Hopf Fork

In mathematics , a Hopf bifurcation (also called a Poincaré-Hopf-Andronov bifurcation or Andronov-Hopf bifurcation) refers to the disappearance or local appearance of an equilibrium of a periodic solution (self-excited oscillation … a limit cycle ) of ordinary differential equations that describe a dynamic system with which that equilibrium (critical values) changes stability when, and only when, the parameter varies in value by the “right” or “left”, or mathematically, through a pair of purely imaginary eigenvalues ​​(eigenvalues) that cross the imaginary axis in the complex plane to the right (the real part). [one]

The bifurcation can be supercritical or subcritical , being stable or unstable (within a variety , “manifold” in English, two-dimensional invariant) limit cycle. The limit cycle is a set that attracts with which orbits and paths converge and which paths are periodic. [2] A variety is the geometric object that generalizes the intuitive notion of curve and surface to any dimension and on diverse bodies.

Hopf bifurcation appear in systems of differential equations with 2 minimum dimensions, and as it has been said, when a point changes its stability and a limit cycle appears. That is, a Hopf fork generates a limit cycle of a fixed point.


[ hide ]

  • 1 Story
  • 2 Definition
  • 3 Hopf’s theorem
  • 4 Shape
  • 5 Lyapunov coefficient
  • 6 Eigenvalue
  • 7 Example 1
    • 1 Other examples
  • 8 See also
  • 9 References


In general, the name “bifurcation” (abbreviated “bif’n”) was first introduced by Henri Poincaré in 1885 in the first mathematical document showing such behavior [3] . Poincaré also named various types of stationary points and classified them [4] .

The Hopf fork is also known as the Poincaré-Hopf-Andronov fork (by Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov). Be careful not to confuse Heinz Hopf (1894-1971) who was a German mathematician in the fields of topology and geometry with Eberhard Hopf.

Eberhard Frederich Ferdinand Hopf (1902, Salzburg, Austria-Hungary – 1983, Bloomington, Indiana) was a mathematician and astronomer, one of the founding fathers of ergodic theory, and a pioneer of bifurcation theory who also made significant contributions to the Topics in partial differential equations and integral equations, fluid dynamics, and differential geometry. The maximum Hopf principle is an early result of his (1927), which is one of the most important techniques in the theory of elliptic partial differential equations.

Aleksandr Aleksandrovich Andronov (Russian: Александр Александрович Андронов; 1901, Moscow – 1952, Gorky) was a Soviet physicist and member of the Soviet Academy of Sciences (1946). The Andronov Crater on the Moon is named after him. Intensive work was done on the stability theory of dynamic systems, the introduction (with Lev Pontryagin) of the notion of structural stability. In that context, he also contributed to the mathematical theory of self-oscillation (a term he named).


The definition in words of a Hopf bifurcation is the appearance or disappearance of a periodic orbit through a local change in the stability properties (parameter) of an equilibrium point. That is, the Hopf bifurcation is a critical point at which the stability of a system changes and a periodic solution emerges [5] . Tabor (1989) says it in other words: it is the bifurcation of a fixed point to a limit cycle [6] . Weisstein (2004), more or less repeats it: a bifurcation (change) from a fixed point to a limit cycle.

Definition: Conditions of appearance of limit cycles of a dynamic system.

dx / dt = f (x, C) with x ∈ R and C ∈ R and with f (x o , C o ) = 0

A limit cycle appears when a pair of complex Jacobian eigenvalues

J (C) = Dxf (x or (C), C) that cross the imaginary axis in the complex plane ,

and that does not satisfy the transversality condition,

that is, where, d / dc [ R {(λ (C)}] = 0, is not valid (it is not zero).

where C or the value for which J (C o ) has only a pair of eigenvalues ​​imagine pure. [7]

Hopf’s theorem

The theorem Hopf bifurcation using a pair of complex numbers (which are conjugate points) as a condition in which the phenomenon occurs Hopf bifurcation.

Theorem: Let Jo be the Jacobian of a dynamic system evaluated at an equilibrium point, and suppose that all the eigenvalues ​​(eigenvalues) of Jo have real parts that are negative, except for a purely imaginary non-zero conjugate pair, then a Hopf bifurcation arises when this pair of eigenvlators cross the imaginary axis due to parameter variation.

