Introductory Chapter: Chaos in Nature and Life

1. Introduction

This chapter provides a short overview related to chaos theory, its basic, the impact of the prominent world scientists to develop the chaos theory, and some examples of dynamical phenomena, which are characterized by chaotic nature. J.C. Maxwell was one of the authors who started discussion about a chaos theory referring to the “butterfly effect” between the 1860s and 1870s (see [1, 2]). H. Poincaré was an early adherent of chaos theory as well. He observed that there can exist the nonperiodic orbits, which are increasing not forever and not approaching to a fixed point, at study of the three-body problem in the 1880s [3, 4]. J. Hadamard published a high-powered work about the chaotic motion of free particle, which glides frictionlessly on some surface with negative constant curvature in 1898 [5]; and this phenomenon got the name “Hadamard's billiards.” It was shown in this work that all trajectories are unstable, and trajectories of all particles deviate exponentially one from one another, and this deviation is characterized by a Lyapunov exponent, which is positive.

An ergodic theory initiated the development of the chaos science. Later, the progress of the chaos theory was connected with studying the nonlinear differential equations. Many prominent scientists, namely Birkhoff [6], Kolmogorov [7], Cartwright and Littlewood [8], and Smale [9] dealt with this problem. A little bit later, chaos studies were inspired in physics: the Birkhoff research on three-body problems, the Kolmogorov turbulence and astronomical problems, and the Cartwright and Littlewood studies related to radio engineering. In the area of planetary mechanics, fluid dynamics, and electromagnetics, the scientists had observed a chaotic planetary motion, a turbulence in fluid flow, and a nonperiodic oscillation in the radio circuits in spite of that the theory cannot explain what they were seeing.

In spite of an initial understanding in the first half of the twentieth century, chaos theory was fully formalized as specific theory in its second half; when the linear theory, the theory of prevailing system could not explain clearly the observed properties of some experiments such as the logistic map, for example. Such misunderstanding phenomena were referred to the inaccuracy of measurement and appearing “noise” was counted by scientists as an inherent component of the systems under study. B. V. Chirikov introduced a criterion for the genesis of the chaos classical phenomena in the Hamiltonian systems, known as the Chirikov criterion, in 1959. This criterion was applied by him to explain series experimental results related to plasma imprisonment in the open mirror catches [10]. This research can be considered as a pioneer physical chaos theory, which allows to explain this specific phenomenon. In this regard, B. V. Chirikov is ranked as one of the classical and quantum chaos pioneers [11].

Electronic computers were served as the main contributor to the evolution of chaos theory. This is explained by the feature that the mathematical description of chaos theory assumes the repeated calculation of mathematical relations that cannot be done “by hand.” The computers realize such repeated computations easily; additionally, the plots and figures provide with the visualization of these researches. Y. Ueda, being with C. Hayashi’s laboratory at the Kyoto University, studied the analog computers and observed “randomly transitional phenomena” on Nov. 27, 1961. This impact of computers on the development of the chaos theory was published some later in 1970 [12].

2. Chaos theory in applications

Although theory of chaos was appearing with the observation of the weather phenomena, it is applicable for a variety of fields in the nature and life. Such areas, become a profit from chaos theory nowadays, are biology, mathematics, geology, economics [13], computer science, engineering [14, 15], finance [16], meteorology, anthropology [17], philosophy, physics [18, 19], politics [20], the population dynamics [21], and robotics. We illustrate several examples below, but this no means an in-depth summary of new appearing applications.

For example, the biology scientists checked the populations of different families with the population models over hundred years. Most of such models were continuous, but recently they are able to implement chaotic models for some families [22]. The chaos theory is applicable to the ecological systems, for example, a hydrology. Although a chaotic model for this field has some shortcomings, a series of the useful results was adopted by application of the chaos theory [23].

An application of chaos theory can be improved many economic models, despite that prediction of the reliability of economic systems and taking into account the influencing factors is an extremely complex problem [24].

One more application is using a particle swarm optimization (PSO) for predicting gas solubility to manufacture polymers in chemistry, but the solutions can converge to the wrong points. To avoid this drawback, a perfected version of PSO has been implemented based on chaos theory, which allows getting the satisfactory results [25]. Applying chaos theory allows to predict better the motion of asteroids when these objects are approaching to Earth or other planets [26] in celestial mechanics. It is known that four of the five moons of Pluto have chaotically rotated orbits. The research of large arrays of Josephson junctions in quantum physics and electrical engineering becomes successful owing the chaos theory [27]. In a fluid and gas dynamics, a chaos theory is close to the phenomena of turbulence. From the point of view of chaos theory, study of the critical points, at which a system becomes turbulence, is important; this was investigated by L. Landau; and the subsequent results in this area got the name of the Landau-Hopf theory of turbulence. Against Landau, D. Ruelle, and F. Таkens predicted later that fluid turbulence could appear through a strange attractor that Ls the main idea of chaos theory. The example of such turbulence is shown in Figure 1.

Figure 1.

The turbulence in a tip vortex from airplane wing. https://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Airplane_vortex_edit.jpg/220px-Airplane_vortex_edit.jpg.

