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Dynamical (Deterministic) Models of Evolution

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Extracting Knowledge From Time Series

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Abstract

Dynamical modelling requires specification of a D-dimensional state vector \(\textbf{x}=(x_{1},x_{2},\ldots,x_{\mathit{D}})\), where x i are dynamical variables, and some rule Φ t allowing unique determination of future states \(\textbf{x}(t)\) based on an initial state \(\textbf{x}(0)\):

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Notes

  1. 1.

    If A and B are number sets, one gets algebraic or transcendental equations. If they are function sets, one gets differential, integral and other equations depending on the kind of the maps.

  2. 2.

    A recurrent formula is the relationship of the form \(x_{n+p}=f(x_n, x_{n+1},\ldots,x_{n+p-1})\) allowing calculation of any element in a sequence if its p starting elements are specified.

  3. 3.

    It is intermediate between a microscopic level, when one studies elements of a system separately (e.g. molecules of a fluid), and a macroscopic one, when an entire system is considered as a whole (e.g. in terms of some averaged characteristics).

  4. 4.

    “Superposition (composition) of functions is arranging a composite function (function of function) from two functions” (Mathematical dictionary, 1988). Here, the terms “superposition” and “composition” are synonyms. However, physicists often call superposition of functions f 1 and f 2 their linear combination \(\mathit{af}_1+\mathit{bf}_2\), where a and b are constants. Then, the meanings of the terms “superposition” and “composition” become different. To avoid misunderstanding, we use only the term “composition” in application to composite functions.

  5. 5.

    Exponential rise of a population observed at \(\alpha>0\) is called the Malthusian rise, since a catholic monk Malthus in the sixteenth century was the first who got this result. It is valid until population gets too large so that there is no longer enough food for everybody.

  6. 6.

    This is a diode with a p – n junction whose capacity depends on voltage, i.e. an electrically controlled capacitor. Circuits with such diodes are used in radioengineering for more than half a century. They were even suggested as memory elements for computers. Different kinds of such circuits are widely presented in contemporary radio sets and TV sets.

  7. 7.

    When a charge is accumulated on the capacitor plates and a current flows in the wires, electric and magnetic forces appear and tend to compress or stretch the wires. Therefore, if substances of the coil and capacitor are not hard enough, their size (and, hence, C and L) can depend on the current and voltage (dynamical variables) implying emergence of nonlinearity.

  8. 8.

    The branch A corresponds to the evolution of in-phase regimes \((m=0),\ B-D\) to the others.

  9. 9.

    Complex dynamics of this non-linear system is illustrated by a computer program available at http://www.nonlinmod.sgu.ru and in research papers Astakhov et al. (1989, 1991a).

  10. 10.

    Map (3.34) is taken for simplicity. Thus, multistability in a set of quadratic maps is formed on the basis of a period-doubled cycle, while in a set of maps (3.34) it is observed already for the period-1 cycles. When an isolated map (3.34) has two period-1 states, there are four period-1 oscillation kinds in a set of two maps (Bezruchko and Prokhorov, 1999).

  11. 11.

    A distribution of the values of a characterising quantity over an automaton workspace.

  12. 12.

    It is a ring consisting of a non-linear amplifier (characterised by a function f), an inertial element (a filter with a response time determined by ε) and a delay line (with a delay time τ).

  13. 13.

    As well, there are purely electric mechanisms of neuron coupling.

  14. 14.

    In von Neumann’s model of computations (realised in a usual computer), memory access is possible only via an address, which does not depend on the memory contents. Associative memory is accessible based on the current contents. Memory contents can be called even by partial or distorted contents.

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Authors and Affiliations

  1. Saratov State University, Astrakhanskaya Street 83, 410012, Saratov, Russia

    Prof. Boris P. Bezruchko

  2. Russian Academy of Science, V.A. Kotel’nikov Institute of RadioEngineering and Electronics, Saratov Branch, Zelyonaya Str. 38, 410019, Saratov, Russia

    Dr. Dmitry A. Smirnov

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  1. Prof. Boris P. Bezruchko
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Correspondence to Boris P. Bezruchko .

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Bezruchko, B.P., Smirnov, D.A. (2010). Dynamical (Deterministic) Models of Evolution. In: Extracting Knowledge From Time Series. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12601-7_3

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