Exothermic reaction: stationary solutions Dynamic equations x – reactant conversion y – rescaled temperature Stationary temperature: find its dependence on m from the plot at the left; multiple solutions at h > 4 multiplicity region h = 3 h = 5 h = 4 h ln m m y

Exothermic reaction: stability multiplicity region h ln m Use static relations Compute Jacobi matrix Det(Jacobi)=0 Trace(Jacobi)=0 Hopf bifurcation Parametric plot: Cusp at y=2

Dynamics near a saddle-node bifurcation NB: near the emerging saddle–node pair the trajectories go along the eigenvector x=y/h – almost 1d dynamics h = 4.5  = approaching trajectories escaping trajectory upper state

Exothermic reaction: stability diagram at g=0.5 In the region A outside the cusped region there is a single stationary state, which is unstable. In the region B there are one stable (lower) and two unstable (upper and intermediate) stationary states. In the region C both the upper and lower stationary states are stable, and the intermediate one is unstable h ln m The lower state may also lose stability as a result of a Hopf bifurcation at lower values of g

Stability diagram at different values of g Solid line: supercritical Dashed line: subcritical h  lower state upper state Re  vs. g upper state intermediate state lower state

Periodic orbits: from Hopf to saddle–loop Periodic orbits surrounding the upper state. The amplitude and period of oscillations T increase with decreasing g T Map of supercritical and subcritical Hopf bifurcations parametrized by the value of the parameter g and the dimensionless temperature at the bifurcation point y h=4.5 m =7 y g

Unstable periodic orbits and basin boundaries Unstable periodic orbits surrounding the lower state; the amplitude and period grow with increasing g (h=4.5, m=7) from Hopf at g =0.246 to SL at g =0.269 The trajectories starting outside the attraction basin of the lower stationary state are attracted to a large-amplitude limit cycle. Below: trajectories at g = 0.3. The basin is bounded by two trajectories connecting the upper state (unstable node) with the saddle

Excitable dynamics & Summary The lower stationary state and the large- amplitude cycle coexist as alternative attractors in the interval 0.269<g<0.329, with the attraction basin of the stationary state gradually expanding to fill a larger part of the interior of the limit cycle. Summary: attractors at different values of g and Hopf (H) and saddle-loop (SL) bifurcations Excitable dynamics at g = 0.32 (between two SL bifurcations)

single s.s Bifurcation diagram for catalytic CO oxidation