Gradient dynamical system

In the previous equation, the sum runs over all critical points of the gradient dynamical system. In the Bonding Evolution Theory, the critical points form the molecular graph. In this graph, they are represented according to the dimension of their unstable manifold. Thus, critical points of / = 0, are associated with a dot, these with I = 1 are associated with a line, these with / = 2 by faces, and finally these with 7=3 by 3D cages. 

The gradient dynamical system and the catastrophe theories are two very useful and complementary mathematical tools for the study of the energetic and mechanisms of chemical reactions. We propose a classification of the potential functions and of the control space parameters. It emerges that the structural stability is a central concept for the understanding of chemical reactions and of chemical reactivity. 

Gradient dynamical system. The vector field of a gradient dynamical system is the gradient of a function called potential function, i.e. X(m) = VV( x ca ), where x implicitly denotes the set of the q variables of R4 defining the point m of the manifold M and where ca stands for the control space parameters. 

Application of the gradient dynamical system theory allows delineation of non-overlapping basins forming a partition of the molecular space in from which the basin adherences are removed. This partition defines a topology (unions of basin interiors completed by the empty set), which is metric in the Lewis sense (the connectivity between basins being related to their synapticity), and which is also a... 

The gradient dynamical system theory appears to be a method of choice for partitioning the molecular space into non overlapping volumes on the basis of energetic or statistical criteria. 

A gradient dynamical system is the gradient vector field W(m) of a function, called potential function, the first and second derivatives of which are defined at any point of M. 

The index of a critical point m of a gradient dynamical system is the number of positive eigenvalues of the matrix of the second derivatives of the potential function at m. In this case, a critical point is said to be hyperbolic if none of the eigenvalues are zero. In the case of a gradient dynamical system, the index of a critical point is the number of positive eigenvalues of the matrix of the second derivatives of its potential function at me. A critical point is said hyperbolic if none of the eigenvalues are zero. 

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