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Bifurcation in Dynamics 📂Dynamics

Bifurcation in Dynamics

Definition 1 2

In dynamical systems, a bifurcation occurs when a phase portrait exhibits a topological nonequivalence due to changes in parameters. A point in the parameter space where the system’s topological type changes, i.e., where a bifurcation occurs, is called a bifurcation point.

Local and Global 3

A bifurcation that can be detected by examining a small neighborhood of a fixed point is said to be local. Conversely, a bifurcation that cannot be detected by only looking at a small neighborhood of a fixed point or cycle is called global.

Explanation

Usually, the concept of bifurcation is not so much ‘understood’ as ‘accepted.’ This is partly because, while it’s intuitively understandable, expressing it in a mathematical statement is not simple. The definition provided above is relatively mathematical, yet the phrase topological nonequivalence ‘appears’ still leans on natural language.

The typical concern of nonlinear dynamics is the analysis of systems, where the natural question becomes how the future evolves over time. If the system accurately reflects reality, then dynamic analysis alone can have practical applications. On the other hand, if the interest lies in how the trajectory of a state changes over time, then focusing on bifurcations means being interested in how the system itself changes. This is about dealing with the set of systems at a higher dimension, which is fundamental knowledge in the world of dynamics.

Intuitive Example

Although it’s not the most mathematically obvious example, let’s try to understand how bifurcations can attract interest through a visual representation.

The figure above depicts a scenario where a weight is placed on top of a column in a two-dimensional space with downward gravity acting only on the x and z axes. If the weight is light, the column stands firm without any issues, but if the weight becomes too heavy, the column will buckle. The red marker in the figure has moved slightly to the left, but it could equally well buckle to the right, and the schematic would be as follows.

In this diagram, the marker’s position remains upright when the weight is light, but as the weight exceeds the blue line, it buckles to the left or right. In theory, it might not buckle if it were perfectly centered, but any tiny difference of… $\varepsilon \approx 0$ would cause it to deviate from the center. In this sense, the marker’s position transitions from one stable fixed point when the weight is light to two stable fixed points when the weight passes the blue line. Since a topological nonequivalence (vector field has changed) appears as the parameter ‘weight of the weight’ changes, this is a bifurcation, and the blue line corresponds to the bifurcation point. This schematic is called a bifurcation diagram.

Stepping beyond this example, if the column were modeling a road… or a bridge, the reason for understanding bifurcations would be immediately apparent. There are countless examples of bifurcations, and they can be applied to problems beyond the physical realm, such as exploring the threshold or critical point in the basic reproduction number of an epidemic model.

Types

Given the extensive academic interest, many types of bifurcations have been discovered. Some of the well-known ones include:

Local Bifurcations

Global Bifurcations

  1. Kuznetsov. (1998). Elements of Applied Bifurcation Theory(2nd Edition): p57. ↩︎

  2. Strogatz. (2015). Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering(2nd Edition): p43. ↩︎

  3. Kuznetsov. (1998). Elements of Applied Bifurcation Theory(2nd Edition): p58. ↩︎