The concept of a<unk>  has its origins in Newtonian mechanics.  There, as in other natural sciences and engineering disciplines, the evolution rule of<unk> s is given implicitly by a relation that gives the state of the system only a short time into the future.  (The relation is either a differential equation, difference equation or other  time scale.)  To determine the state for all future times requires iterating the relation many times—each advancing time a small step.  The iteration procedure is referred to as solving the system or integrating the system.  Once the system can be solved, given an initial point it is possible to determine all its future points, a collection known as a trajectory or orbit.
Before the advent of fast computing machines, solving a<unk>  required sophisticated mathematical techniques and could only be accomplished for a small class of<unk> s.  Numerical methods executed on computers have simplified the task of determining the orbits of a<unk> .
For simple<unk> s, knowing the trajectory is often sufficient, but most<unk> s are too complicated to be understood in terms of individual trajectories.  The difficulties arise because:
The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations.  The approximations used bring into question the validity or relevance of numerical solutions.  To address these questions several notions of stability have been introduced in the study of<unk> s, such as Lyapunov stability or structural stability.  The stability of the<unk>  implies that there is a class of models or initial conditions for which the trajectories would be equivalent.   The operation for comparing orbits to establish their equivalence changes with the different notions of stability.
The type of trajectory may be more important than one particular trajectory.  Some trajectories may be periodic, whereas others may wander through many different states of the system.   Applications often require enumerating these classes or maintaining the system within one class.  Classifying all possible trajectories has led to the qualitative study of<unk> s, that is, properties that do not change under coordinate changes.   Linear<unk> s and systems that have two numbers describing a state are examples of<unk> s where the possible classes of orbits are understood.
The behavior of trajectories as a function of a parameter may be what is needed for an application.   As a parameter is varied, the<unk> s may have bifurcation points where the qualitative behavior of the<unk>  changes.  For example, it may go from having only periodic motions to apparently erratic behavior, as in the transition to turbulence of a fluid.
The trajectories of the system may appear erratic, as if random.  In these cases it may be necessary to compute averages using one very long trajectory  or many different trajectories.  The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of<unk> s has helped establish the foundations of statistical mechanics and of chaos.
It was in the work of Poincaré that these<unk> s themes developed.
Dynamical system
Circulatory system
Haptic perception
Cardiac cycle
Gastrointestinal physiology
Coagulation
Visual perception
Autonomic nervous system