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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.

MSC:
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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