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

Krasnosel’skii, Alexander; Rachinskii, Dmitrii

NoDEA, Nonlinear Differ. Equ. Appl. 9, No. 1, 93-115 (2002).

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 \(\psi(\lambda)\varphi(\lambda_0)> 0\).

Reviewer: Peter Zabreiko (Minsk)

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

Keywords:

Hopf bifurcation at infinity; large cycles; hysteresis operators; control theory equation; Prandtl-Ishlinski hysteresis nonlinearity; hysteresis operator; bifurcation of solutions; large periodic solutions