Chaos for a class of linear kinetic models

J. Banasiak*

School of Mathematical and Statistical Sciences, University of Natal, Durban, South Africa

M. Lachowicz†

Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, Warsaw, Poland

Abstract. In the recent years it was observed that the chaotic behaviour can occur in some infinite–dimensional linear systems. One of the first examples of this type, related to death processes, was discussed in [PA]. In this paper we generalize the results of [PA] to variable coefficients case showing that the property of being chaotic can be in certain sense stable. On the other hand we show that the ”opposite” birth process cannot be chaotic.

Key words: chaos, birth and death process, hypercyclic operators, kinetic model

AMS Subject Classification: 34G10, 47D06, 47N60

1. Introduction

In the recent decade it has been observed that chaos, typically associated with nonlinear phenomena, can occur also in linear infinite-dimensional systems. To make this statement more precise, let us recall that the (semi-)dynamical system {T (t)}_{t≥0} on a metric space X is said to be topologically chaotic (in the sense of Devaney [D]) if it is topologically transitive (meaning that for any two non-empty open sets U, V ⊂ X there is t_{0} ≥ 0 such that T (t_{0})U ∩ V ≠ ∅) and its set of periodic points is dense in X.

Historically speaking, the original Devaney’s definition contained also the intuitive notion of sensitive dependence on initial conditions (sdic) but it was proved in [Bal] that in fact the two conditions given above yield sdic (the original proof for a discrete in time systems can be repeated with slight modifications for continuous in time systems — see [B]).

A link between chaos theory and linear operator theory has been established when it was realized that the topological transitivity was really equivalent to the notion of hypercyclicity: a strongly continuous semigroup of bounded linear operators {T (t)}_{t≥0} on a Banach space X is called hypercyclic if for some x ∈ X the orbit {T (t)x}_{t≥0} is dense in X (see e.g. [DSW], [GS]). This path was followed later in [H] (for discrete in time systems) and several other papers.

* The research was partially supported by National Research Foundation of South Africa.

† The research was completed during the visit of the second author M.L. at the University of Natal in Durban, which was supported by the National Research Foundation of South Africa and the University of Natal Research Fund. M.L. was also partially supported by the Polish State Committee for Scientific Research under the Grant 2P03A00717.

The significance of these results for the applied mathematics (and continuous in time systems) was probably first noted in [PA] where the authors showed that a class of linear systems of kinetic type (described in more detail in Section 2) that can model e.g. the death part of the birth-and-death process, scattering with energy loss, and many other phenomena, is chaotic. However, it should be mentioned that in order to be chaotic, the system must contain a ”creation” part (β > α in the model), as no conservative system (such as e.g. the standard birth-and-death process) can be chaotic in the whole Banach space.

This paper was followed by [W] where it was shown that a linear transport equation modeling certain cell population dynamics is chaotic.

The recent paper [DSW] summarizes and considerably extends many of the existing results. The authors show, among others, that the convection-diffusion on the half-line is chaotic, rendering thus another important example of linear chaotic system. They also provide two important theorems allowing the identification of certain classes of chaotic and non-chaotic systems. As these two theorems will be the main tool in our investigations here we shall quote them for readers’ convenience.

Theorem 1 ([DSW] — Theorem 3.1). Let X be a separable Banach space and let A be the infinitesimal generator of a strongly continuous semigroup {T (t)}_{t≥0} on X. Suppose that the point spectrum of A, σ_{p}(A), has a nonempty interior U such that U ∩iR ≠ ∅. For each λ ∈ U we denote by x_{λ} a corresponding nonzero eigenvector: Ax_{λ} = λx_{λ} and define the function F_{Φ}(λ) =< Φ, x_{λ} > where Φ ∈ X^{∗}. Assume that for any Φ ∈ X^{∗} the function F_{Φ} is analytic in U and doesn’t vanish identically on U unless Φ = 0. Then {T (t)}_{t≥0} is chaotic.

Theorem 2 ([DSW] — Theorem 3.3). Let {T (t)}_{t≥0} be a hypercyclic semigroup generated by A in a Banach space X. Then the adjoint A^{∗} of A and the dual semigroup {T^{∗}(t)}_{t≥0} have the following properties: a. for any 0 ≠ ϕ ∈ X^{∗} the orbit {T^{∗}(t)ϕ}_{t≥0} is unbounded; b. the point spectrum σ_{p}(A^{∗}) is empty.

A survey of chaotic models coming from the applied sciences together with some discussion of links between discrete and continuous time theory can be found also in [B].

