Andronov-Hopf bifurcation

Fig. 1.22. Period doubling and Hopf bifurcations. Top enlarged part at low coupling strength K and noise intensity Tioc Bottom Global view of selected bifurcations. AH Andronov-Hopf bifurcation In, onset of regime with n subthreshold oscillations between two spikes h onset of non-intermittent spiking). Parameters e = 0.01, a = 1.05. [41]...

The transition from vortex motion to turbulence is accompanied by a corresponding sequence of instabilities or bifurcations, which results in a qualitative change of the domain pattern. Prom the theory of dynamical systems [127] the instability called Andronov-Hopf bifurcation [58] is well known. In this case, an oscillating regime appeared between two or more states having almost equal free energy. 

Fig 11.5.1. Soft loss of stability of a stable focus at the origin through a supercritical (Li < 0) Andronov-Hopf bifurcation. 

Fig 11.5.4. supercritical Andronov-Hopf bifurcation in R . The stable focus (the leading manifold jg two-dimensional) in (a) becomes a saddle-focus in (b). A stable periodic orbit is the edge of the unstable manifold W. ... 

Fig 11.5.5. A subcritical Andronov-Hopf bifurcation, (a) An attraction basin of a stable focus is bounded by a stable manifold of a saddle periodic orbit, (b) The periodic orbit narrows to the stable focus at /x = 0, and the latter becomes a saddle-focus (1,2). 

Fig. 14.2.1. Super-critical Andronov-Hopf bifurcation occurs on the safe boundary.

Observe that when q > 0, the surface iJ = 0, corresponds to the Andronov-Hopf bifurcation, whereas the part of the surface where q < —corresponds to the vanishing of the sum of one leading exponent and a non-leading one of opposite sign. 

This corresponds to the Andronov-Hopf bifurcation. When R > 0 the equilibria Oi,2 are stable foci, and when i < 0, they are saddle-foci (1,2). The... 

Fig. C.2.1. A part of the (a, 6)-bifurcation diagram of the Chua s circuit AH denotes the Andronov-Hopf bifurcation curve cr = 0 corresponds to the vanishing of the saddle value when the origin is a saddle.

Fig. C.2.2. The Andronov-Hopf bifurcation curve AH and a pitch-fork curve r = 1 in the (r, (r)-plane of the Lorenz model at 6 = 8/3.

Fig. C.2.5. The (a, 6)-bifurcation diagram in the Shimizu-Morioka system derived from a linear stability analysis. AH labels the Andronov-Hopf bifurcation curve a = 0 corresponds to zero saddle-value HB — H8 corresponds to the change of the leading direction at the origin.

On the stability boundary, the inequality cr > b- -1 is fulfilled. Upon substituting (j = a + 6 +1, the expression for B becomes a polynomial of tr and b with positive coefficients. Hence, if g >0 and 6>0, then Li > 0. Thus, both equilibria 0x 2 are imstable (saddle-foci) on the stability boundary. The boimdary itself is dangerous in the sense of the definition suggested in Chap. 14. Therefore, the corresponding Andronov-Hopf bifurcation of Oi 2 is sub-critical. ... 

Let us derive next the equation for the curve AH which corresponds to the creation of an invariant curve (the Andronov-Hopf bifurcation for maps). Since eigenvalues of such a point are Ax 2 = it follows that the Jacobian... 

Thus, when a and are small, the sign of the first Lyapunov value equals the sign of the difference (/ — 2a). If it is negative, the stable invariant curve is born through the super-critical Andronov-Hopf bifurcation when crossing the curve AH towards larger (3. ... 

Fig. C.6.6. A part of the bifurcation diagram. AH labels the Andronov-Hopf bifurcation of the non-trivial equilibrium state Qi PD labels a flip-bifurcation of the stable periodic orbits that generates from Oi.

Let us describe the essential bifurcations in this system on the path 6 = 2 as fjL increases. On the left of the curve AH, the equilibrium state 0 is stable. It undergoes the super-critical Andronov-Hopf bifurcation on the curve AH. The stable periodic orbit becomes a saddle through the period-doubling bifurcation that occurs on the curve PD. Figure C.6.7 shows the unstable manifold of the saddle periodic orbit homeomorphic to a Mobius band. As a increases further, the saddle periodic orbit becomes the homoclinic loop to the saddle point 0(0,0,0,) at a 5.545. What can one say about the multipliers of the periodic orbit as it gets closer do the loop Can the saddle periodic orbit shown in this figure get pulled apart from the double stable orbit after the fiip bifurcation In other words, in what ways are such paired orbits linked in in R ... 

Fig. C.7.5. Period T of the periodic orbit born through a sub-critical Andronov-Hopf bifurcation versus the parameter a (6 = 1), as the cycle approaches the homoclinic loop. The origin is a saddle with 0.

The last comment on the Chua circuit concerns the bifurcations along the path 6 = 6 (see Fig. C.7.4). Notice that this sequence is very typical for many synunetric systems with saddle equilibrium states. We follow the stable periodic orbit starting from the super-critical Andronov-Hopf bifurcation of the non-trivial equilibrium states at a 3.908. As a increases, both separatrices tend to the stable periodic orbits. The last ones go through the pitch-fork bifurcations at a 4.496 and change into saddle type. Their size increases and at a 5.111, they merge with the homoclinic-8. This, as well as subsequent bifurcations, lead to the appearance of the strange attractor known as the double-scroll Chua s attractor in the Chua circuit. ... 

We will be seeking homoclinic bifurcations by starting from the Andronov-Hopf bifurcation at the non-trivial equilibria Oi,2 that takes place on the curve AH b = (see Sec. C.2). This bifurcation can be super-critical — the first... 

Gonchenko, S. V. and Gonchenko, V. S. [2000] On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies , Preprint No. 556, WIAS, Berlin. 

An interesting case is the so-called Poincare-Andronov-Hopf bifurcation, which implies that for a single unstable steady state in a two-dimensional case when the variables (concentrations of intermediates) cannot be infinite or negative, the only possible attractor is a Hmit cycle. Such limit cycle is then manifested in selfroscillations of concentrations. 

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