Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part I: Controlled hysteresis

Abstract

Abstract. The study of nonlinear vibrations/oscillations in mechanical and electronic systems has always been an important research area. While important progress in the development of mathematical chaos theory has been made for finite dimensional second order nonlinear ODEs arising from nonlinear springs and electronic circuits, the state of understanding of chaotic vibrations for analogous infinite dimensional systems is still very incomplete.The 1-dimensional vibrating string satisfying wtt − wxx = 0 on the unit interval x ∈ (0, 1) is an infinite dimensional harmonic oscillator. Consider the boundary conditions: at the left end x = 0, the string is fixed, while at the right end x = 1, a nonlinear boundary condition wx = αwt − βw 3 t , α, β > 0, takes effect. This nonlinear boundary condition behaves like a van der Pol oscillator, causing the total energy to rise and fall within certain bounds regularly or irregularly. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary condition. Since the solution of the first order hyperbolic system depends completely on this nonlinear relation and its iterates, the problem is reduced to a discrete iteration problem of the type u n+1 = F (un), where F is the nonlinear reflection relation. We say that the PDE system is chaotic if the mapping F is chaotic as an interval map. Algebraic, asymptotic and numerical techniques are developed to tackle the cubic nonlinearities. We then define a rotation number, following J.P. Keener [11], and obtain denseness of orbits and periodic points by either directly constructing a shift sequence or by applying results of M.I. Malkin [17] to determine the chaotic regime of α for the nonlinear reflection relation F , thereby rigorously proving chaos. Nonchaotic cases for other values of α are also classified. Such cases correspond to limit cycles in nonlinear second order ODEs. Numerical simulations of chaotic and nonchaotic vibrations are illustrated by computer graphics.

Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions

“…In recent twenty years, there have been lots of papers studying the chaotic oscillations in the systems governed by one-dimensional wave equations, see [1,2,3,4,5,9,10,11,12,14] and the references therein. Interestingly, Chen et al [6] studied chaotic dynamics of hyperbolic PDE with constant coefficients and van der Pol boundary condition.…”

“…(1) where d i (x) > 0 and k i (x), i = 1, 2, are C 1 on [0, 1]. For clarity, let us consider (1) with initial conditions w(x, 0) = w 0 (x) ∈ C 1 ([0, 1]), w t (x, 0) = w 1 (x) ∈ C([0, 1]), x ∈ (0, 1), (2) and IBCs B 1 (w t (0, t), w x (0, t), w(0, t)) = 0, t > 0, B 2 (w t (1, t), w x (1, t), w(1, t)) = 0, t > 0,…”

“…, n be a nonnegative integer and t = τ + n(l 1 + l 2 ). Applying method of characteristics as described in [1] and the two reflections (8) and (9), u(1, t) and v(0, t) can be solved as follows…”

Functional envelopes relative to the point-open topology on a subset

“…This kind of equations comes from a class of one-dimensional wave equations which has been extensively studied by a series of papers. See [3]- [9] and [13]- [16]. We will give a simple example in Section 5.…”

“…So the dynamical behaviors of the solutions of BVP (9)-(11) can be completely determined by two interval maps F α,β • G η and G η • F α,β which are topologically conjugate [3]. In particular, it is proved that for fixed β > 0 and α : 0 < α < 1, there exist three critical values η 0 , η H and η B : 0 < η 0 < η H < η B < 1 such that G η • F α,β has bounded invariant intervals if and only if η ∈ (0, η B ) ∪ [ 1 η B , +∞) [4].…”