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Complex Systems: Dynamical Systems & Chaos


The book by Gleick is a classic early popularization.  The others are good introductions to the topic.


The expression “dynamical systems” in the literature refers often to both real systems (physical, chemical, biological, etc.) that follow a well-defined evolution in time (a trajectory) and also to the systems of differential equations that are used to model them. Complexity studies are often led to consider dynamical systems that are nonlinear, chaotic or sometimes indeterministic.

Linear systems are described by equations that comply with the principle of linear superposition: if X1 and X2 are both solutions of an equation, then X1 + X2 is also a solution of that equation. If this is not the case the equations (or the system they represent) are said to be nonlinear. If a linear equation represents the functional dependency of a variable x on another variable y, in general if x is increased by a constant factor the variable y is also increased by a proportional amount. Linear equations are relatively easy to solve and analyze, while nonlinear equations are usually difficult and often impossible to solve.

A dynamical system is deterministic if given the exact mathematical description of its state at one time there is only one possible future trajectory. If the system is placed again in the same state the same sequence of states will be exactly repeated.

Chaotic systems are deterministic systems that, paradoxically, are highly unpredictable. This unpredictability is a consequence of an extreme sensitivity to initial conditions found in some nonlinear dynamical systems. The smallest variation in the quantities that specify the state of the system at one given time leads to rapidly divergent trajectories in its future evolution. Since the determination of the initial conditions is always limited by a non-eliminable degree of error, chaotic systems become immediately uncontrollable and unpredictable.


Articles on dynamical systems and chaos theory are found in specialized journals and in mathematics and physics journals dealing with nonlinear systems. Among them the following are available at Linda Hall: