Abstract
Not just nonlinear systems but infinite-dimensional linear systems can exhibit complex behavior. It has long been known that twice the backward shift on the space of square-summable sequencesl2displays chaotic dynamics. Here we construct the corresponding operatorCon the space of2π-periodic odd functions and provide its representation involving a Principal Value Integral. We explicitly calculate the eigenfunction of this operator, as well as its periodic points. We also provide examples of chaotic and unbounded trajectories ofC.
Highlights
Linear systems have commonly been thought to exhibit relatively simple behavior
In addition to 2B there are many examples of chaotic linear operators including weighted shifts [8], composition operators [9], and differentiation and translations [10,11,12]. It has been argued in [13, 14] that nonlinearity is not necessarily required for complex behavior; an infinite-dimensional state space can provide the ingredients of chaotic dynamics
We show that utilizing the representation of operator C one can obtain principal values of certain integrals
Summary
Linear systems have commonly been thought to exhibit relatively simple behavior. Surprisingly, infinite-dimensional linear systems can have complex dynamics. While chaoticity of linear operators is at first puzzling and the backward shift example seems contrived, these operators are not rare. In addition to 2B there are many examples of chaotic linear operators including weighted shifts [8], composition operators [9], and differentiation and translations [10,11,12]. It has been argued in [13, 14] that nonlinearity is not necessarily required for complex behavior; an infinite-dimensional state space can provide the ingredients of chaotic dynamics. We show that utilizing the representation of operator C one can obtain principal values of certain integrals
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