difference eqns

Phil hasnip mathematical modelling introduction

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Phil Hasnip Mathematical Modelling
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Introduction Discrete systems Population analysis Savings account | r | < 1 stable equilibrium different a 0 converge to a | r | > 1 unstable equilibrium different a 0 diverge only get equilibrium if a 0 = a Phil Hasnip Mathematical Modelling
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Introduction Discrete systems Population analysis Non-linear case The logistic equation and Verhulst equations are non-linear, e.g.: a n + 1 = r ( 1 - a n ) a n Their behaviour is interesting: 0 < r < 3 stable equilibrium r = 3 oscillation between 2 different values r = 3 . 6 oscillation between 4 different values – period doubling r = 3 . 7 chaos! No pattern or long-term prediction possible Phil Hasnip Mathematical Modelling
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Introduction Discrete systems Population analysis Summary We can use difference equations to model discrete processes approximate continuous processes Long-time behaviour is often of interest Does the model decay or grow? Does the model tend to a limit? Does the model oscillate? Non-linearity -→ chaos! Phil Hasnip Mathematical Modelling
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