In mathematics , bifurcations of differential equations are qualitative changes in the structure of the dynamic system described by such a differential equation when one or more parameters of the equation are varied.
Summary
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- 1 Story
- 2 Theory
- 3 Introduction
- 4 Limit cycle
- 5 Linearization
- 6 Forks
- 1 Definition
- 2 Local bif’n
- 2.1 Example 1
- 2.2 Bif’n Hopf
- 2.2.1 Example 2
- 2.3 Asymptote
- 3 Global Bif’n
- 7 Diagrams
- 8 Own value
- 9 See also
- 10 References
History
The name “bifurcation” (abbreviated “bif’n”) was first introduced by Henri Poincaré in 1885 in the first mathematics document showing such behavior [1] . Poincaré also named various types of stationary points and classified them [2] .
Theory
Bifurcation theory is a mathematical field focused on the study of changes in the qualitative or topological structure of the behavior of a set of equations. Theory is of practical importance in engineering and physics .
In a dynamic system, a bifurcation is a period of doubling, quadrupling, etc., that accompanies the appearance of chaos , or vice versa. Represents the sudden appearance of a qualitatively different solution for a nonlinear system when some parameter is varied.
Bifurcation theory studies the behavior of families of solutions of differential equations, such as integral curves of a vector field. In reference to dynamic systems, a fork occurs when a small variation in the values of the parameters of a system causes a qualitative or topological change . Branches can occur in either continuous or discrete systems.
Introduction
There are many cases of forks, and they are divided between two types or classifications: local forks and global forks . Bifurcations can exist in one, two, … etc dimensions of systems of differential equations (vector space of any base). As said, bifurcations occur in both systems continuous (described by ordinary differential equation , differential equations delayed , partial differential equation , linear differential equation or nonlinear, and differential equations of several degrees) and systems discrete (described by maps).
When the parameters are changed, the bifurcations that appear may be stable or unstable , and may also be periodic upon reaching stabilization or while in instability. Although there are an infinite number of ways for a system to change (via its parameters), there are only several types of stable structural changes [3] . René Thom proposed seven types of elemental catastrophes in the three dimensions of space and the dimension of time. The seven catastrophes are the fold , the cusp , the butterfly , the swallowtail , the elliptical navel , the hyperbolic naveland the parabolic navel . A special state that is stable in the local system but can become unstable in the presence of diffusion is called Turing instability , named by mathematician Alan Turing .
Limit cycle
Important in bifurcation phenomena is the concept of a limit cycle . A limit cycle is a closed and isolated path ; This means that the paths of their neighbors are not closed, the neighbors move away or approach the limit cycle. They are distinguished in being isolated. Limit cycles can only occur in nonlinear systems. Linear systems have closed paths but are not isolated.
A stable limit cycle is one that attracts all neighboring paths. A system with a stable limit cycle can present self-sustained oscillations, as for example in two-dimensional biological dynamic systems.
In the case where all neighbor paths approach the limit cycle as time tends to infinity, it is called a stable or attractive limit cycle. If the neighbors approach with time tends to negative infinity, then it is an unstable limit cycle.
There are also cycle limits that are neither stable, for example, a neighbor may approach the limit cycle from the outside, but the inside of the limit cycle is addressed by other cycles. Stable limit cycles are examples of attractors , and as stated, involve self-sustained oscillations. In this case the closed path describes stable periodic behavior of the system, and any small disturbance of this closed path returns the system to its original state … like the rhythm of the heart after running.
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 . Subcritical bifurcation can be dangerous, discontinuous, they are called (“hard”) because many times it has no attractor (it has nowhere to go). While the supercritical fork is not dangerous, it is continuous, and is said to be “soft” fork.
Linearization
The analytical linearization process is comprehensive to analyze the fixed points. The dynamics are examined near a fixed point, but not at the point. For example, is there a disturbance on the “left” of the point and is it seen how the function behaves, converges or diverges? This is one way to determine stability. The same is done for the “right” of the fixed point. The word “linearization” is used because during the analytic process, higher order terms (> 2) from the Taylor series of the function are ignored .
