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Adil Gürsel Karaçor (Atilim University, Turkey) and Turan Erman Erkan (Atilim University, Turkey)

Copyright: © 2016
|Pages: 17

DOI: 10.4018/978-1-5225-0075-9.ch014

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TopPrice movements of financial instruments, whether they are random, chaotic, or of any other stochastic process, are definitely non-linear. Prior to going further into predictability analysis, it should be a good idea to define some terminology about non-linear dynamics.

First of all, a *dynamic system* can be defined as a deterministic mathematical prescription for evolving the state of a system forward in time where time may be either a continuous or discrete variable. (Ott E. (1993)). A *dissipative system* is a dynamic system, in which the phase space volume contracts along a trajectory.

The term *non-linear* is defined as the opposite of linear. In a linear system the variables appear in the first degree, they are not multiplied by one another; they are only multiplied by constants, and are combined only by addition or subtraction, while non-linear systems can occur in various forms such as division, multiplication and powers. The vast majority of systems we come across in real life are actually non-linear.

Collection of all possible states of a dynamic system is called the *phase space*. A *map* is a discrete function in the phase space that gives the next state of the system as a function of its current states. In a similar fashion, a *flow* is a continuous function that describes the time derivative of the state variables as a function of the present state values.

A state **x*** is an *equilibrium point* of the system if once **x**(t) is equal to **x***, it remains equal to **x*** for all future times. When the system is exactly on the *periodic orbit* it will move on it forever and pass through the same points periodically. Both flows and maps can have equilibrium points and periodic orbits. Equilibrium points and periodic orbits can be either *stable* (even if the system slightly deviates from the equilibrium point or periodic orbit, it returns), *unstable* (if the system slightly deviates from the equilibrium point or periodic orbit, it does not come back) or *a saddle* (if the system slightly deviates from the equilibrium point or periodic orbit in some direction, it returns; if it deviates in some other direction it diverges; such equilibrium points or periodic orbits are also considered unstable).

As some people call non-linear financial dynamics as chaos, let us explain some more terminology on chaotic systems.

Until the last few decades of 20th century, the term *chaos* only meant disordered formless matter, complete disorder, utter confusion, randomness, or uncertainty. However since then it has gained a special scientific meaning, and started being used designate *deterministic chaos* which is a specific type of behavior that can be observed in non-linear dynamic systems and can be expressed by a set of discrete-time or continuous-time equations. Although its mathematical description is deterministic, a chaotic system has unpredictability in the long run. Another interesting point about chaotic dynamics is that the phase trajectories strongly depend on initial conditions. In addition, a dissipative chaotic system neither converges to a stable point or a stable periodic orbit, nor diverges to infinity; instead it wanders around in a fractal region. Such kind of chaotic behavior can be observed in various systems in different areas ranging from chemistry to electronics. However, it is not usually desired for a non-linear system to exhibit chaotic behavior, this is where control comes into the picture: due to its unpredictable nature chaos can give rise to problems. On the other hand, by taking advantage of certain properties of chaotic behavior, much can be achieved with little control effort. The founders of the OGY method; Ott, Grebogi and Yorke took advantage of the fact that there are quite a number of unstable limit cycles and equilibrium points within the strange attractor once the system comes close enough to one of these points of choice.

As the term *strange attractor* is mentioned, it needs to be explained. A dissipative chaotic system is such a system that neither converges to a stable point or a stable periodic orbit, nor diverges to infinity; instead it wanders around in a fractal region. Such a system contains many unstable or saddle type equilibrium points and so called *strange attractors.* A strange attractor actually consists of an infinitely long single trajectory that does not cross itself, and does not repeat itself periodically. Yet any trajectory attracted to a strange attractor sticks with it forever.

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