Normal Form of Vector Field in Dynamics 📂Dynamics

Normal Form of Vector Field in Dynamics

Definition

Let $p(x;r)$ be some polynomial function. $$ \dot{x} = p(x; r) $$ To describe the properties of a dynamical system, the simplified vector field as above is called a Normal Form.

Explanation

One of the challenging aspects of studying dynamics, such as chaos, bifurcation, fractals, etc., is the feeling that there is a lack of mathematical rigor and generality. Whether this perception is due to one’s own lack of understanding or an actual ambiguity in the subject, the approach of learning ‘phenomena’ through ’examples’ can be quite disconcerting. Unlike typical mathematics books that start with definitions, followed by relevant theorems, their proofs, examples, and practice problems, the intertwined order in dynamics can be confusing.

Normal forms are not inherently complicated and can actually simplify a series of inquiries depending on the objective. However, from a student’s perspective, they might contribute to the confusion mentioned earlier.

Example ¹

$$ \dot{x} = r - x^{2} $$ For instance, the saddle-node bifurcation is often explained using the differential equation above. The change in the vector field happens around $r = 0$, where at $r > 0$ there exist two fixed points $x = \pm \sqrt{r}$, at $r = 0$ there is only one $x = 0$, and at $r < 0$ there are none. Clearly, $r = 0$ is a bifurcation point, and the phenomenon where two fixed points become saddle nodes and then disappear is called ‘saddle-node bifurcation’. However, this is just one example of explaining saddle-node bifurcation, and $\dot{x} = r - x^{2}$ itself is not the entirety of it, which can clash with ‘mathematical intuition’. A mathematician would want to define saddle-node bifurcation clearly and explore its generalized form, but often, the discussion moves on to the next examples without elaborating further.

Here’s another example of a saddle-node bifurcation. $$ \dot{x} = (r-x) - e^{-x} $$ Geometrically, the right side of the system is represented by the sum of a line $\dot{x} = r-x$ and the curve of an exponential function $\dot{x} = e^{-x}$. The number of intersections between them determines the number of fixed points. This system, while slightly different in its mathematical expression, demonstrates the saddle-node bifurcation described earlier. From the perspective of bifurcation, this system is not much different from $\dot{x} = r - x^{2}$, especially when considering the Taylor expansion of $e^{-x}$ near $x = 0$: $$ \begin{align*} \dot{x} =& r - x - e^{-x} \\ =& r - x - \left( 1 - x + {{ x^{2} } \over { 2 }} + \cdots \right) \\ =& (r - 1) - {{ x^{2} } \over { 2 }} + O \left( x^{3} \right) \end{align*} $$

Regardless of how many examples like $\dot{x} = (r-x) - e^{-x}$ exist for saddle-node bifurcation, the simplest form it can be represented in is called the Normal Form for Bifurcation, and in this context, $\dot{x} = r - x^{2}$ is seen not just as a single example.