(Color) Typical breathing pattern [(a), experiments (Ref. 15); (b), simulations (Ref. 11)] in the case of strong gas-phase coupling and multiwave patterns [(c, d), present work] in the case of weak coupling. Each row represents equally intervaled snapshots of the temperature on a disk surface during one quasiperiod. Simulations in row (b) were conducted with fixed-temperature boundary conditions and , , . Simulations in rows (c, d) were conducted with no-flux boundary conditions and , , (c), and (d); ( is a convective velocity). (a, b), 6.4 cm (c, d); reactor length (a, b), 0.3 cm (c, d).
Typical bifurcation diagram of the lumped oscillatory model [Eqs. (3) and (4) with ] showing the homogeneous steady state solutions of (a) and temperature (b) as functions of with various gas temperatures [ (1); 483 K (2), 486 K (3), and 493 K (4)]. Solid and dashed lines denote stable and unstable steady states, respectively. Points mark the Hopf bifurcation points .
Typical bifurcation diagrams showing (a) the homogeneous steady state and oscillatory branches, and (b) the periods of oscillations of the lumped system [Eqs. (3) and (4)] as functions of . Dashed-dotted lines correspond to a constant-velocity rotating pulse solution of the ODE system [Eqs. (6) and (7)] on a ring with . . Other notations as in Fig. 2.
Bifurcation map of the lumped system bounded by Hopf bifurcation lines showing the subdomains (i–iv) of the distributed system [Eqs. (3) and (4)] behavior.
Typical spatial profiles in a moving pulse solution (a) and schematic phase plane (b) showing null curves (solid and dashed lines) and (dash-dotted line) of learning model (5). In (b) thick solid and dashed lines are calculated with a fixed value that correspond to the ignited and the extinguished fronts (denotes by arrows), respectively.
Typical neutral curves corresponding to the Hopf points (lower boundary of the oscillatory domain, solid line) and (upper boundary, dashed line). , ; .
Effect of the rotation velocity on the period-one rotating pulse solution [Eqs. (9) and (10)] showing (a) the maximal and (b) the minimal concentration , (c) the rotation period , and (d) the ratio. Points in (c) denote the results of numerical simulations of the system [Eqs. (3) and (4)] on a ring. Dotted line in (d) denotes the asymptotic value of the temporal period of the corresponding lumped model [Eqs. (3) and (4)]. , .
Effect of the ring perimeter on pattern selection within the oscillatory domain: homogeneous oscillations [(a) ], pseudohomogeneous oscillations [(b) ], period-one rotating waves [(c) ], multiperiodic rotating waves [(d) ], patterns with various numbers of sink∕source points [(f) , ; (g) , ; (h) , ]. Patterns (a)–(c) and (e)–(g) are simulated with gradually varying IC; IC for pattern (d) are obtained by reproducing a period-one solution (c); pattern (h) is simulated with a stepwise IC. Time . , .
A typical sequence of pattern transformation on a ring of fixed perimeter with increasing : (a) moving waves , (b) irregular moving waves (60 Pa), (c) a multiperiodic pattern with two pairs of sinks∕sources (200 Pa), (d) a condensed sinks∕sources pattern (440 Pa), (e) a multispot irregular pattern (450 Pa), (f) homogeneous oscillations (500 Pa). . In plates (a) and (b) only a half of the ring is shown. Time (a), 25 s (b, c), 50 s (d), 100 s (e), and 150 s (f).
A typical sequence of 2D spatiotemporal pattern transformation on a wide cylinder with increasing : (a) an irregular cellularlike pattern , (b) an irregular targetlike pattern (100 Pa), (c) a regular two-target pattern (300 Pa), (d) a clusterlike pattern (480 Pa). , , . Row (a) shows a fragment of the surface.
Effect of initial conditions [(a) symmetric and (b) asymmetric] on spatiotemporal patterns on a disk. , , .
(Color) Effect of the long-range interaction strength on pattern selection showing the pattern suppression with decreasing convective velocity . Each row represents pattern transformation during one period of oscillations for conditions when the relevant four-variable system exhibits an imperfect (a) and regular (b)–(d) target patterns. , (a), 300 Pa (b)–(d); (a), (b), (c), (d). , , (annular reactor width). Flow in horizontal direction from left.
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