echecs16.info Theory CHAOS THEORY BOOK PDF

CHAOS THEORY BOOK PDF

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chaos. James Gleick's book, Chaos: making a new science and a who have been touting" chaos theory, on the one hand, and most scientists (and, . Why did the character Malcolm keep talking about chaos in the book and movie, Jurassic Park? The science of chaos deals with these frustrating and seemingly. CHAOS THEORY IS BASED ON THE. OBSERVATION THAT SIMPLE RULES WHEN ITERATED. CAN GIVE RISE TO APPARENTLY COMPLEX BEHAVIOR.


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echecs16.info edition, Mar 22 .. Local theory: “Golden mean" renormalization .. this book, are credited in the acknowledgments. PDF | On Jan 1, , Peter I. Kattan and others published Chaos Theory Simply Explained. Most books on chaos theory start with. Introduction to Chaos Theory. Book · December with 5, Reads. Publisher: Publisher: University of Belgrade Faculty of Transport.

Introduction[ edit ] Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days unproven ; the inner solar system, 4 to 5 million years. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time.

He postulated that these particles were in perpetual random motion. It is from these considerations that Boltzmann gave a mathematical expression to entropy.

In physical terms, entropy is the measure of the uniformity of the distribution of energy, also viewed as the quantification of randomness in a system.

Since the particle motion in gases is unpredictable, a probabilistic description is justified. Changes over time within a system can be modelized using the a priori of a continuous time and differential equation s , while the a priori of a discontinuous time is often easier to solve mathematically, but the interesting idea of discontinuous time is far from being accepted today.

One can define the state of the system at a given moment, and the set of these system states is named phase space see Table I. A very small cause, which eludes us, determines a considerable effect that we cannot fail to see, and so we say that this effect Is due to chance. If we knew exactly the laws of nature and the state of the universe at the initial moment, we could accurately predict the state of the same universe at a subsequent moment.

But even If the natural laws no longer held any secrets for us, we could still only know the state approximately. If this enables us to predict the succeeding state to the same approximation, that is all we require, and we say that the phenomenon has been predicted, that It Is governed by laws.

But this is not always so, and small differences in the initial conditions may generate very large differences in the final phenomena. A small error in the former will lead to an enormous error In the latter.

Prediction then becomes impossible, and we have a random phenomenon. Moser In , and Vladimimlr Igorevltch Arnold In , he showed further that a quasiperiodic regular motion can persist in an Integrable system Table I even when a slight perturbation Is Introduced Into the system.

The theorem also describes a progressive transition towards chaos: When a more significant perturbation is introduced, the probability of a quasiperiodic behavior decreases and an increasing proportion of trajectories becomes chaotic, until a completely chaotic behavior is reached. In terms of physics, in complete chaos, the remaining constant of motion is only energy and the motion is called ergodic.

Kolmogorov led the Russian school of mathematics towards research on the statistics of dynamical complex system called the ergodic theory. In a linear system Table I , the sum of causes produces a corresponding sum of effects and it suffices to add the behavior of each component to deduce the behavior of the whole system. Phenomena such as a ball trajectory, the growth of a flower, or the efficiency of an engine can be described according to linear equations.

In such cases, small modifications lead to small effects, while Important modifications lead to large effects a necessary condition for reductionism. The nonlinear equations concern specifically discontinuous phenomena such as explosions, sudden breaks In materials, or tornados.

Although they share some universal characteristics, nonlinear solutions tend to be individual and peculiar. In contrast to regular curves from linear equations, the graphic representation of nonlinear equations shows breaks, loops, recursions all kinds of turbulences. Using nonlinear models, on can identify critical points in the system at which a minute modification can have a disproportionate effect a sufficient condition for holism.

The above observations from the field of physics have been applied in other fields, in the following manner: When should one analyze rhythmic phenomena with reductionist versus holistic models? This is a question that one can ask in the field of chronobiology.

He first observed the phenomenon as early as and, as a matter of irony, he discovered by chance what would be called later the chaos theory, in , 18 while making calculations with uncontrolled approximations aiming at predicting the weather. The anecdote is of interest: By the way, this occurs when using computers, due to the limitation of these machines which truncate numbers, and therefore the accuracy of calculations.

