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On bifurcation of large-amplitude stable cycles for equations with hysteresis. (English. Russian original) Zbl 1086.34042
Autom. Remote Control 65, No. 12, 1915-1937 (2004); translation from Avtom. Telemekh. 2004, No. 12, 62-84 (2004).
Consider the scalar differential equation
\[ L\left(\lambda,\frac {d}{dt}\right)x=M\left(\lambda,\frac d{dt}\right)(f(x)+\xi(t)),\tag{*} \] where \(L\) and \(M\) are polynomials in the second variable whose coefficients depend on the scalar parameter \(\lambda\). \(\xi(t)\) is the output
\[ \xi(t)=\int^R_0{\mathcal U}_\rho[t_0,\theta_0(\rho)] x(t)d\mu(\rho),\quad t\geq t_0, \] to the input \(x(t)\), where \({\mathcal U}_\rho\) is the hysteresis stop-nonlinearity with width \(2\rho\). \(\lambda=\lambda_0\) is a bifurcation value related to Andronov-Hopf bifurcation from infinity. The paper establishes conditions for the existence of a continuous branch of limit cycles whose amplitude tends to infinity as \(\lambda\to\lambda_0\). The stability of these cycles is also investigated.

34C55 Hysteresis for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
47J40 Equations with nonlinear hysteresis operators
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