2. Death model with constant coefficients

Following [PA] we consider an immobile medium that hosts particles characterized by an internal degree of freedom, indexed by a positive integers n ∈ ℕ_{0} = {0, 1, 2 …}, and related to internal levels of energetic excitation. These particles collide and interact with the host medium as follows. Upon collision, particles of internal energy level n ≥ 1 are absorbed by the medium at a rate α > 0 and re-emitted as particles of internal energy level n − 1 at a rate β > α. Particles with internal energy n = 0 are absorbed and cease to exist. The process underlying this interaction can be viewed as a chemical reaction, a biological process, or as a generalized automaton rule.

Particles of internal energy n are then described by a one-particle distribution function, f_{n}(t), that satisfies the linear rate equation

(1)  \frac{df_{n}}{dt} = (Lf)_{n} = −αf_{n} + βf_{n+1},  n ∈ ℕ .

The proper space for this process is l^{1} so that the norm of any elements in the positive cone: f = (f_{0}, f_{1} …) ≥ 0, given by

∥f∥ = ∑_{n=0}^{∞} f_{n} ,

represents the total number of particles.

The operator L on the right-hand side is bounded so that there exists a semigroup {T (t)}_{t≥0} solving the above equation. It follows that the point spectrum of L is given by

σ_{p}(L) = {−α + βμ ; |μ| < 1} ,

with eigenvectors given by h_{μ} = (μ, μ^{2}, μ^{3}, …), which satisfies the assumptions of Theorem 1.

3. Death model with variable coefficients

In this section we consider a generalization of the model discussed earlier. The generalization consists in allowing variable coefficients α_{n}, β_{n}, n ∈ ℕ_{0}.

We consider in l^{1} the infinite system of equations

(2)  \frac{df_{n}}{dt} = (Lf)_{n} = −α_{n}f_{n} + β_{n}f_{n+1} ,  n ∈ ℕ_{0} .

Note that some motivation for studing Eq. (2) comes also from the theory of kinetic integral equations (generalized Boltzmann models or nonlocal models) — see [BL] and [LW]. In fact consider the followig linear equation

(3)  ∂_{t}f = γ ∗ f − ηf ,

in L^{2}(𝕋), where 𝕋 is the 1–dimensional torus, γ, η are given function on 𝕋 and ∗ is the convolution. The bilinear version of this type were studied in [LW]. Eq. (3) in Fourier transform reads

(4)  ∂_{t}f_{n} = γ_{n}f_{n} − ∑_{m=−∞}^{+∞} η_{n−m}f_{m} ,  n = 0, ±1, ±2, …,

where f_{n} is the n-th Fourier coefficient of function f, Eq. (4), under the assumption η_{n} = 0, ∀ n ≠ −1, 0 leads to Eq. (2) with α_{n} = η_{0} − γ_{n} and β_{n} = −η_{−1} for n ∈ ℕ_{0}.

In this paper we assume that 0 < α_{n} < β_{n} for any n ∈ ℕ_{0} and

(A1) α_{n} = α + a′_{n}, (n ∈ ℕ_{0}), for some α ≥ 0 and with lim_{n→∞} a′_{n} = 0;

(A2) β_{n} = βb_{n}, (n ∈ ℕ_{0}), for some β > α and with lim_{n→∞} b_{n} = 1.

The assumption lim_{n→∞} b_{n} = 1 is largely for technical convenience.

Solving the eigenvalue problem

(5)  λf_{n} = −α_{n}f_{n} + β_{n}f_{n+1} ,  n ∈ ℕ_{0} ,

gives formally

(6)  h(λ) = {h_{n}(λ)}_{n∈ℕ_{0}} ,

where h_{0}(λ) = 1, and for n ≥ 1

(7)  h_{n}(λ) = ∏_{i=0}^{n−1} \frac{λ + α_{i}}{β_{i}} .

Lemma 1. Under the assumptions (A1) and (A2) the circle {λ ∈ ℂ; |λ+α| < β} belongs to the point spectrum σ_{p}(L) of the operator L and λ = 0 belongs to the interior of σ_{p}(L).

Proof. Assume that |λ + α| < β; then for some ε > 0 we have |λ + α| = β − ε. There is n_{0} such that for all n > n_{0} we have |β_{n} − β| < ε/2 so that

|\frac{λ + α_{n}}{β_{n}} | ≤ |\frac{λ + α}{β_{n}} | + |\frac{a′_{n}}{β_{n}} | ≤ \frac{β − ε}{β − ε/2} + |\frac{a′_{n}}{β_{n}} | .

