Hopf Bifurcation in Symmetric Networks of Coupled Oscillators with Hysteresis

Guaranteed estimates for the length of branches of periodic orbits for equivariant Hopf bifurcation

“…From ( 22) it follows that a −1 ξ (0) = b −1 (0), hence a ξ and b are Ω 1 -admissibly homotopic. To complete the proof, it remans: to take a homemorphism of Ω 1 onto O (see (1)), replace the above function ξ by |λ|( v C − 1) + v C + 1 (see (2)) and apply formula (3).…”

Dynamics of systems with Preisach memory near equilibria

“…We note that the evolution of the state S(t) of the Preisach operator is uniquely defined by the initial state S 0 and the input x(t) of this operator according to the definition of the previous section, hence it is legitimate to consider triplets (x(t), y(t), S 0 ) instead of (x(t), y(t), S(t)). 3 The projection of solutions on the (x, y)-plane plays the main role for the results presented below.…”

“…In particular, various aspects of dynamics of forced oscillators have been studied in [39], [22], [6], [41], [42], [32], [55], [60], [56], [7], including applications to accurate modelling and optimization of parameters of power electronics systems in the presence of the ferroresonance phenomenon [57] and input-to-state stability of control systems. Cycles stemming from the Hopf bifurcation in a model of Van der Pol type oscillator, a network of such oscillators and other systems with hysteresis were studied in [1], [3], [33], [20], [21]. In the electrical circuitry context, this model describes a circuit consisting of an LCR contour and a negative feedback loop (which can be implemented, for instance, by using tunnel diodes) and involving a ferromagnetic core in the inductor.…”

Hopf bifurcation of relative periodic solutions in a system of five passively mode-locked lasers

“…This method is equally applicable to smooth and non-smooth setting (assuming differentiability at selected points; cf. [6], where a system of coupled van der Pol oscillators was considered in the presence of hysteretic non-linearities). Also, it is insensitive to violations of genericity assumptions [16,22] (including the simplicity of purely imaginary eigenvalues at the bifurcation point, transversality of the eigenvalue crossing, and non-resonance conditions).…”