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What is Dynamics about? In broad terms, the goal of Dynamics is to
describe the long term evolution of systems for which an
"infinitesimal" evolution rule is known. Examples and applications
arise from all branches of science and technology, like physics,
chemistry, economics, ecology, communications, biology, computer
science, or meteorology, to mention just a few. These systems have
in common the fact that each possible state may be described by a
finite (or infinite) number of observable quantities, like
position, velocity, temperature, concentration, population density,
and the like. Thus, m the space of states (phase space) is a subset
M of an Euclidean space M . Usually, there are some constraints
between these quantities: for instance, for ideal gases pressure
times volume must be proportional to temperature. Then the space M
is often a manifold, an n-dimensional surface for some n < m.
For continuous time systems, the evolution rule may be a
differential eq- tion: to each state x G M one associates the speed
and direction in which the system is going to evolve from that
state. This corresponds to a vector field X(x) in the phase space.
Assuming the vector field is sufficiently regular, for instance
continuously differentiable, there exists a unique curve tangent to
X at every point and passing through x: we call it the orbit of x.
What is Dynamics about? In broad terms, the goal of Dynamics is to
describe the long term evolution of systems for which an
"infinitesimal" evolution rule is known. Examples and applications
arise from all branches of science and technology, like physics,
chemistry, economics, ecology, communications, biology, computer
science, or meteorology, to mention just a few. These systems have
in common the fact that each possible state may be described by a
finite (or infinite) number of observable quantities, like
position, velocity, temperature, concentration, population density,
and the like. Thus, m the space of states (phase space) is a subset
M of an Euclidean space M . Usually, there are some constraints
between these quantities: for instance, for ideal gases pressure
times volume must be proportional to temperature. Then the space M
is often a manifold, an n-dimensional surface for some n < m.
For continuous time systems, the evolution rule may be a
differential eq- tion: to each state x G M one associates the speed
and direction in which the system is going to evolve from that
state. This corresponds to a vector field X(x) in the phase space.
Assuming the vector field is sufficiently regular, for instance
continuously differentiable, there exists a unique curve tangent to
X at every point and passing through x: we call it the orbit of x.
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