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The idea here is that if a model is faithful in reproducing the behavior of the target system, refining the model will produce an even better fit with the target system's behavior. This is to say that if a model is faithful, successive improvements will lead to its behavior monotonically converging to the target system's behavior. Again, the import of the faithful model assumption is that if one were to plot the trajectory of the target system in an appropriate state space, the model trajectory in the same state space would monotonically become more like the system trajectory as the model is made more realistic.

What both of these basic approaches have in common is that piecemeal monotonic convergence of model behavior to target system behavior is a mark for confirmation of the model Koperski By either improving the quality of the initial data or improving the quality of the model, the model in question reproduces the target system's behavior monotonically better and yields predictions of the future states of the target system that show monotonically less deviation with respect to the behavior of the target system.

In this sense, monotonic convergence to the behavior of the target system is a key criterion for whether the model is confirmed. If monotonic convergence to the target system behavior is not found by pursuing either of these basic approaches, then the model is considered to be disconfirmed. For linear models it is easy to see the intuitive appeal of such piecemeal strategies.

Encounters with Chaos and Fractals

After all, for linear systems of equations a small change in the magnitude of a variable is guaranteed to yield a proportional change in the output of the model. So by making piecemeal refinements to the initial data or to the linear model only proportional changes in model output are expected. However, both of these basic approaches to confirming models encounter serious difficulties when applied to nonlinear models, where the principle of linear superposition no longer holds.

In the first approach, successive small refinements in the initial data used by nonlinear models is not guaranteed to lead to any convergence between model behavior and target system behavior.

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Any small refinements in initial data can lead to non-proportional changes in model behavior rendering this piecemeal convergence strategy ineffective as a means for confirming the model. The small refinement in data quality may very well lead to the model behavior diverging away from the system's behavior. In the second approach, keeping the data fixed but making successive refinements in nonlinear models is also not guaranteed to lead to any convergence between model behavior and target system behavior.

With the loss of linear superposition, any small changes in the model can lead to non-proportional changes in model behavior again rendering the convergence strategy ineffective as a means for confirming the model. The small refinement in the model may very well lead to the model behavior diverging away from the system's behavior. So whereas for linear models piecemeal strategies might be expected to lead to better confirmed models presuming the target system exhibits only stable linear behavior , no such expectation is justified for nonlinear models deployed in the characterization of nonlinear target systems.

Intuitively, piecemeal convergence strategies look to be dependent on the perfect model scenario. Given a perfect model, refining the quality of the data should lead to monotonic convergence of the model behavior to the target system's behavior, but even this expectation is not always justifiable for perfect models cf. Judd and Smith ; Smith On the other hand, given good data, perfecting a model intuitively should also lead to monotonic convergence of the model behavior to the target system's behavior. By making small changes to a nonlinear model, hopefully based on improved understanding of relevant features of the target system e.

The loss of linear superposition, then, leads to a similar lack of guarantee of a continuous path of improvement as the lack of guarantee of piecemeal confirmation. And without such a guaranteed path of improvement, there is no guarantee that a faithful nonlinear model can be perfected. Of course, we do not have perfect models. But even if we did, they are unlikely to live up to our intuitions about them Judd and Smith ; Judd and Smith If there is either no perfect model for a target system, or the perfect model still does not guarantee monotonic improvement with respect to the target system's behavior, the traditional piecemeal confirmation strategies will fail.

Merely faithful nonlinear models are not guaranteed to converge to nonlinear target system behavior under piecemeal confirmation strategies. The bottom line for modeling nonlinear systems, then, is that piecemeal monotonic convergence of nonlinear models to target system behavior is not guaranteed. This is the upshot of the failure of the principle of linear superposition.


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No matter how faithful the model, no guarantee can be made of piecemeal monotonic improvement of a nonlinear model's behavior with respect to the target system of course, if one waits for long enough times piecemeal confirmation strategies will also fail for linear systems. Furthermore, problems with these confirmation strategies will arise whether one is seeking to model point-valued trajectories in state space or one is using probability densities defined on state space.

One possible response to the piecemeal confirmation problems discussed here is to turn to a Bayesian framework for confirmation, but similar problems arise here for nonlinear models.

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Given that there are no perfect models in the model class to which we would apply a Bayesian scheme and given the fact that imperfect models will fail to reproduce or predict target system behavior over time scales that may be short compared to our interests, there again is no guarantee that monotonic improvement can be achieved for our nonlinear models I leave aside the problem that having no perfect model in our model class renders many Bayesian confirmation schemes ill-defined. For nonlinear models, faithfulness can fail and perfectibility cannot be guaranteed, raising questions about scientific modeling practices and our understanding of them.

However, the implications of the loss of linear superposition reach father than this.

Encounters with Chaos and Fractals, Second Edition

Policy assessment often utilizes model forecasts and if the models and systems lying at the core of policy deliberations are nonlinear e. Suppose administrators are using a nonlinear model in the formulation of economic policies designed to keep GDP ever increasing while minimizing unemployment among achieving other socio-economic goals. While it is true that there will be some uncertainty generated by running the model several times over slightly different data sets and parameter settings, assume that policies taking these uncertainties into account to some degree can be fashioned.

Once in place, the policies need assessment regarding their effectiveness and potential adverse effects, but such assessment will not involve merely looking at monthly or quarterly reports on GDP and employment data to see if targets are being met.

