Philosophical Reflections in Physics and Math

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Philosophical Reflections in Physics and Math

Chaotic Attraction - Part 1

In the traditional or classical approach to science, as exemplified in the post-Newtonian world, one assumes that if one begins with an approximate knowledge of the initial conditions of a given system, then, one will be able to produce an approximate picture of how the system will behave in the future. This presupposes, of course, one has possession of formulae capable of capturing the laws governing such a system.

In the traditional approach to science, a further assumption is made. This assumption states one can ignore fluctuations of small magnitude involving initial conditions. These sort of fluctuations are considered to be incapable of affecting the general properties and behavior of the system to any appreciable degree. In other words, such minor fluctuations are assumed to fall below a level of intensity or strength capable of interfering with one's ability to predict how the system will manifest itself in the future if one starts from a given set of initial conditions.

In the case of chaotic systems, however, the so-called minor fluctuations occurring in a given system are part of a complex dialectic. This dialectic magnifies the character of these fluctuations over time.

When such fluctuations are magnified beyond certain critical values, they become the source of turbulence. As a result, the traditional linear approach, with its underlying assumptions, becomes an ineffective means of describing or accounting for the long-term behavior of a chaotic system.

Lorenz, aperiodic phenomena and the Butterfly effect

In the early 1960's, Edward Lorenz was trying to develop models capable of reflecting the behavior of weather systems. Unfortunately, the system of equations he was using to model that behavior was problematic. When small errors of measurement, which inevitably occur, were introduced into the equations, these errors soon became magnified to catastrophic levels as they were processed through the various functions and operations inherent in the equations.

For example, suppose one had obtained a measurement value extending to 6 decimal points, but, for whatever reason (e.g., to save space on a printout, for ease of calculations, etc.) the measurement was shortened to just three decimal points. When this value was plugged into the equations being used to model the behavior of weather, the missing decimal values had a sort of multiplier effect as one processed the equations. As a result, the answers produced by the equations were at considerable variance with the observed values in the actual behavior of weather.

However, the essential problem of model building with respect to systems such as the weather was not just a matter of the lack of precision inherent in the process of measurement. Furthermore, the essential problem surrounding nonlinear systems was not due to the distorting effect ensuing from the over-simplifications of the underlying assumptions of classical physics. The crucial issue at the heart of model building in relation to nonlinear systems concerned the aperiodic character of such systems.

Aperiodicy refers to systems that do not gravitate toward a uniform, steady state in which certain structural themes repeat themselves, more or less exactly, from one cycle to the next on a regular or periodic basis. Aperiodic systems seem to fluctuate in an unpredictable fashion about certain values. Sometimes, such systems generate a spectrum of fluctuating values similar to, but not replicas of, one another.

On the surface, aperiodic phenomena appear to be a manifestation of random elements and local 'noise'. This local noise prevents such systems from reaching a steady, uniform state of regular periodicy.

Lorenz sensed, however, there was an underlying structure to the apparent "randomness". He felt an essential structure was being camouflaged by the spectrum of variable fluctuations characteristic of aperiodic systems.

According to Lorenz, one manifestation of the relationship between structure and variability is the Butterfly effect. This refers to the tendency of aperiodic or chaotic systems to be very sensitive to, and dependent on, the character of initial conditions. Small magnitudes of fluctuation inherent in initial conditions of aperiodic systems come to have a vectoral multiplier effect on the way the system behaves over time.

Linear and nonlinear systems

Linear equations can be described by a straight line on a graph. The straightness of the line indicates that the relationship between x and y, which is being plotted, is a proportional one. Therefore, determinate changes undergone by one of the two variables will be reflected in the proportionate character of the determinate changes undergone by the other variable under the same functional conditions.

Another characteristic of linear systems is their amenability to being broken down analytically. The components generated by such analysis, then, can be reassembled to produce the original linear system, complete in all its properties and aspects.

None of the foregoing characteristics of linear systems, however, are true of nonlinear systems. Thus, the variables being linked in a nonlinear system are not proportionate to one another.