Lorenz considered, as did many mathematicians of his time, that a small variation at the start of a calculation would Induce a small difference In the result, of the order of magnitude of the initial variation. This was obviously not the case, and all scientists are now familiar with this fact. In order to explain how important sensitivity the to initial conditions was, Philip Merilees, the meteorologist who organized the conference session where Lorenz presented his result, chose himself the title of Lorenz's talk, a title that became famous: Yorke, in The figure that appeared was his second discovery: The Belgian physicist David Ruelle studied this figure and he coined the term strange attractors in It is also Ruelle who developed the thermodynamic formalism.

There are four types of attractors. Figure 1 describes these types: According to Newton's laws, we can describe perfectly the future trajectories of our planet. However, these laws may be wrong at the dimension of the universe, because they concern only the solar system and exclude all other astronomical parameters. Then, while the earth is indeed to be found repetitively at similar locations in relation to the sun, these locations will ultimately describe a figure, ie, the strange attractor of the solar system.

A chaotic system leads to amplification of initial distances in the phase space; two trajectories that are initially at a distance D will be at a distance of 10 times the value of D after a delay of once the value of characteristic Lyapunov time Table I.

Advances in Chaos Theory and Intelligent Control

If the characteristic Lyapunov time of a system is short, then the system will amplify its changes rapidly and be more chaotic.

However, this amplification of distances is restricted by the limits of the universe; from a given state, the amplification of the system has to come to an end.

It is within the amplification of small distances that certain mathematicians, physicists, or philosophers consider that one can find randomness. The solar system characteristic Lyapunov time is evaluated to be in the order of 10 years.

The terms of negative and positive feedback Table I concern interactions that are respectively regulations and amplifications. An example of negative feedback is the regulation of heat in houses, through interactions of heating apparatus and a thermostat. Biology created negative feedback long ago, and the domain of endocrinology is replete with such interactions. An example of positive feedback would be the Larsen effect, when a microphone is placed to close to a loud-speaker.

In biology, positive feedbacks are operative, although seemingly less frequent, and they can convey a risk of amplification. Negative and positive feedback mechanisms are ubiquitous in living systems, in ecology, in daily life psychology, as well as in mathematics.

A feedback does not greatly influence a linear system, while it can induce major changes in a nonlinear system. Thus, feedback participates in the frontiers between order and chaos. Mitchell Jay Feigenbaum proposed the scenario called period doubling to describe the transition between a regular dynamics and chaos. His proposal was based on the logistic map introduced by the biologist Robert M. May in The logistic map is a function of the segment [0,1] within itself defined by:.

The dynamic of this function presents very different behaviors depending on the value of the parameter r:. This function of a simple beauty, in the eyes of mathematicians I should add, has numerous applications, for example, for the calculation of populations taking into account only the initial number of subjects and their growth parameter r as birth rate. When food is abundant, the population increases, but then the quantity of food for each individual decreases and the long-term situation cannot easily be predicted.

He described his theory in a book, 27 where he presented what is now known as the Mandelbrot set. A characteristic of fractals is the repetition of similar forms at different levels of observation theoretically at all levels of observation. Thus, a part of a cloud looks like the complete cloud, or a rock looks like a mountain. Fractal forms in living species are for example, a cauliflower or the bronchial tree, where the parts are the image of the whole.

A simple mathematical example of a fractal is the so-called Koch curve, or Koch snowflake. This is then repeated for each of the smaller segments obtained. Fractal objects have the following fundamental property: A second fundamental property of fractal objects, clearly found in snowflakes, is that of self similarity, meaning that parts are identical to the whole, at each scaling step.

A few years later, Mandelbrot discovered fractal geometry and found that Lorenz's attractor was a fractal figure, as are the majority of strange attractors. He defined fractal dimension Table I. Mandelbrot quotes, as illustration of this new sort of randomness, the French coast of Brittany; its length depends on the scale at which it is measured, and has a fractal dimension between 1 and 2. This coast is neither a one-dimensional nor a two-dimensional object. For comparison the dimension of Koch snowflake is 1.

