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We consider autonomous systems with a nonlinear part depending on a parameter and study Hopf bifurcations at infinity. The nonlinear part consists of the nonlinear functional term and the Prandtl-Ishlinskii hysteresis term. The linear part of the system has a special form such that the close-loop system can be considered as a hysteresis perturbation of a quasilinear Hamiltonian system. The Hamiltonian system has a continuum of arbitrarily large cycles for each value of the parameter. We present sufficient conditions for the existence of bifurcation points for the non-Hamiltonian system with hysteresis. These bifurcation points are determined by simple characteristics of the hysteresis nonlinearity. 2000 Mathematics Subject Classification: 34C23, 37G25. 94 Alexander Krasnosel'skii and Dmitrii Rachinskii NoDEAEquations (1) with arbitrary polynomials L(p), M (p) satisfying condition (i) below are usual in control theory (see, e.g., [3,11,12]). If M ≡ 1, then (1) is an ordinary autonomous higher-order differential equation. The standard definition of solutions for equation (1) is as follows. Consider the systemFor any L, M , and f there exist a matrix A and vectors b, c (non-unique) such that this system is equivalent to equation (1).Everywhere below it is assumed that the polynomials L(p) and M (p) and their degrees and m satisfy the following conditions:(i) The polynomials L(p) and M (p) are coprime and > m;(ii) The polynomials L(p) and M (p) are even;(iii) The polynomial L(p) has a pair of simple imaginary roots ± iw 0 (w 0 > 0) and L(iw 0 n) = 0 for n = 0, 2, 3, . . . Condition (ii) implies that (1) is a Hamiltonian equation [6]. Suppose that the function x −1 f (x) vanishes as x → ∞ (e.g., f (x) is uniformly bounded). Then due to (iii) equation (1) has a continuum of large-amplitude periodic solutions (cycles) x r = x r (t) of all amplitudes r = x r C with r ≥ r 0 ; the period T r of x r goes to 2π/w 0 as r →∞.If the function f (x, λ) depending on the parameter λ is used in place of f (x) in equation (1), then this equation has a continuum of arbitrarily large cycles for each parameter value. If equation (1) with f (x, λ) in place of f (x) is perturbed by some hysteresis term, then large periodic solutions may exist for parameter values accumulating near some bifurcation points only. To be precise, the following definition from [4] is used.Definition 1 A parameter value λ 0 is called a Hopf bifurcation point at infinity (shortly, a bifurcation point or HBP) with a frequency w 0 for some equation depending on the parameter λ if for any sufficiently large r > 0 there exists a λ r such that the equation with λ = λ r has a T r -periodic solution x r = x r (t) and λ r → λ 0 , x r C →∞, T r → 2π/w 0 as r → ∞.