Transient chaos and unbounded dynamics are two outstanding phenomena that dominate in chaotic systems with large regions of positive and negative divergences. Here, we investigate the mechanism that leads the unbounded dynamics to be the dominant behavior in a dissipative flow. We describe in detail the particular case of boundary crisis related to the generation of unbounded dynamics. The mechanism of the creation of this crisis in flows is related to the existence of an unstable focus-node (or a saddle-focus) equilibrium point and the crossing of a chaotic invariant set of the system with the weak-(un)stable manifold of the equilibrium point. This behavior is illustrated in the well-known Rössler model. The numerical analysis of the system combines different techniques as chaos indicators, the numerical computation of the bounded regions, and bifurcation analysis. For large values of the parameters, the system is studied by means of Fenichel's theory, providing formulas for computing the slow manifold which influences the evolution of the first stages of the orbit.
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