Since the first term is strictly smaller than 1 and the second tends to 0, by picking possibly larger n_{0} we obtain that for some q < 1 and n > n_{0} we have

|\frac{λ + α_{n}}{β_{n}} | ≤ q ,

thus h(λ) ∈ l^{1} and λ ∈ σ_{p}(L). Assumption (A2) gives that λ = 0 is an interior point of this circle.

Using the ideas of Theorem 1 we take an arbitrary sequence {Φ̃_{k}}_{k∈ℕ_{0}} ∈ l^{∞} and consider the function

FΦ̃(λ) = ∑_{k=0}^{∞} Φ̃_{k}h_{k}(λ) ,

where the series is uniformly convergent at least in the circle |λ+α| < β. We shall rearrange the above expression as

F_{Φ}(μ) = FΦ̃(λ) = ∑_{k=0}^{∞} \frac{Φ̃_{k}}{∏_{i=0}^{k−1} b_{i}} ∏_{i=0}^{k−1} (\frac{λ + α}{β} + \frac{a′_{i}}{β} ) = ∑_{k=0}^{∞} Φ_{k} ∏_{i=0}^{k−1}(μ + a_{i}) ,

where we introduced the notation

Φ_{k} = \frac{Φ̃_{k}}{∏_{i=0}^{k−1} b_{i}} ,  μ = \frac{λ + α}{β} ,  a_{n} = \frac{a′_{n}}{β} ,

and the convention that ∏_{i=0}^{−1} = 1. The function F_{Φ}(μ) is analytic for |μ| < 1. By As-

sumption (A2), if {Φ̃_{k}}_{k∈ℕ_{0}} ∈ l^{∞}, then also {Φ_{k}}_{k∈ℕ_{0}} ∈ l^{∞}. Thus, showing that F_{Φ} ≡ 0 yields Φ_{k} = 0 for k ∈ ℕ_{0} for any bounded {Φ_{k}}_{k∈ℕ_{0}} proves the same statement for FΦ̃. Consequently we shall concentrate on F_{Φ}.

To be able to use the properties of analytic functions we shall write F_{Φ}(μ) as a power series in μ by multiplying out the brackets and rearranging the terms - this is possible as the series is absolutely convergent (see e.g. [K], p. 142). By induction one can prove that

∏_{j=0}^{n}(μ + α_{j}) = ∑_{j=0}^{n} μ^{j}A_{jn} ,

where

A_{jn} = \{ \begin{matrix}0 & for & j > n , \\ 1 & for & n = j , \\ ∑_{0≤i_{1}<...<i_{n−j}≤n−1} a_{i_{1}} · . . . · a_{i_{n−j}} & otherwise & . \end{matrix}

Thus we have

F_{Φ}(μ) = ∑_{j=0}^{∞} μ^{j} ∑_{n=j}^{∞} Φ_{n}A_{jn} ,

and to be able to use Theorem 1 we have to show that the only bounded solution of the infinite upper-triangular linear system

Φ_{0} + A_{01}Φ_{1} + A_{02}Φ_{2} + … = 0,
Φ_{1} + A_{12}Φ_{2} + … = 0,
Φ_{2} + … = 0,
… = 0,

is the zero solution.

Let us denote by A the operator generated by the matrix {A_{ij}}_{0≤i,j≤∞} in l^{∞}. We have the following result.

Lemma 2. Let us assume that there exist: q < 1, and k_{0} such that |a_{k}| ≤ q^{k+1} for k ≥ k_{0}. Then A is a bounded operator in l^{∞}.

Proof. Let f = (f_{1}, f_{2}, …) ∈ l^{∞}; then Af = (g_{1}, g_{2}, …) where g_{n} = ∑_{j=n}^{∞} A_{nj}f_{j} and

∥Af∥_{∞} = sup_{n∈ℕ_{0}} | ∑_{j=n}^{∞} A_{nj}f_{j}| ≤ ∥f∥_{∞} sup_{n∈N_{0}} ∑_{j=n}^{∞} |A_{nj}| = ∥f∥_{∞} sup_{n∈ℕ_{0}} S_{n} ,

where

S_{n} = 1 + | ∑_{0≤i≤n} a_{i}| + | ∑_{0≤i_{1}<i_{2}≤n+1} a_{i_{1}}a_{i_{2}} | + … + | ∑_{0≤i_{1}<...<i_{j−n}≤j−1} a_{i_{1}} · . . . · a_{i_{j−n}} | + ….

It is clear that we can assume that for k < k_{0} we have |a_{k}| ≤ Mq^{k+1} for some constant M. With this assumption we get the estimate

S_{n} ≤ M^{k_{0}}(1 + ∑_{1≤i≤n+1} q^{i} + ∑_{1≤i_{1}<i_{2}≤n+2} q^{i_{1}+i_{2}} + … ∑_{1≤i_{1}<...<i_{j−n}≤j} q^{i_{1}+...+i_{j−n}} + …) .