A psychology is research field attracting the chaos theory as well. As a result, one has found that the collective dynamic is a result of the dynamics of individuals; by this, each individual brings into the group dynamics its different contribution. Finally, the chaotic doing of group has a response in each member (according to the Wilfred Bion’s theory) [28].

In 1992, Redington and Reidbord tried to show that the human heart could indicate chaotic traits. This was confirmed by the changing between heartbeat intervals for a variety psychotherapy patient while periods of varying the emotional intensity pending therapy sessions. The observed results were conceivably indeterminate. There appeared the ambiguities in trying to purportedly show facts of chaotic dynamics, namely spectral analysis, phase trajectories, and autocorrelation plots, additionally, when they try to calculate the Lyapunov exponent, as the definitive parameter of chaotic behavior, they found that this is not reliable [29].

In 1995, Metcalf and Allen [30] endorsed that in animal behavior, there appears a pattern of doubling period resulting in chaos. They studied a well-known response termed as schedule-induced polydipsia. The fixed parameter operating in this process was the length of interval between the feeding. A variety of behavior states of a large number of animals was checked, and they made the decision that changes in response patterns were result of the different starting places, which confirms me chaotic behavior of the animal movement.

3. Chaotic dynamics

In usual understanding, the term “chaos” corresponds to “a state of disorder” [31]. But the chaos theory defines this term more accurately. In fact, the universal mathematical definition of chaos does not exist, therefore a usually used definition, formulated by R.L. Devaney, foresees that dynamical phenomenon must possess the properties:

• It is sensitive to initial conditions.

• It is topologically transitive.

• It has the dense periodic orbits

• in order to be chaotic [32].

Particularly, the last two properties denote actually my sensitivity to initial conditions [33]. This is correct for all possible continuous maps in metric spaces for the case of discrete time. In these cases, when some practically property is significant, “sensitivity to initial conditions” does not apply for the definition. If the consideration applies to the limited domains, the second property changes the rest ones. A weaker definition of the chaos is based on the first two properties only.

Sensitivity to the initial conditions foresees that every state in a chaotic system is as much as closely approached by other states, which have the different forthcoming trajectories or paths. In such conditions, an arbitrarily small perturbation or change of the current trajectory can evoke the different forthcoming behavior [34]. The sensitivity to initial conditions is known usually as the “butterfly effect;” this was termed using the title of a paper Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? [35], published by E. Lorenz in 1972. The flapping wing characterizes a small change of initial condition of the object, which results in series of events, which prevents the prediction of the forthcoming phenomena.

In Lorenz’s book The Essence of Chaos, published in 1993 [36], it was suggested that “sensitive dependence can serve as an acceptable definition of chaos.” The butterfly effect was defined there as: “The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration.” This definition is in the correspondence with the sensitive dependence of solutions on the initial conditions (SDIC). In order to confirm the sensitivity of time-varying paths on the initial positions, the idealized sport skiing model was developed. A horizon of predictability can be sought for prior the start of SDIC, namely prior to the valuable separations of the initial close trajectories [37].

Topological mixing, or by other words, me weaker condition of the topological transitivity denotes that the system maturates timely in such a way that any fixed domain (or open set of its phase space) after all coincides with any other given domain. Such mathematical idea of “mixing” relates with the standard intuition, and the mixing of color fluids or dyes is a visible example of chaotic system. Usually, the topological mixing is excluded often from usual examples of chaos, which define chaos with sensitivity to initial conditions only. But the individual sensitive dependence on initial conditions does not produce a chaos. The example of topological mixing is shown in Figure 2. The blue region is transformed by the dynamics to the purple region firstly, to the pink and red regions after this, and to a cloud of vertical lines scattered across the space at the end.

Figure 2.

The map $x\to 4x\left(x-1\right)$ and $y\to \left(x+y\right)$by (mod 1) indicate the topological mixing as well. https://upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Chaos_Topological_Mixing.png/220px-Chaos_Topological_Mixing.png.

A chaotic system has dense periodic orbits if each point in the respective space can be approximated by arbitrarily closed periodic orbit s [38]. One-dimensional map $x\to 4x\left(x-1\right)$ represents the simplest systems with density of periodic orbits. For example, $\frac{5-\sqrt{5}}{8}\to \frac{5+\sqrt{5}}{8}\to \frac{5-\sqrt{5}}{8}$, that is. approximately to 0.3454915 → 0.9045085 → 0.3454915, is one of unstable orbits with period equal to 2; and there are the similar orbits at periods 4, 8, 16, etc. This is true for all periods obtained in Sharkovskii’s theorem. This theorem is the basis of the Li and Yorke proof [39] (1975), which states that any one-dimensional continuous system, which has a regular cycle of period equal to three, has the regular cycles of more length, and has the completely chaotic orbits.

4. Conclusions

A short history of chaos theory explaining its significant importance in the better understanding of the complex nature and life phenomena is presented in chapter. A role of the prominent scientists, including Maxwell, Hadamard, and Lorentz in the development of chaos theory, is accentuated. A series of examples explains a significance of chaos theory at the correct description of complex dynamical systems.