(no subject)
The behavior of many non-linear dynamical systems as a function of time is found to
be chaotic. The characteristic feature of chaos is that the system never repeats its past
behavior. Chaotic systems nonetheless obey classical laws of motion which means that the
equations of motion are deterministic.
PoincarÈ was the first to study the effects of small changes in the initial conditions
on the evolution of chaotic systems that obey non-linear equations of motion. In a chaotic
system, the erratic behavior is due to the internal, or intrinsic, dynamics of the system.
Let a dynamical system be described by a set of first-order differential equations:
dx/dt = f (x,x,x,...x ) (11.1)
1 1 1 2 3 n
dx/dt= f(x,x,x,...x )
2 2 1 2 3 n
. .
. .
dx /dt= f (x,x,x,...x )
n n 1 2 3 n
where the functions f are functions of n-variables.
n
The necessary conditions for chaotic motion of the system are
1) the equations of motion must contain a non-linear term that couples several of the
variables.
A typical non-linear equation, in which two of the variables are coupled, is therefore dx/dt= ax + bx + cxx + ... rx , (a, d, c, ...r are constants)
1 1 2 1 2 n
and
2) the number of independent variables, n, must be at least three.
be chaotic. The characteristic feature of chaos is that the system never repeats its past
behavior. Chaotic systems nonetheless obey classical laws of motion which means that the
equations of motion are deterministic.
PoincarÈ was the first to study the effects of small changes in the initial conditions
on the evolution of chaotic systems that obey non-linear equations of motion. In a chaotic
system, the erratic behavior is due to the internal, or intrinsic, dynamics of the system.
Let a dynamical system be described by a set of first-order differential equations:
dx/dt = f (x,x,x,...x ) (11.1)
1 1 1 2 3 n
dx/dt= f(x,x,x,...x )
2 2 1 2 3 n
. .
. .
dx /dt= f (x,x,x,...x )
n n 1 2 3 n
where the functions f are functions of n-variables.
n
The necessary conditions for chaotic motion of the system are
1) the equations of motion must contain a non-linear term that couples several of the
variables.
A typical non-linear equation, in which two of the variables are coupled, is therefore dx/dt= ax + bx + cxx + ... rx , (a, d, c, ...r are constants)
1 1 2 1 2 n
and
2) the number of independent variables, n, must be at least three.