  The concept of nearby trajectories diverging or growing away from each other plays an important role in discussions of chaos. Three useful benchmarks for characterizing trajectory divergence are linear, exponential and geometric growth rates. Linear growth can be represented by the simple expression \(y = ax+b\), where \(a\) is an arbitrary positive constant and \(b\) is an arbitrary constant. A special case of linear growth is illustrated by stacking pennies on a checkerboard \((a = 1\), \(b = 0)\). If we use the rule of placing one penny on the first square, two pennies on the second square, three pennies on the third square, and so forth, we will end up with 64 pennies stacked on the last square. The total number of pennies on the checkerboard will be 2080. Exponential growth can be represented by the expression \(y = n_{0}e^{ax}\), where \(n_{0}\) is some initial quantity (say the initial number of pennies to be stacked) and \(a\) is an arbitrary positive constant. \((n_{0}\) is called ‘initial’ because when \(x = 0\) (the ‘initial time’), we get \(y = n_{0}\).) Going back to our penny stacking analogy \((a = 1)\), we again start with placing 1 penny on the first square, but now about 2.7 pennies are stacked on the second square, about 7.4 pennies on the third square, and so forth, and we finally end up with about \(6.2 \times 10^{27}\) pennies staked on the last square! Clearly, exponential growth outpaces linear very rapidly. Finally, we have geometric growth, which can be represented by the expression \(y = a^{bx}\), where \(a\) and \(b\) are arbitrary positive constants. Note that in the case \(a = e\) and \(b = 1\), we recover the exponential   case.[1] 