Catastrophe theory is interesting in that it places much emphasis on explanation rather than measurement. Thom was at the origin of a renewed debate on the issue of determinism. I'd like to say straight away that this fascination with randomness above all bears witness to an unscientific attitude. It is also to a large degree the result of a certain mental confusion, which is forgivable in authors with a literary training, but hard to excuse in scientists experienced in the rigors of rational enquiry.

What in fact is randomness? Only a purely negative definition can be given: Ilya Prigogine, author of a theory of dissipative structures in thermodynamics, considers that the universe is neither totally deterministic nor totally stochastic.

Initial conditions can no longer be assimilated to a point in the phase space, but they correspond to a region described by a probability distribution. It is a nonlocal description, a new paradigm. The mathematician Alan Turing, famous for his work on cybernetics and artificial intelligence, showed that the synergy between reaction and diffusion could lead to spontaneous modes of concentrations.

A morphogen is a substance participating in reactions generating forms. Morphogens growth factors, transcription factors, or other endogenous compounds influence in a spatial and temporal manner, the expression of series of genes; this influence is very precise, possibly because morphogens are rapidly synthesized, but diffuse more slowly, and this discrepancy would lead to periodical maximal values of concentration. This model proposed by Turing enables to explain several phenomena: The phenomenon of spontaneous exchange of information was used by biologists such as Meinhartdt and Gierer 33 in their explanation of the periodic structure of leaves.

This kind of self-organized chemical reactions could also explain the emergence of the zebra skin or a quantity of biological phenomena that illustrate selforganizing structures. A century later, deterministic chaos, or the chaos theory, is much debated. Biologists, economists, specialists in social sciences, and researchers in medicine call themselves chaoticists. Moreover, debates on the chaos theory are no longer limited to groups of scientists having an extended knowledge in mathematics, but is widely found through the media, with participation from philosophers, psychoanalysts, journalists, or movie makers.

This suggests that the concepts of chaos theory find some resonance in present social and philosophical preoccupations.

Book pdf theory chaos

A superficial analysis might lead to the conclusion that the success of the chaos theory has only a semantic origin: Some researchers in the field of social sciences even propose that the chaos theory offers a revolutionary new paradigm, away from the materialistic Utopia, and that social system should be maintained at the edge of chaos, between too much and too little authoritarian control.

This comment concerns politics rather than physics.

Classical and Quantum Chaos

The specificity of present time physics, with entropy, chaos, and fractal dimensions, confers reality to phenomena as we can perceive and measure them, and it somehow invalidates the idea of a fundamental, or true, reality that might be explained by an elegant model.

The use of such models entails too many simplifications, and may lead for instance to the reversibility of time that is imposed by the mathematical structure of mechanics. The initial conditions of the universe with mass, charge of particles, size of atoms, fundamental forces, speed of light, combination of carbon and oxygen, and many others happened to be organized in such a way that life could appear, and with it consciousness.

This could suggest that the destiny of the universe is not towards an inevitable and generalized chaos. On the contrary, this destiny might be oriented towards complexity. Many discoveries in medicine can be seen as indications that organs function in a linear and deterministic manner, and that the causality principle applies to normal or abnormal physiology: In chronobiology, destruction of the suprachiasmatic nucleus alters Orcadian rhythms, and genetic crossing of insects strains with different circadian clock gene modifies the period of circadian rhythms in a predictable manner, etc.

These obvious findings are numerous and they might hide, to some extent, the fact that bodily functions and their temporal coordination are probably under laws that are inherently complex.

Indeed, living species are capable of increasing their complexity, to organize orderly functions from disorder in terms of physics, not medicine , and they do this without external informational input. Thus, living species exhibit some complex chaotic systems. However, in order to confirm this, it is necessary to have access to a huge number of data points; only then does it become possible to describe a biological phenomenon in its phase space and to study its evolution over time, using the methods of nonlinear dynamic systems.

These methods have shed light on a few aspects of organ functioning, in particular the cardiovascular system. Indeed, the cardiac rhythm is sensitive to initial conditions and to the fractal dimension of its attractor, and it was found that when the heart rate becomes highly regular, the heart is less capable of adaptation to demands, and that this condition predisposes to arrhythmias and myocardial infarction.

