Density evolution under delayed dynamics. An open problem (Q2007483)

The book is devoted to the problem of evolution of densities in dynamical systems described by delay differential equations. The authors give a survey of all attempts done for solving the problem and try to provide an appropriate framework for further study. After a short introduction and some motivations given in Chapter 1, the density evolution of finite-dimensional dynamics and of a general dynamical system \((S_t)\) on a measure space \((X, \mathcal{A}, m)\) is presented in Chapter 2. The main role is covered by the Frobenius-Perron operator \(R^t\) defined on \(L^1\) by the implicit formula \(\int_AP^{t}fdm=\int_{S^{-1}_t(A)}fdm\). Then some examples are given for the exact form of the operator. In Chapter 3 the evolution of densities is described by some examples, as, for instance, the Mackey-Glass equation \(x'(t)=-2x(t)+4x(t-1)/(1+[x(t-10)]^{10}),\) with constant initial functions. The formulation of the evolution problem is described in Chapter 4, starting with the delay differential equation \(x'(t)=\mathcal{F}(x(t),x(t-\tau)),\) \(\tau\geq 0,\) \(x_0=\phi.\) This equation induces a flow \((S_t)\) on the space \(C([-\tau,0],\mathbb{R}^n)\) which defines a strongly continuous semigroup. If the initial distribution of states \(u\) is described by a density \(f(u)\) with respect to some measure \(m\), then after some time \(t\), the density will be evolved to \(P^tf\), where \(P^t\) is the Frobenius-Perron operator which corresponds to the system \((S_t).\) The first part of Chapter 5 is devoted to an exposition of Hopf functionals as well as to some known considerations of Hopf characteristic functionals. In the second part the delay differential equation is transformed to an infinite chain of linear partial differential equations. Then information about the evolution equation for the density functionals is given. In Chapter 6 the method of steps is applied to a delay differential equation of the form \(x'(t)=\mathcal{G}(x(t))\), if \(t\in[0,1)\), and \(x'(t)=\mathcal{F}(x(t), x(t-1)),\) if \(t\geq 1,\) in order to transform it to a system \(y'=\mathcal{F}(y,t)\) of ordinary differential equations. This investigation is continued by using the method of characteristics that gives an alternative way of looking at the problem. Then the authors provide a geometric interpretation using the Mackey-Glass equation as an example. Chapter 7 is based on a published work of the first two authors [Phy. Rev. E 52, No. 1, 115--128 (1995)]. Here the delay differential equation \({x}'(t)=\alpha x(t)+S(x(t-1)),\) is considered and it is shown that an approximate form of it is a system of difference equations \(x_{n+1}=T_m(x_n)\), where \(T_m\) is produced by known square matrices. This induces an asymptotically periodic Frobenius-Perron operator. In the last Chapter 8, the authors transform the delay differential equation \(x'(t)=-x(t)+S(x(t-\tau)),\) into a difference equation of the form \(x_{n+1}=T(x_n,y_n),\) and the form of the Frobenius-Perron operator is given explicitly. To close the chapter, three examples are presented. The book is interesting and introduces the reader to the problem of the evolution of the densities. Indeed, formulating (if possible) such an evolution one is able to guess the asymptotic behavior of the solutions with known density of their initial values. The examples in the book are carefully selected in order to illustrate the main theoretical results.