## On a bifurcation governed by hysteresis nonlinearity.(English)Zbl 1013.34036

The authors study autonomous systems of the type $L\Biggl({d\over dt}\Biggr)x= M\Biggl({d\over dt}\Biggr)(f(x,\lambda)+ H_\lambda x),\tag{1}$ where $$L(\cdot)$$ and $$M(\cdot)$$ are even and coprime polynomials of degrees $$l$$ and $$m,l>m$$, the polynomial $$L(\cdot)$$ has a pair of simple imaginary roots $$\pm w_0$$ $$(w_0>0)$$, $$L(iw_0n)\neq 0$$. $$n= 0,2,3,\dots$$, $$f(x,\lambda)$$ is a function satisfying the condition $$x^{-1}f(x,\lambda)\to 0$$ as $$x\to\infty$$ locally uniformly with respect $$\lambda$$, and, at last, $$H_\lambda x$$ is a Prandtl-Ishlinski hysteresis nonlinearity that can be written in the form $H_\lambda x(t)= \int^\infty_0 U_\rho[t_0, \xi_0(\rho)] x(t) d\mu(\rho, \lambda),$ where $$U_p[t_0, \xi_0]$$ is an elementary hysteresis operator (called stop) defined, for monotone $$x(t)$$, by the formulas $U_\rho[t_0, \xi_0]x(t)= \begin{cases} \min\{\rho, \xi_0+ x(t)- x(t_0)\}\quad &\text{if }x(t)\text{ increases}\\ \max\{-\rho, \xi_0+ x(t)- x(t_0)\}\quad &\text{if }x(t)\text{ decreases}\end{cases}$ for piecewise monotone $$x(t)$$ by semigroup identity, and for continuous $$x(t)$$, by the continuity property.
The main results are three theorems on the bifurcation of solutions to (1) from infinity in terms of the functions $\psi(\lambda)= \int^\infty_0 \rho\mu(\rho,\lambda),\quad \varphi(\lambda)= \int^\infty_0 \rho^2 d\mu(\rho, \lambda).$ The condition $$\psi(\lambda_0)= 0$$ is necessary to $$\lambda_0$$ to be a bifurcation point. The phenomen of bifurcation takes place for $$\lambda= \lambda_0$$ if $$\lambda_0$$ is a zero of $$\psi(\cdot)$$ and $$\psi(\cdot)$$ takes both positive and negative values in any neighborhood of $$\lambda_0$$. If $$\lambda_0$$ is an isolated zero of $$\psi(\cdot)$$ and $$\varphi(\cdot)$$ is continuous at $$\lambda_0$$ and $$\varphi(\lambda_0)\neq 0$$ then large periodic solutions to (1) exist for the values of $$\lambda$$ close to $$\lambda_0$$ and satisfy the inequality $$\psi(\lambda)\varphi(\lambda_0)> 0$$.

### MSC:

 34C23 Bifurcation theory for ordinary differential equations 34C55 Hysteresis for ordinary differential equations 37G25 Bifurcations connected with nontransversal intersection in dynamical systems 47J40 Equations with nonlinear hysteresis operators
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