DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

non-linear and non-stationarity. New tools and techniques are required for their analysis such that
the structure, function and growth of networks can be considered simultaneously. In this regard
we discuss “Network Dynamics” in Section 9, which deals with such issues and can be used to
study the structure of supply chain and its implication for its functionality. Understanding the
behavior of large complex networks is the next logical step for the field of nonlinear dynamics,
because they are so pervasive in the real world. We begin with a brief introduction to dynamical
systems theory, in particular nonlinear dynamics in next section.
5 Dynamical Systems Theory
Many physical systems that produce continuous-time response can be modeled by a set of
differential equations of the form:
dy = f ( y,
a)
, (I)
dt
where, y = y ( t),
y ( t),......
y ( )) represents the state of the system and may be thought of as a
(
1 2
n
t
point in a suitably defined space S-which is known as phase space and
a = a ( t),
a ( t)
L,
a ( )) is a parameter vector. The dimensionality of S is the number of
(
1 2
m
t
apriori degrees of freedom in the system. The vector field f(y,a) is in general a non-linear operator
acting on points in S. If f(y,a) is locally Lipschtiz, above equation defines an initial value problem
in the sense that a unique solution curve passes through each point y in the phase space. Formally
we may write the solution at time t given an initial value y0 as y( t)
= ϕ
t
y0
. ϕ
t
represents a oneparameter
family of maps of the phase space into itself. We can perceive the solutions to all
possible initial value problems for the system by writing them collectively as ϕ . This may be
thought of as a flow of points in the phase space. Initially the dimension of the set ϕ t
S will be
that of S itself. As the system evolves, however, it is generally the case for the so-called
dissipative system that the flow contracts onto a set of lower dimension known as attractor. The
attractors can vary from simple stationary, limit cycle, quasi-periodic to complicated chaotic ones
(Strogatz 1994, Ott 1996). The nature of attractor changes as parameters (a) are varied, a
phenomena studied in bifurcation analysis. Typically a nonlinear system is always chaotic for
some range of parameters. Chaotic attractors have a structure that is not simple; they are often not
smooth manifolds, and frequently have a highly fractured structure, which is popularly referred to
as Fractals (self–similar geometrical objects having structure at every scale). On this attractor,
stretching and folding characterize the dynamics; the former phenomenon causes the divergence
of nearby trajectories and latter constraints the dynamics to finite region of the state space. This
accounts for fractal structure of attractors and the extreme sensitivity to changes in initial
conditions, which is hallmark of chaotic behavior. System under chaos is unstable everywhere
never settling down, producing irregular and aperiodic behavior which leads to a continuous
broadband spectrum. While this feature can be used to distinguish chaotic behavior from
stationary, limit cycle, quasi-periodic motions using standard Fourier Analysis it makes it
difficult to separate it from noise which also has a broadband spectrum. It is this “deterministic
randomness” of chaotic behavior, which makes standard linear modeling and prediction
techniques unsuitable for analysis.
5.1 Nonlinear Models for Supply Chain
Understanding the complex interdependencies, effects of priority, nonlinearities, delays,
uncertainties and competition/cooperation for resource sharing is fundamental for prediction and
control of supply chains. System dynamics approach often leads to models of supply chains,
which can be described in the form of equation (I). Dynamical systems theory provides a
powerful framework for rigorous analysis of such models and thus can be used to supplement the
S t