This chaotic behavior of the cardiac rhythm raises the essential question of the role of chaos in biology in general. In fact, the cardiac system could not function without chaos, since the power of self-organization participates in the capacity of the heart to adapt to physiological demands. Dynamical models of the brain are also a domain of research, notably into the artifical neural networks. There is, however, a risk linked to self-reference stating that we try to understand the brain using our brains, a somewhat problematic circular approach.

Brain disorders are accompanied by measurable changes in the electroencephalogram EEG , most obvious in a series of epileptic fits, where high-amplitude synchronized waves are observed. The various types of EEG patterns present different attractors and different fractal dimensions.

Handbook of Applications of Chaos Theory

When a healthy person stands with his or her eyes open, the EEG shows low-amplitude high-requency alpha waves, and the corresponding attractor has a high dimension. When the eyes are shut, EEG wave amplitude increases, frequency decreases, and the corresponding fractal dimension is lower. It is small during slow-wave sleep, and even more so during an epileptic fit or a coma.

Thus one can conclude that the cognitive power, defined as the capacity to perceive and analyze information, parallels the fractal dimension of the EEG. Arnold Mandell was the first psychiatrist to combine abstract mathematical models and quantitative experimental finding in an effort to approach clinical questions, for example that of the structure of personality.

For example, it was proposed that schizophrenia was characterized by nonlinear phenomena alternating with pure randomness in the brain function architecture, a proposal in line with the early theoretical work of Prigogine. Finally, mathematical models have been used to describe biological rhythms and to explore the biological clocks functional rules, eg, to explore how biological clocks interact. In several cases, this conclusion seems to apply to chronobiology.

National Center for Biotechnology Information , U. Journal List Dialogues Clin Neurosci v. Dialogues Clin Neurosci. Author information Copyright and License information Disclaimer.

This is an open-access article distributed under the terms of the Creative Commons Attribution License http: This article has been cited by other articles in PMC. Abstract Whether every effect can be precisely linked to a given cause or to a list of causes has been a matter of debate for centuries, particularly during the 17th century when astronomers became capable of predicting the trajectories of planets.

Table I. Definitions of concepts related to the history of chaos theory. Every effect has an antecedent proximate cause. A philosophical proposition that every event is physically determined by an unbroken chain of prior occurrences. A pattern, plan, representation, or description designed to show the structure or workings of an object, System, or concept. A System that changes over time in both a causal and a deterministic manner; ie, its future depends only on phenomena from its past and its present causality and each given initial condition will lead to only one given later state of the System determinism.

Systems that are noisy or stochastic, in the sense of showing randomness, are not dynamical Systems, and the probability theory is the one to apply to their analysis. An abstract space in which ail possible states of a System are represented which, each possible state of the System corresponding to one unique point in the phase space.

This is when a change in one variable has the consequence of an exponential change in the system. In mathematics, this refers to a System of differential equations for which solutions can be found. In mechanics, this refer to a system that is quasiperiodic. The characteristic time of a system is defined as the delay when changes from the initial point are multiplied by 10 in the phase space. A response to information, that either increases effects positive feedback , or decreases them negative feedback , or induces a cyclic phenomenon.

This means that an object is composed of subunits and sub subunits on multiple levels that statistically resemble the structure of the whole object, However; in every day life, there are necessarily lower and upper boundaries over which such self-similar behavior applies. Is a geometrical object satisfying two criteria: Let an object in a n-dimensions space be covered by the smallest number of open spheres of radius r.

For example, the adjective linear pharmacokinetics describes a body clearance of a constant fraction per unit of time of the total amount of a substance in the body while a nonlinear pharmacokinetics describes the elimination of a constant quantity of cormpound per unit of time.

AIso feedback is a well-known term in biology or medicine, while its use in physics is less familiar to nonphysicists. Open in a separate window.

Book pdf theory chaos

The roots of modern science Newton and causality Johannes Kepler published the three laws of planetary motion in his two books of 1 and , 2 and Galileo Galilei wrote, in , 3: Laplace and determinism Determinism is predictability based on scientific causality Table I.

This was the birth of chaos theory. Ruelle and strange attractors The Belgian physicist David Ruelle studied this figure and he coined the term strange attractors in Figure 1. Fixed point: Limit cycle: Examples include the swings of a pendulum clock and the heartbeat while resting.