Basically, the theorem shows that the amplitude and frequency of a periodic solution of a dynamic system can be roughly calculated when an important parameter is varied. [8]


The normal form of a fork is a simple dynamic system that is equivalent to all the systems that show this fork. [9]

The normal way for the Hopf fork is:

dx / dt = x ((C + i) + a | x | 2 , where x, a are complex with C being the parameter.

If written with a = L + ib, then L is the first coefficient of Lyapunov.

if L is negative, then there is a stable limit cycle and the bifurcation is called supercritical. if L is positive, then there is an unstable limit cycle and the bifurcation is called subcritical.

Lyapunov coefficient

The Hopf bifurcation is subcritical or supercritical depending on the signal of the first Lyapunov coefficient .

A limit cycle is stable orbital if the first Lyapunov coefficient is negative and the bifurcation is supercritical. Otherwise it is unstable and the fork is subcritical.

A limit cycle appears as the focus changes stability. The direction of the cycle is determined by the first Lyapunov coefficient:

1.) supercritical (smooth, not catastrophic) if the first Lyapunov coefficient is negative

2.) subcritical (hard, catastrophic) if the first Lyapunov coefficient is positive [10]


The importance of the proper value (or eigenvalue), the Jacobian (Jacobian matrix), and complex numbers can be seen in the example of the Hopf theorem which says that in the imaginary plane bifurcations occur when both eigenvalues cross the right side of the plane (the real part that is positive).

Example 1

An example of dimension 2 (subcritical), the differential equations:

dx / dt = A –CX – X + X 2 Y;

dy / dt = CX – X 2 Y

where C is the parameter that is varied to observe a qualitative change (in this example it will be a Hopf bifurcation) in the dynamic system (which in this case is a chemical reaction ).

The equilibrium points of the system of differential equations are calculated by solving the equations:

a – cx – x + x 2 y = 0;

cx – x 2 y = 0

adding the two equations results in x = a

So the only point of balance is (a, c / a)

The Jacobian is used to determine the stability of the system:

D [f (x, y)] = | D (1,1) = -c-1 + 2xy; D (1,2) = x 2 ; D (2,1) = c-2xy; D (2,2) = -x 2 |

D [f (a, c / a)] = | D (1,1) = c-1; D (1,2) = a 2 ; D (2,1) = -c; D (2,2) = -a 2 |

Trace (D [f (a, c / a)]) = – a 2 + c – 1; stable spiral if trace is negative but unstable

Det (D [f (a, c / a)]) = a 2

Since the determinant is always positive, never zero, the equilibrium point is never chair-node [11] .

when a = 2, c <a 2 + 1, c <5 stable, so there are no limit cycles;

when a = 2, c> 5 if there are limit cycles because the spiral moves away from zero (Figure 2).

So point c = a 2 + 1 is a Hopf fork and is subcritical.

To prove that there is a Hopf fork use the Hopf fork theorem (that the eigenvalues ​​are pure imaginary and not null). In this example it is also possible in the presence of diffusion to have Turing instability.

You are reminded that the limit cycle is stable orbital if the first Lyapunov coefficient is negative and the bifurcation is supercritical. Otherwise it is unstable and the fork is subcritical.

This example demonstrates the Brusselator, which is a type of diffusion reaction that is autocatalytic (a process by which a chemical compound induces and controls a chemical reaction on itself). The Brusselator is a chemical reaction that is oscillating and nonlinear (being autocatalytic) and was controversial in the 1900s and peak between chemists. It was proposed by Ilya Prigogine at the Free University of Brussels . The name is an acronym for Brussels (‘Brussels’) and oscillator (‘oscillator’). The “smallest bifurcation chemical reaction of Hopf” was found in 1995 in Berlin, Germany [12] .

Other examples

Hopf’s bifurcation also occurs in ordinary differential equations with infinitely dimensions generated by the partial differential and delay equations, to which the Multiple Center Theorem is applied , which is a way to simplify dynamic systems by reducing the dimension of the system . An analog of the Hopf bifurcation, called the Neimark-Sacker bifurcation , occurs in dynamic systems generated by iterated maps when the critical fixed point has a pair of eigenvalues ​​e ± iθ .

The bifurcation of Bautin is a fork of a balance in a two – parameter family of ordinary differential equations regions to which the critical balance has a pair of purely imaginary eigenvalues and the first Lyapunov coefficient for the Hopf bifurcation is gone. This phenomenon is called Hopf’s generalized bifurcation .


Leave a Comment