Subsections

• Definitions.
• Dynamical system:
• Asymptotic behaviour:
• Attractor:
• Basin and separatrix:
• Limit points and characteristic exponents:
• Structural stability:
• Domain:
• Gradient dynamical system:
• Number of critical points:
• Structural stability of gradient dynamical system:
• Control space:
• Catastrophe:

The Mathematical model.

The dynamical system theory has been pioneered at the beginning of the last century by Poincaré, Lyapounov and Morse. It has been further developed after the fifties by Peixoto, Arnol'd, Palis, Smale, Thom and other mathematicians. It is now acknowledged as a general body of Mathematics.

Definitions.

Dynamical system:

A dynamical system is a vector field bound on a manifold $M$. If this vector field is of class $C^1$, the solutions of the system of equations $dm/dt=\mathbf X(m)$ possess the property of local unicity, therefore the vector field has no discontinuity. This system of equations interprets $\lbrace \mathbf X\rbrace$ as a velocity field in which $p$is the time variable. This formal analogy is at the origin of the name ``dynamical system''.
 

                                                 

                                              This representation of a dynamical system shows the vectors in a
                                              plane. This kind of representation is called garden representation.

Asymptotic behaviour:

One and only one vector is bound to each point $h_t(p)$ of the manifold. Therefore, the integration of $dm/dt=\mathbf X(m)$ with respect to the variable $p$ generates only one curve or trajectory. If $p$ is a point belonging to $M$, the trajectory passing through$p$ is denoted by$h_t(p)$. The limit set of $p_t$ in $M$ corresponding to $p$ tending to$-\infty$ and $+\infty$ are called $\alpha$- and $\omega$-limits, respectively. The set of points defining the trajectories having the same $\omega$-limit set is called stable manifold of the limit set. In the same way, the set of points defining the trajectories having their same $\alpha$-limit set is the unstable manifold of the $\alpha$-limit set. In a dynamical system there are two types of points: the wandering points where $\mathbf X\neq 0$ which do not belong to limit sets and the non wandering points ($\mathbf X=0$which belong to limit sets. The set of the trajectories of a dynamical system is called the phase portrait of this dynamical system.
 

                                                      

                                   In this phase portrait some representative trajectories are represented.

Attractor:

An attractor is a limit set which constitutes the$\omega$-limit set of any points in its neighbourhood. An attractor has a stable manifold and it does not possess an unstable manifold.

Basin and separatrix:

The basin of an attractor $K$ is the stable manifold of the attractor. The attractor does not belong to its basin. The separatrix is the set of wandering points which do not belong to a basin A separatrix is the stable manifold of a limit set which is not an attractor.
 
                                    

                         The two red points are the attractors of the dynamical system and the trajectories belonging to their basins
                          are drawn in blue. The magenta dot is not an attractor, the green trajectories belong to its stable manifold
                          and constitute the separatrix of the dynamical system.

Limit points and characteristic exponents:

When limit sets are single points they are characterized by their index and by their characteristic exponents. Consider first an one dimensional dynamic system (definition set of dimension 1). The graph of the vector field is obtained by rotating each vector by $\pi/2$ in the trigonometric sense. The Lyapounov characteristic exponent is the slope of the curve representing the vector field at the critical point ($\mathbf X=0$).

                        

The generalization to $n$-dimensional definition sets is straightforward and the number of critical exponents is $n$. The index is the dimension of the unstable manifold of the critical point. It is zero for an attractor. In the two-dimensional cases, there are two types of limit points, the nodal and focal ones. The characteristic exponents of a nodal limit point are real whereas those of a focal critical point are a pair of complex conjugate numbers.

type	index	characteristic exponents ($a, b>0$)
focal attractor	0	$-a\pm\imath b$
nodal attractor	0	$-a$	$-b$
saddle point	1	$a$	$-b$
focal repulsor	2	$a\pm\imath b$
nodal repulsor	2	$a$	$b$

Three-dimensional cases are more complicated.

type	index	characteristic exponents ($a, b, c>0$)
focal nodal attractor	0	$-a\pm\imath b$	$-c$
nodal attractor	0	$-a$	$-b$	$-c$
focal nodal saddle point	1	$-a\pm\imath b$	$c$
nodal saddle point	1	$-a$	$-b$	$c$
focal nodal saddle point 2	$a\pm\imath b$	$-c$
nodal saddle point	2	$-a$	$b$	$c$
focal nodal repulsor	3	$a\pm\imath b$	$c$
nodal repulsor	3	$a$	$b$	$c$

A limit point is hyperbolic if all its characteristic exponents are non zero, degenerated otherwise.

Structural stability:

The structural stability of a dynamical system characterizes the response of the dynamical system to a weak perturbation. A weak perturbation is a weak vector field which is added to the dynamical system. This perturbation yields a new dynamical system (perturbed). If the phase portraits of the perturbed and unperturbed dynamical systems are topologically equivalent (homeomorphism), i.e. if it is possible to go from one to the other by a continuous deformation, the dynamical system is structurally stable.

Domain:

Consider a subset $M_A$ of the manifold $M$. If two any points $a$ and $b$ of $M_A$ can be joined by a path totally contained in $M_A$,$M_A$ is a domain.

Gradient dynamical system:

In a gradient dynamical system, the vector field is the gradient field of a function called potential function: $$\mathbf X(m)=\nabla V(m)$$

A generic property (i.e. true except for exceptional cases) of gradient dynamical system is that the dimension of their limit sets is 0 because the potential function is continuous and differentiable at any point belonging to the definition set. In $\mathbb{R} ^3$, these exceptions are encountered for potential functions which transforms according to a continuous symmetry group. In the case the relevant dimension of the space to be considered is 2 for the cylindrical symmetry, 1 for the spherical symmetry and 0 for the group of all the translations. The critical points all nodal points where:

$$\nabla V(\mathbf r)\vert_{\mathbf r=\mathbf r_c}=0$$

The critical exponents are the eigenvalues of the Hessian matrix.

Number of critical points:

The number of critical points fulfills the Poincaré-Hopf relation: $$\sum\limits_P (-1)^{I_P}=\chi(M)$$

where $I_P$ is the index of the critical point $P$ and $\chi(M)$ the Euler characteristic of the manifold. For aperiodic definition sets in $\mathbb{R} ^3$,$\chi(M)=1$, for a periodic one $\chi(M)=0$.

Structural stability of gradient dynamical system:

In $\mathbb{R} ^3$a gradient dynamical system is structurally stable if all its critical points are hyperbolic.

Control space:

The potential function often depends upon of parameters $c_\alpha$ called the control parameters which are the elements of the set $W$ referred to as the control space of dimension k. In the case of applications related to physics or chemistry such parameters may be related to external constraints (i.e electric or magnetic field, external pressure, temperature).

Catastrophe:

Upon the variation of these parameters the behaviour of the the potential function changes and therefore the phase portrait of the dynamical system may be changed. A subset of $W$ within which no change occurs is called domain of structural stability, within a domain of structural stability the Hessian matrix is regular. If the control space parameters are dropped from a domain of structural stability to another the phase portrait of the dynamical system changes, nevertheless the Poincaré-Hopf relation is always fulfilled. The configuration of the control parameter at which the change occurs is called a bifurcation point. At a bifurcation point the Hessian matrix of the potential function at a given critical point is singular, therefore this point becomes non hyperbolic and the structural stability is lost.
2002-04-01