If two of these frequencies form an irrational ratio, the trajectory is no longer closed, and the limit cycle becomes a limit torus. Strange attractor: The dynamics of satellites in the solar system is an example. This figure shows a plot of Lorenz's attractor. The golden age of chaos theory Felgenbaum and the logistic map Mitchell Jay Feigenbaum proposed the scenario called period doubling to describe the transition between a regular dynamics and chaos.

The logistic map is a function of the segment [0,1] within itself defined by: The dynamic of this function presents very different behaviors depending on the value of the parameter r: Figure 2.

The horizontal axis shows the values of the parameter r while the vertical axis shows the possible long-term values of x. Figure 3. The Mandelbrot set a point c is colored black if it belongs to the set and white if not. Figure 4. Turing and self-organization The mathematician Alan Turing, famous for his work on cybernetics and artificial intelligence, showed that the synergy between reaction and diffusion could lead to spontaneous modes of concentrations.

Chaos theory and medicine Many discoveries in medicine can be seen as indications that organs function in a linear and deterministic manner, and that the causality principle applies to normal or abnormal physiology: Kepler J.

Pdf book chaos theory

Astronomia Nova, Donahue WH, trans. Cambridge, UK: Cambridge University Press; 1 Am J Physiol. Harmonice Mundi. Field J, trans. The Harmony of the World. Philadelphia, Pa: Galileo Galilei. II Saggiatore.

The Controversy on the Comets of , Philadelphia, Pa: University of Pennsylvania Press; [ Google Scholar ]. Descartes R. The Philosophical Writings Of Descartes. Cambridge University Press; [ Google Scholar ]. Newton I. Philosophiae Naturalis Principia Mathematica London, 1 ; facsimile edition: London, UK: Jones; [ Google Scholar ].

Nova Methodis pro Maximis et Minimis, itemque Tangentibus, quae necfractas nee irrationales quantitates moratur, et singulare pro illis calculi genus. Struik, DJ.

Classical and Quantum Chaos

A Source Book in Mathematics, Cambridge, Mass: Harvard University Press; Koestler A. Every effect has an antecedent proximate cause. A philosophical proposition that every event is physically determined by an unbroken chain of prior occurrences. A pattern, plan, representation, or description designed to show the structure or workings of an object, System, or concept. A System that changes over time in both a causal and a deterministic manner; ie, its future depends only on phenomena from its past and its present causality and each given initial condition will lead to only one given later state of the System determinism.

Systems that are noisy or stochastic, in the sense of showing randomness, are not dynamical Systems, and the probability theory is the one to apply to their analysis. An abstract space in which ail possible states of a System are represented which, each possible state of the System corresponding to one unique point in the phase space. This is when a change in one variable has the consequence of an exponential change in the system. In mathematics, this refers to a System of differential equations for which solutions can be found.

In mechanics, this refer to a system that is quasiperiodic. The characteristic time of a system is defined as the delay when changes from the initial point are multiplied by 10 in the phase space. A response to information, that either increases effects positive feedback , or decreases them negative feedback , or induces a cyclic phenomenon.

This means that an object is composed of subunits and sub subunits on multiple levels that statistically resemble the structure of the whole object, However; in every day life, there are necessarily lower and upper boundaries over which such self-similar behavior applies. Is a geometrical object satisfying two criteria: self-similarity and fractional dimensionality.

Let an object in a n-dimensions space be covered by the smallest number of open spheres of radius r. For example, the adjective linear pharmacokinetics describes a body clearance of a constant fraction per unit of time of the total amount of a substance in the body while a nonlinear pharmacokinetics describes the elimination of a constant quantity of cormpound per unit of time.

AIso feedback is a well-known term in biology or medicine, while its use in physics is less familiar to nonphysicists. Open in a separate window The review is focused on those strictly deterministic dynamic Systems that present the peculiarity of being sensitive to initial conditions, and, when they have a propriety of recurrence, cannot be predicted over the long term.

