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Consider the function

Since for any **x**, it follows that the
Lyapunov exponent of the linear map is the same
for any orbit, namely . The
Lyapunov exponent is negative if **0<a<1**, in which
case all orbits converge to zero. When **a>1**,
the Lyapunov exponent is positive and the distance
between neighbouring orbits becomes arbitrarily large.
Hence, is related to the sensitivity
of orbits to perturbations. For ,
neighbouring orbits remain close. However, when
**a>1**, even though , the
system is not chaotic. This is due to the fact that the
system is unbounded for **a>1**. (Boundedness is
another requirement for chaos; this is better
illustrated by another example.)

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