A didactic implication of “existence of periodic point of period three implies chaos”

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Abstract

Mathematicians use neuron models to better explain the mental behaviour and cognitive structure of humane thinking (brain activity). These activities consist of pattern recognition, memory, mental calculation, and interpretation, which are exhibited in the form of dynamical characteristics such as stability, bifurcation, and chaos, etc. Using the finite difference method, we can discretize a given differential equation representing the neuron model and verify that there exists a periodic point of period three in the discrete system and hence ensure that chaotic behaviour exhibits in a single effective neuron model. The finite difference scheme is obtained by a simple linearization using Taylor series expansion of the nonlinear term where the terms with power four or higher are truncated (LeVeque, 2007). This method, which was already known by Euler (1768), was one of the classical methods to simplify differential equations that can be understood by secondary students. Our investigation demonstrates why elementary mathematics teacher, for example, secondary school teachers, should pay effort to learn more advanced mathematics during their preparation years of teacher profession. Possession of advanced subject knowledge is very helpful in interpreting the taught contents, and teachers could be capable to unpack mathematics knowledge for students’ learning.
Original languageEnglish
Publication statusPublished - 2016

Citation

Leung, K. C. I. (2016, May). A didactic implication of “existence of periodic point of period three implies chaos”. Seminars conducted at MIT Departmental Seminar, The Hong Kong Institute of Education, Hong Kong, China.

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