# What do you mean by Hopf bifurcation?

## What do you mean by Hopf bifurcation?

The term Hopf bifurcation (also sometimes called Poincaré-Andronov-Hopf bifurcation) refers to the local birth or death of a periodic solution (self-excited oscillation) from an equilibrium as a parameter crosses a critical value.

What is supercritical Hopf bifurcation?

A Hopf Bifurcation occurs when a periodic solution or limit cycle, surrounding an equilibrium point, arises or goes away as a parameter µ varies. When a stable limit cycle surrounds an unstable equilibrium point, the bifurcation is called a supercritical Hopf bifurcation.

### What do you mean by the Hopf bifurcation for a two dimensional dynamical system?

Hopf bifurcation occurs when a pair of complex conjugate eigenvalues of the Jacobian matrix of the dynamical system crosses the imaginary axis with a non-zero speed (all the other eigenvalues being in the left half plane) as the control parameter is varied slowly.

Who developed Hopf bifurcation?

Yuri A. Kuznetsov
Yuri A. Kuznetsov (2006), Scholarpedia, 1(10):1858. Figure 1: Supercritical Andronov-Hopf bifurcation in the plane. Figure 2: Subcritical Andronov-Hopf bifurcation in the plane.

#### What is backward bifurcation?

The phenomenon of backward bifurcation in disease transmission models, where a stable endemic equilibrium co-exists with a stable disease-free equilibrium when the associated reproduction number is less than unity, has been observed in a number of disease transmission models.

What does a bifurcation diagram look like?

A Bifurcation Diagram is a visual summary of the succession of period-doubling produced as r increases. Bifurcations occur at r=3, r=3.45, 3.54, 3.564, 3.569 (approximately), etc., until just beyond 3.57, where the system is chaotic. However, the system is not chaotic for all values of r greater than 3.57.

## Which is the best description of a Hopf bifurcation?

In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system’s stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues -…

Is the Hopf limit cycle stable or unstable?

Supercritical and subcritical Hopf bifurcations. The limit cycle is orbitally stable if a specific quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical. Otherwise it is unstable and the bifurcation is subcritical.

### Which is bifurcation branches from a fixed point?

Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point. A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Aleksandr Andronov and Eberhard Hopf .

Which is the first Lyapunov coefficient for Hopf?

The number α is called the first Lyapunov coefficient. r = − λ / α and ω = 1 + β r 2 . {\\displaystyle r= {\\sqrt {-\\lambda /\\alpha }} { ext { and }}\\omega =1+\\beta r^ {2}.\\,} The bifurcation is then called supercritical. If α is positive then there is an unstable limit cycle for λ < 0. The bifurcation is called subcritical. 