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Some small improvements to scalar_dynam.rst #783

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17 changes: 7 additions & 10 deletions source/rst/scalar_dynam.rst
Original file line number Diff line number Diff line change
Expand Up @@ -15,10 +15,10 @@ Overview
In this lecture we give a quick introduction to discrete time dynamics in one
dimension.

This means that the state of the system is described by a single variable.
In one-dimensional models, the state of the system is described by a single variable.

Although most interesting dynamic models need two or more state variables, the
one dimensional setting is a good place to learn the foundations and build
Although most interesting dynamic models have two or more state variables, the
one-dimensional setting is a good place to learn the foundations of dynamics and build
intuition.

Let's start with some standard imports:
Expand Down Expand Up @@ -54,12 +54,10 @@ In the definition,

* time homogeneity means that :math:`g` is the same at each time :math:`t`

* first order means dependence on only one lag (:math:`x_{t+1} = g(x_t,
x_{t-1})$` is a second order difference equation and so on).
* first order means dependence on only one lag (i.e., earlier states such as :math:`x_{t-1}` do not enter into :eq:`sdsod`).


If :math:`x_0 \in S` is given, then :eq:`sdsod` recursively defines a sequence
given by
If :math:`x_0 \in S` is given, then :eq:`sdsod` recursively defines the sequence

.. math::
:label: sdstraj
Expand All @@ -71,14 +69,13 @@ given by

This sequence is called the **trajectory** of :math:`x_0` under :math:`g`.

If we set :math:`g^n := n` compositions of :math:`g` with itself, then we can
write the trajectory more simply as :math:`x_t = g^t(x_0)` for :math:`t \geq 0`.
If we define :math:`g^n` to be :math:`n` compositions of :math:`g` with itself, then we can write the trajectory more simply as :math:`x_t = g^t(x_0)` for :math:`t \geq 0`.


Example: A Linear Model
-----------------------

One easy to work with example is the **linear difference equation**
One simple example is the **linear difference equation**

.. math::
x_{t+1} = a x_t + b, \qquad S = \mathbb R
Expand Down