The research group focuses on understanding processes in various climate-related dynamical systems. We model the atmosphere, the ocean and the climate as a dynamical system. Due to the mathematical complexity of the earth system and our own limited resources we find it necessary to consider only subsystems in which external influences are prescribed by means of exogenous parameters.
The earth system is itself a subsystem of the universe, depending as it does upon external influences like irradiation from the sun. As a matter of modelling economy it is unavoidable to idealize and simplify the dynamical subsystems.
Dynamical system - Wikipedia
It is in this spirit that we analyze tropical cyclones, the ocean circulation, simple climate models with radiative transfer and the hydrologic cycle of the global climate system. The solutions can numerically determined only by high performance computers. A discrete-time , affine dynamical system has the form of a matrix difference equation:.
The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map. As in the continuous case, the eigenvalues and eigenvectors of A determine the structure of phase space. Points in this straight line run into the fixed point. There are also many other discrete dynamical systems. The qualitative properties of dynamical systems do not change under a smooth change of coordinates this is sometimes taken as a definition of qualitative: It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood.
In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates usually unspecified, but computable that makes the dynamical system as simple as possible.
A flow in most small patches of the phase space can be made very simple. This is known as the rectification theorem. The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space.
The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.
Mathematical model
In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. The results on the existence of a solution to the conjugation equation depend on the eigenvalues of J and the degree of smoothness required from h. As J does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of J are not in the unit circle, the dynamics near the fixed point x 0 of F is called hyperbolic and when the eigenvalues are on the unit circle and complex, the dynamics is called elliptic.
The hyperbolic case is also structurally stable. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.
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For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on Bifurcation theory. Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle—Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor.
In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations. In the Hamiltonian formalism , given a coordinate it is possible to derive the appropriate generalized momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure. In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible.
The volume of the energy shell, computed using the Liouville measure, is preserved under evolution. Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space. Then almost every point of A returns to A infinitely often. One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis.
The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable a is a function that to each point of the phase space associates a number say instantaneous pressure, or average height. This idea has been generalized by Sinai, Bowen, and Ruelle SRB to a larger class of dynamical systems that includes dissipative systems.
SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems. Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behavior has been called chaos.
Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system which is often hopeless , but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?
Note that the chaotic behavior of complex systems is not the issue. Meteorology has been known for years to involve complex—even chaotic—behavior. The system relating inputs to outputs depends on other variables too: Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables.
Furthermore, the output variables are dependent on the state of the system represented by the state variables. Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance , as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved computationally as the number increases.
For example, economists often apply linear algebra when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables. Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available.
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A black-box model is a system of which there is no a priori information available. A white-box model also called glass box or clear box is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take. Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly.
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Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model. In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions.
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Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively the NARMAX Nonlinear AutoRegressive Moving Average model with eXogenous inputs algorithms which were developed as part of nonlinear system identification [3] can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise.
The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque. Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition , experience , or expert opinion , or based on convenience of mathematical form.
Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision perhaps by looking at the shape of the coin about what prior distribution to use.
Incorporation of such subjective information might be important to get an accurate estimate of the probability. In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability.
Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a paradigm shift offers radical simplification [ citation needed ]. For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model.
Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model.