By Percival Frost Joseph Wolstenholme

The Authors of the subsequent Treatise have endeavoured to offer sooner than scholars as finished a view of the topic as convinced barriers have allowed them to do. the need of those obstacles has constructed itself during getting ready the paintings in the course of a interval of 4 years. The learn of innumerable papers, by means of the main celebrated mathematicians of all international locations, has confident the authors that the topic is nearly inexhaustible, and that, to have taken care of all components of it with whatever imminent to the fulness with which they've got taken care of the 1st element, might have swelled their paintings in a apprehensive percentage to what it has already attained.

(Typographical mistakes above are as a result of OCR software program and do not ensue within the book.)

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**Example text**

Berry and Houston [6] have also given a schematic diagram of a modeling cycle. 4) [21]. The nodes of the diagram represent information to be collected, sorted, evaluated, and organized. The edges of the diagram represent activities of two-way communication between the nodes and the corresponding source of information. S: Problem statement (arising out of real-world situations). M: Mathematical model (uses concepts/variables, relations, restrictions, defines a goal and decides priorities/quality assignments).

14 Show that x(t) = 0, t ≥ t0, is an unstable solution of the equation x + x sin 2x − x = 0. 27 Introduction to Mathematical Modeling SOLUTION The equivalent system is x1 = x2 , x2 = x1 − x2 sin 2x1 , where x = x1. Choose, V(x1, x2) = x1x2. In every neighborhood of the origin, V(x1, x2) takes both positive and negative values. We obtain V *( x) = x12 + x22 − x1x2 sin(2x1 ). In any neighborhood of the origin, |x1|, |x2| are small and sin(2x1) ≈ 2x1. The third term on the right-hand side is of order of one higher than the first two terms.

Then, V *( x) = 2a x1 f1 + 2bx2 f 2 = 2a x1[− x2 − (1/2)x1x22 ] + 2bx2 [3 x1 − (1/3)x2 ] = 2x1x2 [− a + 3b] − ax12 x22 − (2/3)bx22 . 42 Introduction to Mathematical Modeling and Chaotic Dynamics Set 3b − a = 0. We may choose a = 3, b = 1. We get V ( x) = 3 x12 + x22 and V *( x) = − x22 (3 x12 + 2). Now, V * (x) ≤ 0 at all points (x1, x2) ∈ R 2 and V(x1, x2) → ∞ as x12 + x22 → ∞. Hence, all the solutions are bounded as t → ∞. Also, V * (x) = 0 when x2 = 0, that is, at all points on the x1 axis. E is the set of all points on the x1 axis.