Wave bifurcation

A spatial Hopf bifurcation, commonly called a wave bifurcation, corresponds to a pair of purely imaginary eigenvalues for some 0> Ci = 0 and C2 > 0. According to the stability conditions (10.23), T < 0, and therefore... 

Spatial Hopf bifurcations or wave bifurcations can never occur in two-variable reaction-diffusion systems, see Sect. 10.1.2. This is no longer the case for reaction-transport systems with inertia. As shown in Sects. 10.2.1 and 10.2.2, spatial Hopf bifurcations are in principle possible in two-variable hyperbolic reaction-diffusions... 

The uniform steady state of a DIRW can undergo a spatial Hopf or wave bifurcation only if the right-hand side of (10.115) is positive. In other words, the rate of activation must exceed the loss rate of the activator in the steady state. If a spatial Hopf instability occurs, then all spatial modes with wavenumbers bigger than jj, the positive root of (10.107), are unstable. 

The coefficients C K) of the characteristic polynomial det[J( )—X tl4] = 0 and the Hurwitz determinants A K) are easily obtained using computational algebra software such as Mathematic A (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario). The th mode undergoes a stationary bifurcation when condition (12.41d) is violated, namely c K) = 0, as discussed in Sect. 1.2.3, see (1.36). In other words, a Turing bifurcation of the uniform steady state corresponds to c iki) = 0 with k 0, while the stability conditions (12.41) are satisfied for all other modes with k fej. The feth mode undergoes an oscillatory bifurcation when condition (12.41c) is violated, namely A K) = 0, as discussed in Sect. 1.2.3, see (1.38). A wave bifurcation of the uniform steady state corresponds to A k i) = 0 with k f/ 0, while the stability conditions (12.41) are satisfied for all otha modes with k few As discussed in Sect. 10.1.2, see (10.29), a wave bifurcation cannot occur in a two-variable reaction-diffusion system. 

Given the lengthy expressions for c, (AT) and A,(AT), an exact general derivation of the Turing bifurcation and wave bifurcation thresholds is neither desirable nor analytically feasible for the latter, even with the help of symbolic computation software. Vanag and Epstein have carried out numerical evaluations of the eigenvalues... 

Krbmker, S. Wave bifurcation in models for heterogeneous catalysis. Acta Math. Univ. Comenianae 67(1), 83—100 (1998). http //alf 1.cii.fc.nl.pt/EMIS/joumals/AMUC/ vol-67/ no l/ kromker/kromker.html... 

Shi WT, Goodridge CL, Lathrop DP (1997) Breaking waves bifurcations leading to a singular wave state. Phys Rev E 56(4) 4157-4161... 

It was observed that the yellow wave propagated downward in the presence or absence of natural light at pH 4.68 of lower content. However, a remarkable change in its characteristics could be noticed when the pH was reduced by adding a small amount of acetic acid. At pH<2.72, the yellow wave bifurcates into many alternate red and yellow bands at the lower end of the tube in the presence of natural light. 

Consequently, when D /Dj exceeds the critical value, close to the bifurcation one expects to see the appearance of chemical patterns with characteristic lengtli i= In / k. Beyond the bifurcation point a band of wave numbers is unstable and the nature of the pattern selected (spots, stripes, etc.) depends on the nonlinearity and requires a more detailed analysis. Chemical Turing patterns were observed in the chlorite-iodide-malonic acid (CIMA) system in a gel reactor [M, 59 and 60]. Figure C3.6.12(a) shows an experimental CIMA Turing spot pattern [59]. 

The condition that gives rise to multiple shock fronts (i.e., allows a shock wave to bifurcate as indicated in Fig. 4.10(b)) will occur when the second wave propagation velocity (with respect to the laboratory) is given by (4.39). How-... 

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

Leaving the details, the equation describing the motion of one particle in two electrostatic waves allows perturbation methods to be applied in its study. There are three main types of behavior in the phase space - a limit cycle, formation of a non-trivial bounded attracting set and escape to infinity of the solutions. One of the goals is to determine the basins of attraction and to present a relevant bifurcation diagram for the transitions between different types of motion. 

At some point, gradients may become infinite and form a shock wave. This breaking time represented as stage 3 in the cases (b) and (c) of Figure 8.8 depends on the initial distribution and solution properties and is known mathematically as a bifurcation (e.g., Logan, 1987 Strang, 1986). Further evolution results in a... 

Erom a general experience with wave-packet motion in periodic potentials [237], it may be expected that the complexity of the dynamics is partially caused by the symmetric excitation of the system (i.e., at wave function right from the beginning. To simplify the analysis, it is therefore helpful to invoke an initial preparation that results in a preferred direction of motion of the system. With this end in mind, we next assume that the initial wave packet contains a dimensionless average momentum of po = 23.24, corresponding to an... 

Fig. 9. Linear stability diagram illustrating (21). The bifurcation parameter B is plotted against the wave number n. (a) and (b) are regions of complex eigenvalues < > (b) shows region corresponding to an unstable focus (c) region corresponding to a saddle point. The vertical lines indicate the allowed discrete values of n. A = 2, D, = 8 103 D2 = 1.6 10-3.

Fig. 11. Bifurcation diagram for an even wave number and

These results are thus in agreement with those of bifurcation theory. In the case of odd wave numbers they demonstrate that in general the bifurcation diagrams have to exhibit a subcritical branch. However, there always exists even for odd wave numbers a value of the parameters such that the bifurcation is soft and this value marks the transition from an upper to a lower subcritical branch (see Fig. 21). This feature was less... 

The Hopf bifurcation approach is a mathematically rigorous technique for locating and analysing the onset of oscillatory behaviour in general dynamical systems. Another approach which has been particularly well exploited for chemical systems is that of looking for relaxation oscillations. Typically, the wave form for such a response can be broken down into distinct periods,... 

If the CSTR is fed with both A and B, so p0 > 0, then a fifth pattern of response can also be found over a narrow range of experimental conditions. This is shown in Fig. 6.19(e) and has both a breaking wave and an isola. In total such a bifurcation diagram shows three extinction points and only one ignition. 

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