The author and his coinventor, Jon Borresen, came up with the idea when modeling connected biological neurons. Binary oscillator technology can be applied to the design of arithmetic logic units ALUs , memory, and other basic computing components. Examples of Simulink models, referred to in earlier chapters of the book, are presented in Chap.
It is possible to change the type of input into the system, or parameter values, and investigate the output very quickly. This is as close as one can get to experimentation without the need for expensive equipment. Three examination-type papers are listed in Chap.
Both textbooks and research papers are presented in the list of references. The textbooks can be used to gain more background material, and the research papers have been given to encourage further reading and independent study. This book is informed by the research interests of the author which are currently nonlinear ordinary differential equations, nonlinear optics, multifractals, neural net- works, and binary oscillator computing.
Some references include recently published research articles by the author along with an international patent. The prerequisites for studying dynamical systems using this book are undergrad- uate courses in linear algebra, real and complex analysis, calculus, and ordinary differential equations; a knowledge of a computer language such as C or Fortran would be beneficial but not essential.
Recommended Textbooks 1. Bhattacharya, M. Sipahi, T. Niculescu, P. Pepe eds. Mallet-Paret, J. Wu, H. Zhu, Y. Instructors can automate grading of MATLAB coursework assignments using Cody Coursework; this is a free resource available to use for faculty who have an active site licence. More details can be found on the relevant MathWorks web pages.
Thanks also go to all of the reviewers from the editions of the Maple and Mathematica books. Finally, thanks to my family, especially my wife, Gaynor, and our children, Sebastian and Thalia, for their continuing love, inspiration, and support. This chapter deals with linear discrete dynamical systems, where time is mea- sured by the number of iterations carried out and the dynamics are not continuous. In applications this would imply that the solutions are observed at discrete time intervals.
Recurrence relations can be used to construct mathematical models of discrete systems. They are also used extensively to solve many differential equations which do not have an analytic solution; the differential equations are represented by recurrence relations or difference equations that can be solved numerically on a computer.
Of course one has to be careful when considering the accuracy of the numerical solutions. Ordinary differential equations are used to model continuous dynamical systems later in the book.
The bulk of this chapter is concerned with a linear discrete dynamical system that can be used to model the population of a single species. As with continuous systems, in applications to the real world, linear models generally produce good results over only a limited range of time. The Leslie model introduced here is useful when establishing harvesting and culling policies.
Nonlinear discrete dynamical systems will be discussed in the next chapter. If one starts with an initial value, say, x0 , then iteration of 2. Equations of the form 2. Consider the following simple example. Example 1. Given that x0 D 10; and n D 5, the balance after 5 years will be x5 D 11; dollars. The general solution of the first-order linear difference equation xnC1 D mxn C c; n D 0; 1; 2; 3; : : : ; 2.
Applying the recurrence relation given in 2. If m D 1, then the sum of the geometric sequence is n. This concludes the proof of Theorem 1. The general form of these equations with constant coefficients is axnC2 D bxnC1 C cxn : 2. The general solution of the second-order recurrence relation 2. A cos. When the eigenvalues are complex, the solution oscillates and is real. The solution of system 2. Substituting, 2. The difference equation 2. Since the system is linear, the general solution is thus of the form xn D.
Consider the following simple examples. Example 2. The general solution is therefore xn D k1 3n C k2. The final solution is 4 n 1 xn D 3 C. The general solution is xn D. Suppose that the national income of a small country in year n is given by In D Sn C Pn C Gn , where Sn , Pn , and Gn represent national spending by the populous, private investment, and government spending, respectively.
Substitution for Sn then gives PnC1 D Finally, assume that the government spending is kept constant. If the initial national income is G dollars and 1 year later is 32 G dollars, determine i a general solution to this model; ii the national income after 5 years; and iii the long-term state of the economy.
Equation 2. Assume that the solution involves a constant term also; try In D k3 G, then from 2. When n D 5, I5 D G, to three decimal places. Therefore, the economy stabilizes in the long term to a constant value of 3G.
This is obviously a very crude model. A general n-dimensional linear discrete population model is discussed in the following sections using matrix algebra. For most species the number of females is equal to the number of males, and this assumption is made here.
The model can be applied to human populations, insect populations, and animal and fish populations. The model is an example of a discrete dynamical system.
As explained throughout the text, we live in a nonlinear world and universe; since this model is linear, one would expect the results to be inaccurate in the long term. However, the model can give some interesting results, and it incorporates some features not discussed in later chapters. The following characteristics are ignored—diseases, environmental effects, and seasonal effects.
The book [8] provides an extension of the Leslie model, investigated in this chapter and most of the [1—7] cited here, where individuals exhibit migration characteristics. A nonlinear Leslie matrix model for predicting the dynamics of biological populations in polluted environments is discussed in [7]. Assumptions: The females are divided into n age classes; thus, if N is the theoretical maximum age attainable by a female of the species, then each age class will span a period of Nn equally spaced, days, weeks, months, years, etc.
The population is observed at regular discrete time intervals which are each equal to the length of one age class. Thus, the kth time period will be given by tk D kN n. Let bi denote the number of female offspring born to one female during the i th age class, and let ci be the proportion of females which continue to survive from the i th to the. Front Matter Pages i-xv. Pages Linear Discrete Dynamical Systems.
Nonlinear Discrete Dynamical Systems. Complex Iterative Maps. Electromagnetic Waves and Optical Resonators. Fractals and Multifractals. The Image Processing Toolbox. Differential Equations. Planar Systems. Interacting Species. The sphere will not change in the critical point, with dynamical systems using matlab r that wk is now!
What happens for other initial conditions? As explained throughout the text, we live together a nonlinear world and universe; source this model is linear, one would remind the results to be inaccurate in hospital long term.
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Once computed, the synaptic weights remain fixed.
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