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But, of course, data for the model have now changed and there is no guarantee that the model will produce a forecast with this new data that fits well with the old forecasts used to craft the original policies. Nor is there a guarantee of any fit between the new runs of the nonlinear model and the economic data being gathered as part of ongoing monitoring of the economic policies.

How, then, are policy makers to make reliable assessments of policies? The same problem—that small changes in data or model in nonlinear contexts are not guaranteed to yield proportionate model outputs or monotonically improved model performance—also plagues policy assessment using nonlinear models. Such problems are largely unexplored. One of the exciting features of SDIC is that there is no lower limit on just how small some change or perturbation can be—the smallest of effects will eventually be amplified up affecting the behavior of any system exhibiting SDIC.

The essential point is that the nature of particular kinds of nonlinear dynamics—those which exhibit stretching and folding confinement of trajectories, where there are no trajectory crossings, and which exhibit aperiodic orbits—apparently open the door for quantum effects to change the behavior of chaotic macroscopic systems.

The central argument runs as follows and is known as the sensitive dependence argument SD argument for short :.


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Premise A makes clear that SD is the operative definition for characterizing chaotic behavior in this argument, invoking exponential growth characterized by the largest global Lyapunov exponent. Premise B expresses the precision limit for the state of minimum uncertainty for measuring momentum and position pairs in an N -dimensional quantum system note, the exponent is 2 N in the case of measuring uncorrelated electrons. Briefly, the reasoning runs as follows.


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Since quantum mechanics sets a lower bound on the size of the patch of initial conditions, unique evolution must fail for nonlinear chaotic systems. The SD argument does not go through as smoothly as some of its advocates have thought, however. There are difficult issues regarding the appropriate version of quantum mechanics e.

For instance, just because quantum effects might influence macroscopic chaotic systems doesn't guarantee that determinism fails for such systems. Whether quantum interactions with nonlinear macroscopic systems exhibiting SDIC contribute indeterministically to the outcomes of such systems depends on the currently undecidable question of indeterminism in quantum mechanics and the measurement problem.

There is a serious open question as to whether the indeterminism in quantum mechanics is simply the result of ignorance due to epistemic limitations or if it is an ontological feature of the quantum world. Suppose that quantum mechanics is ultimately deterministic, but that there is some additional factor, a hidden variable as it is often called, such that if this variable were available to us, our description of quantum systems would be fully deterministic.

Under either of these possibilities, we would interpret the indeterminism observed in quantum mechanics as an expression of our ignorance, and, hence, indeterminism would not be a fundamental feature of the quantum domain. It would be merely epistemic in nature due to our lack of knowledge or access to quantum systems. So if the indeterminism in QM is not ontologically genuine, then whatever contribution quantum effects make to macroscopic systems exhibiting SDIC would not violate unique evolution. In contrast, suppose it is the case that quantum mechanics is genuinely indeterministic; that is, all the relevant factors of quantum systems do not fully determine their behavior at any given moment.

Then the possibility exists that not all physical systems traditionally thought to be in the domain of classical mechanics can be described using strictly deterministic models, leading to the need to approach the modeling of such nonlinear systems differently e. Moreover, the possible constraints of nonlinear classical mechanics systems on the amplification of quantum effects must be considered on a case-by-case basis. For instance, damping due to friction can place constraints on how quickly amplification of quantum effects can take place before they are completely washed out Bishop forthcoming.

And one has to investigate the local finite-time dynamics for each system because these may not yield any on-average growth in uncertainties e.

Encounters with Chaos and Fractals

In summary, there is no abstract, a priori reasoning establishing the truth of an SD argument; the argument can only be demonstrated on a case-by-case basis. Perhaps detailed examination of several cases would enable us to make some generalizations about how wide spread the possibilities for the amplification of quantum effects are. Two traditional topics in philosophy of science are realism and explanation. Although not well explored in the context of chaos, there are plenty of interesting questions regarding both topics deserving of further exploration.

Chaos raises a number of questions about scientific realism see scientific realism only some of which will be touched on here. First and foremost, scientific realism is usually formulated as a thesis about the status of unobservable terms in scientific theories and their relationship to entities, events and processes in the real world.

In other words, theories make various claims about features of the world and these claims are approximately true. It seems more reasonable, then, to discuss some less ambitious realist questions regarding chaos: Is chaos a real phenomenon? Do the various denizens of chaos, like fractals, actually exist? Recall this assumption maintains that our model equations faithfully capture target system behavior and that the model state space faithfully represents the actual possibilities of the target system.

Is the sense of faithfulness here that of actual correspondence between mathematical models and features of actual systems? Or can faithfulness be understood in terms of empirical adequacy alone, a primarily instrumentalist construal of faithfulness? Is a realist construal of faithfulness threatened by the mapping between models and systems potentially being one-to-many or many-to-many?

A related question is whether or not our mathematical models are simulating target systems or merely mimicking their behavior. To be simulating a system suggests that there is some actual correspondence between the model and the target system it is designed to capture. On the other hand, if a mathematical model is merely mimicking the behavior of a target system, there is no guarantee that the model has any genuine correspondence to the actual qualities of the target system. The model merely imitates behavior. These issues become crucial for modern techniques of building nonlinear dynamical models from large time series data sets e.

In such cases, after performing some tests on the data set, the modeler sets to work constructing a mathematical model that reproduces the time series as its output. Do such models only mimic behavior of target systems? Where does realism come into the picture?