Moreover, one cannot analytically break down a nonlinear system, as one can with linear systems, due to the way nonlinear values are constantly changing instead of remaining uniform. Finally, nonlinear systems always have proven resistant to yielding solutions to the equations being used in nonlinear contexts.

For example, suppose one wanted to calculate the manner in which a hockey puck accelerates when it is the beneficiary of an input of energy transmitted through a hockey stick. As long as one disregards friction, one can come up with a linear equation capable of describing the acceleration of such a system. However, as soon as one introduces friction into the calculations, the linear character of the system disappears.

The reason for the disappearance of linear character is because of the intimate nature of the relationship among velocity, friction and energy. More specifically, the amount of friction generated by a hockey puck moving across a surface depends on the velocity of the puck. Yet, at the same time, the velocity of the puck will be affected by the friction being generated by the puck's movement.

Therefore, the effect a given input of energy will have on acceleration (i.e., when the puck is struck by a hockey stick) will both affect, as well as be affected by, the existing values of velocity and energy. In short, one cannot assign a uniform, single value to any of the basic variables of the system because they are all mutually reactive and affect one another through a complex dialectic that cannot be captured by a set of stable, constant linear relationships.

One runs into the same kinds of problems in the case of fluid dynamics. The basic method for solving problems in fluid dynamics is the Navier-Stokes equation.

This equation links together variables of pressure, viscosity, velocity and density in one set of functional relationships. However, since the contexts described by the Navier-Stokes equation are nonlinear, the character of the relationships among the variables is constantly subject to change and transition. Consequently, one is, once again, confronted with a complex dialectic of variables which cannot be easily grasped, if at all, by linear techniques.

A time series has traditionally been used to depict the changing values of a given variable as a function of time. The usual way of displaying this is to plot time along the horizontal axis, while using the vertical axis to plot fluctuations in the character of a given variable. When plotted against time, these fluctuations appear as a series of amplitude-like values falling above or below the horizontal axis.

A time series, however, cannot capture the manner in which the relationships among a set of variables change with respect to one another over time. To be able to show this aspect of transition in the relationship among a set of variables, one needs to think of each point that is to be plotted as the product of the vectoral interaction of a set of variables. The interaction of this set of variables establishes or fixes the location of the point in three-dimensional space.

As the relationships among the set of variables changes with time, this element of change will be reflected by the way a point moves about in the three-dimensional space. The plotting of the movement of the point in three-dimensional space gives expression to the continuous character of change of relationship among the set of variables over time.

If one is plotting such a complex point in a nonlinear system, the trajectory described by the movement of that point never intersects itself since the character of the vectoral interaction of the variables never quite repeats itself. The trajectory of the movement of the point in a nonlinear system loops around one or more central tendencies indefinitely.

A chaotic system always stays within a set of parameters or envelope of values, but it never precisely repeats itself. The order which is manifested is always showing a new 'look' or new mode of trajectory within a set of constraints. The trajectories constitute the degrees of freedom of the structural character of the chaotic system in question, whereas, the envelope of values within which the trajectories express themselves constitutes the constraints of the structural character of the chaotic system.

In a Lorenz attractor the graph of the kind of moving point described above is expressed as a pair of concentric-like set of elipsoids which each revolve around a central space. These sets of ellipsoids have a cross-over region intermediate between them.

The point being plotted moves from the sphere of influence of one set of concentric ellipsoids to the sphere of influence of the other set of ellipsoids. When fully graphed, the whole thing looks sort of like a pair of owl's eyes .

Although the microscopic character of chaotic systems often was understood quite well, this understanding could not be translated into an ability to grasp how and why the macroscopic behavior of these chaotic systems had the complex character it did. In short, the problem facing researchers was that the global behavior of a system was different from the local behavior of that same system. As a result, knowledge of the micro-structure of a nonlinear system was not very helpful in permitting one to derive a knowledge of the macro-structure of that nonlinear system.

