2023年4月15日 · A drastic change of this sort (typically a change in the number of equilibrium solutions or a change in the type of stability of an equilibrium) is called a bifurcation. The value of the parameter where a bifurcation occurs is called a bifucation point. Consider the differential equation dP dt = 0.2P − q

A bifurcation occurs in a nonlinear differential equation when a small change in a parameter results in a qualitative change in the long-time solution. Examples of bifurcations are when fixed points are created or destroyed, or change their stability.

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.

When a small change in the parameter leads to large changes in the behavior of the solution, then the system is said to undergo a bifurcation. We will turn to some generic examples, leading to special bifurcations of first order autonomous differential equations.

In this project I will study the di erent types of bifurcations that can arise in ordinary di erential equations, the necessary conditions for each type of bifurcation to occur and the normal form. I will then go on to consider systems of ordinary di erential equations.

In between the fixed points the differential equation x˙ = f(x, a) x ˙ = f (x, a) dictates if x x increases or decreases along a solution. We can indicate this in the drawing by including up– or downward pointing arrows. Definition. A fixed point (x0,a0) (x 0, a 0) is called regular if. ∂f ∂x(x0,a0) ≠ …

Bifurcation Theory The change in the qualitative character of a solution as a control parameter is varied is known as a bifurcation. This occurs where a linear stability analysis yields an instability (characterized by a growth rate of a perturbation of the base solution with Re 0).

A bifurcation diagram for a parameterized family of autonomous differential equations depending on a parameter k, dy/dt = f(y; k), d y / d t = f (y; k), is a diagram in the ky -plane that summarizes the change in qualitative behavior of the equation as the parameter k is varied.

It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations.

In this chapter, we also discuss several types of bifurcations, saddle node, transcritical, pitchfork and Hopf bifurcation. Among these types, we especially focus on Hopf bifurcation. The first three types of bifurcation occur in scalar and in systems of differential equations.