Theorem: Suppose that x o is an equilibrium point of the differential equation dx / dt = f (x) where f is a continuously differentiable function. So,
1.) if df (x o ) / dt <0, then x o is a “sink”;
2.) if df (x o ) / dt> 0, then x o is a “source”;
3.) If df (x o ) / dt = 0, then more information is needed to determine the type of point.
Forks
The qualitative structure of the flux in the vector field can change when the parameters are varied: in particular, fixed points can be created or destroyed, or their stability can change. These qualitative changes in dynamics are called bifurcations , and the values of the parameters at which they occur are called the bifurcation point .
Definition
The definition of a fork is not exact, as it can be said as Captain Hector Barbossa in the movie Pirates of the Caribbean when he said: “And thirdly, the rules are more guidelines than they are definitions.” In general, a fork refers to a qualitative change in the behavior of a dynamic system with some parameter on which the system depends when it continuously varies.
Definition: A point (x 0 , C) is an equilibrium bifurcation point if the number of solutions of the equation f (x; C) = 0 (x with the variant and C being the parameter), and if in each environment of (x 0 , C), is not constantly independent of parameter C.
Local bif’n
Local cases include but are not limited to trident bifurcation (where a new one comes out, called “pitchfork” in English), saddle-node bifurcation (where it disappears, called tangential , or “saddle-node” or “fold” or “tangent” in English), transcritical bifurcation (where they collide), vertical bifurcation , and Hopf bifurcation in minimum of two dimensions, and zip bifurcation . Hopf bifurcations have periodic orbits with limit cycles, and their averages can be calculated with asymptotic expansions of radio and frequency [4]. Trident bifurcations, like Hopf bifurcations, can be supercritical or subcritical, depending on the signal of the third derivative . Bifurcations are found by analyzing the linearized system (determinant of the Jacobian matrix) at the bifurcation point.
Summary of several examples of local bif’n:
- Chair-node bifurcation
- Transcritical bifurcation
- Trident fork
- Bifurcation doubling period
- Hopf Fork
- Zip fork
Example 1
An example of dimension 1 (supercritical). Bifurcations of catastrophe fold or cusp. The differential equation:
dx / dt˙ = 4x – x 3 + C
describes a structure of a physical dynamic system, where C is the parameter that is varied to find a qualitative change (which can be stable or unstable (or vice versa) of stabilization or stabilization) in the dynamic system.
Bif’n Hopf
Definition 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 (or critical point , value where the derivative is zero). 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. The Hopf fork is also known as the Poincaré-Hopf-Andronov fork (Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov).
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 J or the Jacobian of a dynamic system evaluated at an equilibrium point, and suppose that all the eigenvalues (eigenvalues) of J o 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 variation of the parameter.
Example 2
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 [7] .
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 Hopf bifurcation chemical reaction” was found in 1995 in Berlin, Germany [8] .
Asymptote
An asymptotic series expansion , or Poincaré expansion, is a formal series of functions that has the property that truncation of the series after a finite number of terms provides an approximation to a given function as the argument of the function tends toward a given point and often infinite.
A formal series is a generalization of a polynomial , where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number [9] .
Asymptotic expansions can be derived to limit cycle solutions due to a Hopf bifurcation, as for example, in illustrated chemical reaction systems [10] .
Global bif’n
Global cases include but are not limited to heterocline connection bifurcation, homoclinic connection bifurcation, infinite period bifurcation, and Takens-Bogdanov bifurcation , [11] . Global ones are more complex and there is no easy way to determine them [12] .
Summary of several examples of global bif’n:
- Homoclinical connection fork
- Heterocline connection branch
- Takens-Bogdanov fork
- Bifurcation of infinite period
- Bifurcation of blue sky ( catastrophe of the same name with periodic trajectory tends to the critical value without limit, Fuller 1967)
Diagrams
Branch diagrams (Graphs). A bifurcation diagram of a dynamic system is a layering of its parameter space induced by topological equivalence, along with representative phase portraits of each stratum. Stable solutions are usually represented by solid lines, while unstable solutions are represented by dotted lines.
Own value
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). To be stable, the trace must be negative and the determinant positive (necessary conditions). The trace, itself, is the sum of the eigenvalues, while the determinant is the product of the eigenvalues. So clearly, when looking for the phenomena of interest knowing the stable state of the system, we look for the situation where the trace can be positive or the negative determinant.
