Overview of Nonlinear Dynamics and Chaos Theory

Motion near an energetic barrier of a potential energy surface is typically (but not always) chaotic at energies above the barrier height. If chaotic were synonymous with statistical, then reaction dynamics could in such cases be completely understood using standard statistical methods and theories. However, modern work in the theory of dynamical systems has shown that chaotic motion has an underlying structure there is order in chaos. The underlying structure within chaotic motion is the source of nonstatistical behavior. This structure needs to be understood in order to predict how the system will behave. 

It is fair to say that there has been a renaissance in the study of the properties of nonlinear dynamical systems in recent years.This work has brought forth the notion that the equations of classical mechanics describing most systems are fundamentally unsolvable, or nonintegrable, and that as a result all nonintegrable systems have universal scaling laws which underlie their complex and apparently chaotic motions.  

We now delve into the topic of what is meant by chaotic motion. We should first point out that there is no generally agreed upon technical definition of what the word chaotic means for dynamical systems, but it is possible for us to get a sense of what the issues are, nevertheless. It is not a foreign notion to consider that some kinds of systems exhibit regular motion (a pendulum, a onedimensional oscillator, a thrown baseball) and others behave erratically (a balloon with air escaping from its nozzle). It is more unsettling to consider the three ideas that follow. 

According to one point of view, expressed by Laplace, dynamical systems like the Solar System are completely deterministic, so probability theory can have no relevance. But this point of view requL a God-like omniscience in being able to determine initial conditions exactly. This requires an infinite number of digits and is beyond the capacity of anybody or anything ctf finite size, including the observable Universe (Ford 1983) [Ref. 59]. In reality measurement is only able to determine the state of a classical system to a finite number of digits, and even this determination is subject to errors, without quantum mechanics, and whether this determination is made by human or machine. Such measurements limit the known or recorded motion to a range of possible orbits. 

Taking these three considerations together, it is apparent that we must generally resign ourselves to an imperfect ability to assess and predict the dynamical states of most deterministic systems. Coarse-grained descriptions of system dynamics are therefore the subject of much interest. The aim of a coarse-grained description of the dynamics and also transition state theory in the context of reaction dynamics, is to predict the behavior of families df trajectories instead of individual ones. 

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