Traditionally, a great deal of a would-be physicist's education is devoted to the study of how to go about solving differential equations. When one uses differential equations to model some aspect of reality, reality is assumed to have a continuous nature in which all transitions manifested in such a system will be smooth and not discrete. The problem with this approach, however, and leaving aside the difficulties surrounding the idea of what is meant by 'being continuous', is that the majority of differential equations are not solvable.

One of the reasons for the lack of success of differential equations is the following. If one is to hope to have a chance of solving a differential equation, one must be able to find some minimal number (which varies from context to context) of regular invariants that can be fed into a given system of differential equations. Only those phenomena manifesting properties which can be fit into this mold of regular invariance are somewhat amenable to treatment by differential equations.

Although differential equations can handle certain kinds of situations exhibiting change over time, many of the contexts displaying various degrees and instances of change do so in an irregular, erratic, variable manner. Such erratic, irregular modes of change tend to fall beyond the capacity of present differential techniques.

Consequently, often times, the most one can hope to accomplish if one continues to rely on differential equations to solve problems concerning nonlinear systems is to try to come up with linear approximations for nonlinear change. These sorts of technique, however, tend to yield results that are far from satisfying.

Chaos: a special case of nonlinearity

James Yorke, a mathematician who gave the study of the dynamics of nonlinear systems its name (i.e., chaos) saw the need to discover better ways of uncovering regularities in the midst of apparent disorder and erratic behavior. He addressed this issue in his paper entitled: "Period Three Implies Chaos".

The main thesis of his paper was as follows. In any one-dimensional system, cycles of period three will display regular cycles interspersed by chaotic cycles. In other words, Yorke claimed that the idea of a period-three oscillatory system which regularly repeats itself but does not give expression to chaotic behavior is not possible in a one-dimensional system.

If one has a period-three oscillatory system (even one as simple as a one-dimensional system), and if some parameter is changing at a rate, and in a direction which pushes the system deeper and deeper into nonlinearity, this affects the structural character of the system's equilibrium, causing it to bifurcate. The system, then,, will proceed to oscillate between the points of bifurcated equilibrium. As the system is pushed deeper into nonlinearity due to the continued changing character of one of the system's parameters, the rate of the bifurcation process will increase, and the system will oscillate among the different junctures of equilibrium.

Eventually, the accelerating bifurcation process will reach what is referred to as the point of accumulation. At this point, the previously regular periodic character of the system will be replaced with chaotic behavior. However, if the system is further driven into nonlinearity by the changing character of the same parameter, one will observe pockets or envelopes of regularity re-emerging in the midst of the chaotic behavior.

If one vectors parameters other than the one with which one started and, thereby, subjects the system to increasingly nonlinear forces of a different sort, this will lead to the same system displaying a new set of bifurcation/point of accumulation characteristics. Therefore, the dialectic between chaotic behavior and the emergence of pockets of regular periodicy which occurs in the context of the vectoring of different parameters will generate unique arrangements of order and chaos.

Chaos refers to the way in which systems manifesting perturbations or fluctuations never settle down to a single, stable point of equilibrium. Nonetheless, chaotic systems also exhibit envelopes of stability and regular periodicy which, suddenly and unpredictably, emerge, from time to time, in the midst of such fluctuations.

One further feature of chaotic systems concerns the way in which the structure of these systems seems to run indefinitely deep. If one examines any given region of a chaotic system closely enough, that region will display the same structural character as the entire system as a whole does.

In other words, the micro-structure of any given region of a chaotic system is a reflection of the macro-structure of the system. Moreover, if one were to examine the micro-structure of a given region of the chaotic system, one would discover an underlying mini-micro structure that, again, was a reflection of the structural character of the macro level of the chaotic system. This aspect of chaotic systems is an expression of a fractal-like property.

This quality of fractal-like structure in chaotic systems is very crucial and central. Essentially, what it means is this. Even very simple deterministic systems are, under the right circumstances, capable of producing behavior which, on the one hand, appears to manifest random properties yet which is, on the other hand, highly structured on a multiplicity of levels.

| Next | Chaos - Part 3 | Chaos - Part 4 |

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