Qualitative investigations of planetary motions were investigated intensively by BRAHE, KEPLER and GALILEO . Brahe spent 20 years making accurate recordings of planetary motions. Kepler became Brahe's assistant and was given the particular task to calculate the orbit of Mars. No circular orbits (Copernican or the Ptolemaic theory of epicycles) could fit well the data. After years (Brahe had died) there were still big discrepancies of 8 arc minutes = a dinner plate looked at from 100 yards). Kepler, convinced that Brahe did not make any mistake, did more calculations (900 pages during 6 years) and finally came up with his new model.
Later came the fruitful development of calculus and calculus of variations with applications to (especially celectial) mechanics by NEWTON , EULER , LAGRANGE , LAPLACE and HAMILTON.
Question of stability were investigated by LAPLACE, LAGRANGE, POISSON and DIRICHLET and HARETU. All of these people claimed having proven the stability of the solar system, a question which is open until now.
The king of Sweden offered a prize for the discovery of the proof (of stability). POINCARE got this prize for a paper proving that the series expansion (leading to stability) does not converge. POINCARE started a qualitative area leading to the development of modern differential topology and geometry.
The new qualitative approach lead to existence results of stable periodic and quasiperiodic solutions of differential equations like the solar system, particles in a dipol field (responsible for the northern lights) or the heavy top. Contributions of mathematicians like BIRKHOFF, LIAPUNOV, SUNDMAN, SIEGEL, KOLMOGOROV , ARNOLD, MOSER, SMALE were important.
Already POINCARE had found evidence for chaotic motion and mathematicians like JULIA , FATOU, BROLIN had discovered such chaotic behaviour while iterating complex maps at the beginning and in the mid's of the century. Still, the importance was hidden.
Experiments of LORENTZ (a meterologist), HENON (an astronomer), MAY (a biologist), FEIGENBAUM (a physisist) and others brought chaotic motion and so dynamics into wider attention.
The work of RUELLE, TAKENS, BOWEN, LANFORD or SINAI clarified the mathematics behind some of the stories and some deep connections between statistical mechanics and dynamics were revield. The two topics are now closely intermingeled.
Already CANTOR , BESICOVICH or HAUSDORFF had devolped a theory of fractals. More work by ROGERS, TAYLOR, HUTCHINSON in the fifties and sixties. The book of MANDELBROT lead to an explosion of interest in this subject not at least because computers were awailable, which made experiments accessible to more mathematitians. Especially complex dynamics which was pioneered by DOUADY and HUBBARD, got a boost through the beautyful pictures. The interest will certainly continue until the big open question whether the Mandelbrot set is locally connected is answered.
At the other end of chaos are located solvable, so called integrable systems. An infinite dimensional system, the Korteweg de Vries equation, which was already studied 100 years ago is an example of such a system. Some integrable systems have special solutions so called solitons, which raized for some time hope to be candidates for models of elmentary particles. The interest in nonlinear higher dimensional systems raised seriously in the 50's, when FERMI, PASTA and ULAM studied the problem of nonlinear lattices numerically. These experiments were amoung the first computer experiments. Fermi,Pasta and Ulam experimented with quadratic, cubic and piecewise linear functions $f$ and $N=32$ or $N=64$ particles. One expected at that time that such a system would evolve chaotically and one was surprised that recurrence phenomena occured. It was still more surprising that one would get an integrable system when $f$ is an exponential function which gives the Toda system. The recurrence discovery for the quadratic, cubic lattice is today partially explained by the fact that one is not too far from an integrable sytem. More numerical computations, this time from PdE's by KRUSKAL, ZABUSKY, GARDNER, GREENE and MIURA. More recent theortical works of TODA, MOSER, FLASCHKA, KAC, MOERBEKE, FADDEV and others were influencial and revield beautyful connections with the theory of Lie algebras or with algebraic geometry and spectral theory of Schroedinger operators.
The interest of dynamical systems like cellular automata or coupled map lattices was partly motivated by models in statistical mechanics and fluid mechanics. Work of BUNIMOVICH, SINAI and KANEKO on coupled map lattices. Cellular automata were studied by ULAM, VON NEUMANN or (more mathematical) by HEDLUND.
Connections of chaos appear also with other fields like information theory (closely related with ergodic theory), biology (a rich source of models), cryptology (treating random number generators or encryption schemes as dynamical systems), geometry (the dynamics of geodesics) or quantum mechanics (the analysis of Schroedinger operators and the dynamics defined by the Schroedinger equation).
