Bifurcations explosions of limit sets

Here we treat bifurcations as peculiarities in the dependences 0)(x, k) on the initial x and the parameters k and of coT(k) on the parameters %. This is a somewhat original viewpoint since, as usual, the dependence of the limit set on the initial data is not considered when studying bifurcations and attention is concentrated on its dependence on the parameters. The necessity to consider peculiarities of the co(x, k) function is due to our aim of examining slow relaxations. 

The same aim also dictates the overall variety of peculiarities. Among the 

The appearance of new limit points consists of the following. Let there exist a sequence of such x,- - x, kt - k for which co(x, % ) has such a point y that we obtain 

For the co(x, k) and (u(fe)-bifurcations, the following fact will be essential. In the limit set corresponding to the bifurcation point (x, k )(or k ) there exists such a point y which is localized at a finite distance from the whole of the sets co(x , )[or co( )] for a certain sequence (x , kt) - (x, )[or - S )J. The appearance of new co-limit sets consists in the following. Let there exist such sequences x, — x, k, - k that for every yeco(x, k ) 

Q-bifurcations are different for co-bifurcations. For the former at a finite distance from those limit sets that correspond to the converging sequences [(x , ki) or /e,] we can find all the points of a certain limit set co(x, k ) and not only some points of ca(x, )[or co(A )]. 

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