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The highest and lowest values of a function, relative to a reference axis, are known as extreme values. These include absolute maximum and absolute minimum values, which represent the highest and lowest points the function reaches across its entire domain. Within a restricted portion of the function, the highest and lowest values are referred to as local maximum and local minimum values, respectively.Periodic functions, such as sine and cosine, show extreme values at infinitely many points due...
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An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the...
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Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
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A state function is a thermodynamic property that depends solely on the current state of a system, irrespective of its history or how it arrived at that state. These functions are represented by capital letters, such as U, H, and S, which stand for internal energy, enthalpy, and entropy, respectively.For instance, the value of internal energy depends on the system's state variables and remains unaffected by the process path. This means that whether the system underwent a linear process or a...
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Extremes in dynamic-stochastic systems.

Christian L E Franzke1

  • 1Meteorological Institute and Center for Earth System Research and Sustainability, University of Hamburg, Hamburg, Germany.

Chaos (Woodbury, N.Y.)
|February 3, 2017
PubMed
Summary

Predicting extreme weather and climate events is crucial. Dynamic-stochastic models, combining deterministic and random elements, show promise for forecasting these impactful events and understanding their clustering.

Area of Science:

  • Meteorology and Climatology
  • Complex Systems Science
  • Predictive Modeling

Background:

  • Extreme events, particularly meteorological and climatological ones, result in substantial economic losses and fatalities annually.
  • Accurate prediction of these extreme events is of significant societal and economic importance.
  • Existing models may not fully capture the complex dynamics leading to extreme event occurrences.

Purpose of the Study:

  • To survey the predictive skill and predictability of extreme events.
  • To explore the utility of dynamic-stochastic models for understanding and predicting extremes.
  • To investigate how nonlinear dynamics and noise components contribute to extreme event generation.

Main Methods:

  • Utilizing dynamic-stochastic models that integrate deterministic nonlinear dynamics with stochastic components (additive and multiplicative noise).

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  • Analyzing how nonlinear dynamics, multiplicative noise, and heavy-tailed additive noises generate extremes within these models.
  • Assessing the models' capability to replicate observed clustering patterns of extreme events.

    • Main Results:

    • Dynamic-stochastic models demonstrate the ability to generate extreme events through nonlinear dynamics, multiplicative noise, or heavy-tailed additive noises.
    • These models naturally capture the phenomenon of extreme event clustering.
    • The predictive skill and predictability of extremes can be effectively assessed using these integrated modeling approaches.

    Conclusions:

    • Dynamic-stochastic models offer a robust framework for studying and predicting extreme events.
    • The combination of deterministic and stochastic elements is key to capturing the complexity of extreme event generation and clustering.
    • These models hold significant potential for improving forecasts of meteorological and climatological extremes.