On a bifurcation governed by hysteresis nonlinearity
type: Item
instance of: scholarly article
title: On a bifurcation governed by hysteresis nonlinearity
zbMATH Open document ID: 1013.34036
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2002-05-30T00:00:00Z
review text: 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 \textit{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(\…
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C55
Mathematics Subject Classification ID: 37G25
Mathematics Subject Classification ID: 47J40
zbMATH DE Number: 1747960
zbMATH Keywords: Hopf bifurcation at infinity
zbMATH Keywords: large cycles
zbMATH Keywords: hysteresis operators
zbMATH Keywords: control theory equation
zbMATH Keywords: Prandtl-Ishlinski hysteresis nonlinearity
zbMATH Keywords: hysteresis operator
zbMATH Keywords: bifurcation of solutions
zbMATH Keywords: large periodic solutions
author: Alexander Krasnosel'skij
author: Dmitrii Rachinskii
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: s00030-002-8120-2
OpenAlex ID: W2157532438
DOI: 10.1007/S00030-002-8120-2
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description: scientific article; zbMATH DE number 1747960