Art. 07 – Vol. 28 – No. 4 – 2018

Chaos theory, a modern approach of nonlinear dynamic systems

Ioana-Elena ENE
National Institute for Research and Development in Informatics – ICI Bucharest
8-10 Mareșal Alexandru Averescu Av., 011455, Bucharest, Romania

Abstract: Chaos theory is a branch of mathematics focusing on nonlinear dynamic systems. As a relatively new field with a significant applicability area, chaos theory is an active research area involving many different disciplines (mathematics, topology, physics, social systems, population modeling, biology, meteorology, astrophysics, information theory, computational neuroscience, cryptography, robotics etc.). The availability of cheaper and more powerful computers has made a major contribution to the achievement of major advances in nonlinear dynamic systems theory, the interest in deterministic chaos has increased enormously, being reflected in both literature and real life.

Cuvinte cheie: chaos, determinism, dynamical system, phase space, dynamical system, fractal.

Most data analysis methods use linear models that are based on relationships described by linear differential equations because they are easy to manipulate and usually give unique solutions. However, nonlinear behavior occurs frequently in real-life systems due to their complex dynamic nature. This cannot be adequately described by linear models; the use of chaotic systems is an appropriate solution.

A system is a set of components that interact and form a whole; nonlinearity means that due to feedback or multiplicative effects between components, the whole becomes more than the sum of the individual parts. Finally, the dynamic term refers to the fact that the system changes over time depending on its state at some point in time.

In nonlinear systems, the relationship between cause and effect is not proportional and determined, but rather vague and difficult to discern. Nonlinear systems can be characterized by periods of both linear and nonlinear interactions between variables. Thus, dynamic behavior can reveal linear continuity over certain periods of time, while relationships between variables can change, resulting in dramatic structural and behavioral changes in other periods. The dramatic change from a qualitative behavior to another is called “bifurcation.”

Chaotic systems are a simple subtype of nonlinear dynamic systems. They can contain very few interactive parts and they can follow very simple rules, but all these systems have a very sensitive dependence on the initial conditions. Despite their deterministic simplicity, these systems can produce a completely unpredictable and chaotic behavior over time. Studies on nonlinear systems highlight three types of temporal behavior: (1) stable behavior (mathematical equilibrium or fixed point); (2) oscillation between mathematical points in a stable, smooth and periodical manner; (3) seemingly random behavior, lacking model (or non-periodic behavior) dominated by uncertainty and in which predictability decomposes. These behaviors may occur intermittently throughout the “life” of a non-linear system. A regime can dominate at certain times, while other regimes dominate at other times [21]. These characteristics determine a variety of behaviors that represent the dynamics of nonlinear systems.

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Ioana-Elena ENE, Chaos theory, a modern approach of nonlinear dynamic systems, Romanian Journal of Information Technology and Automatic Control, ISSN 1220-1758, vol. 28(4), pp. 89-96, 2018.


A chaotic system has three simple features: it is deterministic (it has a determinant equation that governs its behavior); it is sensitive to the initial conditions (even a very small change of the starting point can lead to major changes of the deterministic trajectory); it is neither random nor disordered.

All chaotic systems are nonlinear and involve certain iterative rules. Numerical analysis is usually the only possibility of analyzing such systems. Depending on the initial value of the control parameter, the system can evolve to stable, constant or periodic orbits, or to non-periodic, or chaotic orbits.

In the next phase of the research the focus will be set to identify and investigate the threedimensional systems Lorentz, Rössler, Chen, Lu, the main algorithms used in the theory of chaos, comparisons between classical deterministic-stochastic methods and methods based on the theory of self-organizing the chaos. A number of applications will be built in order to illustrate the modeling, controlling and synchronization of a chaotic system, including acquisition of signals from the studied process and formation of a time series, analysis of Lyapunov coefficients and their processing, attractors’ detection, drawing bifurcation diagrams for the analysis of possibilities to stabilize the system.


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