution of (A.1)

or (A.2) exists for all t>0and remains in is an equilibrium of

system (A.1) if and only if x D g . x /, and it is an equilibrium of system (A.2) if and
only if g . x / D 0 . If in any time period x .t/ becomes x , then the state remains at the
equilibrium for all future time periods. Therefore equilibria of a dynamical system
are sometimes called the steady states of the system. If x .0/ is selected nearby an
equilibrium, then the state might go away from the equilibrium, it might stay close
to the equilibrium for all future times or it even might converge to the equilibrium
as t !1 : In all cases distances between state vectors have to be defined in order
to decide if a state vector is close to the equilibrium or not. The distance between
any two vectors is usually defined as the norm of their difference. The norm is a
mathematical way to characterize the lengths of real vectors.
A norm k : k in the n-dimensional vector space

D . A vector x 2 D

n is a real valued function defined
on all n-element vectors such that the following conditions are satisfied:

R

n ; and k v kD 0 if and only if v D 0

1. k v k 0 for all v 2 R

n

2. k ˛ v kDj ˛ jk v k for all v 2 R and real numbers ˛

n :

3. k v C w kk v kCk w k for all v ; w 2 R

It is easy to prove that all vector norms are continuous functions, that is, k v k is
continuous in v . The proof is based on conditions (2) and (3), since

k v kk v w kCk w k

and

k w kk w v kCk v kDk v w kCk v k ;

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