Chaos theory has a few applications for modeling endogenous biological rhythms such as heart rate, brain functioning, and biological docks. The roots of modern science Newton and causality Johannes Kepler published the three laws of planetary motion in his two books of 1 and , 2 and Galileo Galilei wrote, in , 3 : Philosophy Is written In this vast book, which continuously lies open before our eyes I mean the universe.

But It cannot be understood unless you have first learned to understand the language and recognize the characters in which it is written. It is written in the language of mathematics, and the characters are triangles, circles, and other geometrical figures.

For example, this principle is accepted a priori in physics. Newton, having developed differential calculus and written the gravitational law, can be seen as the researcher who launched the development of classical science, ie, physics up to the beginning of the 20th century before relativity and quantum mechanics.

Newton wrote his manuscript on differential calculus in Gottfried Wilhelm von Leibniz had another point of view on this theme, and he published his book in Yet, later it was Leibniz' ideas that were used. These steps in the acquisition of human knowledge are described in Arthur Koestler's books on astronomy 8 and mentioned in science dictionaries. One distinguishes schematically between local and universal determinism. Local determinism concerns a finite number of elements.

A good illustration would be ballistics, where the trajectory and the site of impact of a projectile can be precisely predicted on the basis of the propulsive force of the powder, the angle of shooting, the projectile mass, and the air resistance. Local determinism raises no particular problem. Obviously, one cannot.

Theory book pdf chaos

In a whirlwind of dust, raised by elemental force, confused as it appears to our eyes, in the most frightful tempest excited by contrary winds, when the waves roll high as mountains, there is not a single particle of dust, or drop of water, that has been placed by chance, that has not a cause for occupying the place where it is found; that does not, in the most rigorous sense of the word, act after the manner in which it ought to act; that is, according to its own peculiar essence, and that of the beings from whom it receives this communicated force.

A geometrician exactly knew the different energies acting in each case, with the properties of the particles moved, could demonstrate that after the causes given, each particle acted precisely as it ought to act, and that It could not have acted otherwise than It did. However, It was the mathematician and astronomer Pierre-Simon Laplace who most clearly stated the concept of universal determinism shortly after d'Holbach, In 12 : We ought then to regard the present state of the universe as the effect of Its anterior state and as the cause of the one which is to follow.

Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose It - an Intelligence sufficiently vast to submit these data to analysis-- it would embrace in the same formula the motions of the greatest bodies of the universe and those of the lightest atom; for It nothing would be uncertain and the future, as the past, would be present to Its eyes.

The creativity of Laplace was tremendous. He demonstrated that the totality of celestial body motions at his time, the sun and the planets could be explained by the law of Newton, reducing the study of planets to a series of differential equations.

Urbain Jean Joseph Le Verrier discovered the planet Neptune in , only through calculation and not through astronomical observation. He then developed further Laplace's methods by, for example, approximating solutions to equations of degree 7 and concluded 14 : It therefore seems impossible to use the method of successive approximations to assert, by virtue of the terms of the second approximation, whether the system comprising Mercury, Venus, Earth, and Mars will be stable Indefinitely.

It is to be hoped that geometricians, by integrating the differential equations, will find a way to overcome this difficulty, which may well just depend on form. In the middle of the 19th century, it became clear that the motion of gases was far more complex to calculate than that of planets. One of their main postulates was the following: an isolated system in equilibrium is to be found in all its accessible microstates with equal probability. In , Maxwell described the viscosity of gases as a function of the distance between two collisions of molecules and he formulated a law of distribution of velocities.

Boltzmann assumed that matter was formed of particles molecules, atoms an unproven assumption at his time, although Democrites had already suggested this more than years previously.

He postulated that these particles were in perpetual random motion. It is from these considerations that Boltzmann gave a mathematical expression to entropy. In physical terms, entropy is the measure of the uniformity of the distribution of energy, also viewed as the quantification of randomness in a system.

Since the particle motion in gases is unpredictable, a probabilistic description is justified. Changes over time within a system can be modelized using the a priori of a continuous time and differential equation s , while the a priori of a discontinuous time is often easier to solve mathematically, but the interesting idea of discontinuous time is far from being accepted today. One can define the state of the system at a given moment, and the set of these system states is named phase space see Table I.

A very small cause, which eludes us, determines a considerable effect that we cannot fail to see, and so we say that this effect Is due to chance.