Chua’s circuit

Let us consider next a few examples. We will focus our consideration on the Lorenz equation, the Chua s circuit, the Shimizu-Morioka model and some others. 

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.

C.4.9jt47. Let us consider next the following version of Chua s circuit... 

C.5. 60. Prove that infinity is unstable in a Chua s circuit modeled by... 

C.7. 83. nsider the following Z2-synimetric Chua s circuit with cubic nonlinearity [179] ... 

The point NS. This point is of codimension two as <7 = 0 here. The behavior of trajectories near the homoclinic-8, as well as the structure of the bifurcation set near such a point depends on the separatrix value A (see formula (13.3.8)). Moreover, they do not depend only on whether A is positive (the loops are orientable) or negative (the loops are twisted), but it counts also whether A is smaller or larger than 1. If A < 1, the homoclinic-8 is stable , and unstable otherwise. To find out which case is ours, one can choose an initial point close sufficiently to the homoclinic-8 and follow numerically the trajectory that originates from it. If the figure-eight repels it (and this is the case in Chua s circuit), then A > 1. Observe that a curve of double cycles with multiplier 4-1 must originate from the point NS by virtue of Theorem 13.5. 

The point NSF a = 0 corresponds to a neutral saddle-focus. At this codimension-two point the dynamics of the trajectories near the homoclinic loops to the saddle-focus becomes chaotic. This bifurcation indeed proceeds the origin of the chaotic double scroll attractor in Chua s circuit. In the general case, this bifurcation was first considered in [29]. The complete unfolding of... 

Khibnik, A. I., Roose, D. and Chua, L. O. [1993] On periodic orbits and homoclinic bifurcations in Chua s circuit with smooth nonlinearity , Int J. Bifurcation Chaos 3(2), 363-384. 

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. ... 

He also proposed the circuit symbol shown in Figure 9.29 for the memristor. Looking back, it seems the discovery by Leon Chua did not get as much attention as we today realize it deserved. One reason may be that combining magnetic flux and charge, one could not readily see how this new component could be used in practice. However, using Faraday s law of induction and the definition of electric current in Eq.9.45, it simplifies to ... 

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