Proximity Of Bifurcation Research Articles

Homoclinic explosion in the vicinity of bifurcation at the triple-zero eigenvalue

Equations describing small-amplitude motion in the proximity of bifurcation at the triple-zero eigenvalue, after being reduced to a normal form by nonlinear transformations, display a double-scale structure that combines relatively fast conservative orbital motion with slow dissipative evolution of two integrals of motion. The homoclinic explosion occurs when the attractor of the averaged dissipative equations comes close to a homoclinic trajectory of the fast conservative subsystem. Applying the method of matched asymptotic expansions allows one to reduce the dynamical system near this point to a simple next return map possessing an infinite family of invariant manifolds that can turn into attractors of different character under small parametric perturbation. After the attractor of the averaged system has been expelled from the region of closed orbits, one observes large-amplitude dynamics with rare events of anomalous intermittency superimposed upon a quasiperiodic or chaotic attractor of nonuniversal character.

Dynamics of lumped chemically reacting systems near singular bifurcation points

The systematic method of locating and classifying complex dynamical regimes of lumped reactors involves analysis of dynamics near singular bifurcation points at multiple eigenvalues. Amplitude expansion in the proximity of a singular bifurcation combined with multiple time scaling are used to derive universal equations that, though valid quantitatively only in a narrow patch of the parametric space, give full qualitative information on dynamics of a system possessing the respective highest singularity. The principal general result is that an algebraic singularity of a certain order is required to obtain dynamics of a certain dimensionality. In the lowest order, this is cusp singularity for two-dimensional dynamics in the proximity of bifurcation at double zero eigenvalue, and swallowtail singularity for three-dimensional dynamics at triple zero eigenvalue. The basic equation of two-dimensional dynamics describes various transitions between stationary and periodic states as observed, for example, in a first-order exothermic reaction. The equation of three-dimensional dynamics generates more complex phenomena, in particular, chaotic regimes.