Dissipative and Conservative

  The local rate of expansion or contraction of a dynamical system can be calculated directly from the vector field or difference equation without explicitly finding any solutions. We say a system is conservative  if the absolute value of the Jacobian of its map exactly equals one, or if the divergence of its vector field equals zero,

 

for all times and all points. A physical system is dissipative  if it is not conservative. Most of the physical examples studied in this book are dissipative dynamical systems. The phase space of a dissipative dynamical system is continually shrinking onto a smaller region of phase space called the attracting set.