Each power q^{k} with k > 2 can appear several times in the above sum. However, the number of occurences cannot exceed the number P (k) of all possible integer solutions to i_{1} + … + i_{r} = k (it is clear that any particular sequence (i_{1}, …, i_{r}) may appear at most once). Thus we can write

S_{n} ≤ M^{k_{0}} ∑_{k=0}^{∞} P (k)q^{k}

and using the result from combinatorics (see e.g. [VW], p. 140) that

P (k) < \frac{π}{\sqrt{6(k − 1)}} exp(π\sqrt{2k/3}) , for k > 2

we obtain that

\overline{lim_{k→∞}} ^{k} \sqrt{q^{k}P (k)} < 1 .

Thus the Cauchy criterion yields the convergence of each S_{n} with the common bound independent of n. This gives the boundedness of A.

Lemma 3. There is q < 1 such that if |a_{k}| ≤ q^{k+1} for all k ∈ ℕ_{0}, then A is an isomorphism in l^{∞}.

Proof. We split A as

A = I + ( \begin{matrix}0 & A_{01} & A_{02} & A_{03} & … \\ 0 & 0 & A_{12} & A_{13} & … \\ 0 & 0 & 0 & A_{23} & … \\ ⋮ & ⋮ & ⋮ & ⋮ & … \end{matrix} ) = I + A′ ,

where I is the infinite identity matrix. As in the proof of Lemma B we obtain for any f = (f_{1}, f_{2}, …) ∈ l^{∞}

∥A′f∥_{∞} = sup_{n∈ℕ_{0}} | ∑_{j=n+1}^{∞} A_{nj}f_{j}| ≤ ∥f∥_{∞} sup_{n∈ℕ_{0}} ∑_{j=n+1}^{∞} |A_{nj}| = ∥f∥_{∞} sup_{n∈ℕ_{0}} S′_{n} .

where this time we have the estimate

S′_{n}(q) ≤ q + q^{2} + … + q^{n+1} + ∑_{1≤i_{1}<i_{2}≤n+2} q^{i_{1}+i_{2}} + … ∑_{1≤i_{1}<...<i_{j−n}≤j} q^{i_{1}+...+i_{j−n}} + ….

Using the combinatorial argument of Lemma 2 we obtain that there exists a continuous function of q ∈ [0, 1), say S(q) such that S′_{n}(q) ≤ S(q) for any n. Thus

∥A′f∥_{∞} ≤ S(q)∥f∥_{∞} .

Since S is continuous with S(0) = 0 we obtain the existence of q_{0} such that if q < q_{0}, then S(q) < 1. Using the Neumann series theorem we obtain that A is an isomorphism.

Combining the above results we obtain the main result of this paper.

Theorem 3. Suppose that the sequences {α_{n}}_{n∈ℕ_{0}} and {β_{n}}_{n∈ℕ_{0}} satisfy the assumptions (A1) and (A2), and the assumption of Lemma 3. Then the semigroup generated by Eq.

(2) is chaotic in any l_{p}, 1 ≤ p < ∞, and in c_{0}.

Proof. From Lemma 1 we see that σ_{p}(L) contains the open circle |λ + α| < β with λ = 0 being an interior point of σ_{p}(L), that is, σ_{p}(L) contains an open subsets V_{+} ⊂ ℂ_{+} = {λ; ℜλ > 0}, V_{−} ⊂ ℂ_{−} = {λ; ℜλ < 0} and an open interval V_{0} of the imaginary axis. Using the representation of F_{Φ}(μ) in the form of the power series we obtain that if F_{Φ}(μ) vanishes on either set: β^{−1}(V_{±} + α) or β^{−1}(V_{0} + α), then the coefficients of the power series must vanish, which is equivalent to AΦ = 0. Since we proved in Lemma 3 that A is an isomorphism of l^{∞}, the only bounded solution is Φ = 0 which, by Theorem 1, yields that the semigroup {T (t)}_{t≥0} is chaotic in l^{1}.

Let us consider the same problem in c_{0} and l^{p} with 0 < p < +∞. The constructed eigenvectors belong to each of the above spaces. We have c_{0}^{∗} = l^{1} (see e.g. [KF], p. 185), and (l^{p})^{∗} = l^{r} with 1/p + 1/r = 1. Hence, if the only solution Φ ∈ l^{r}, 1 ≤ r < +∞ of the equation AΦ = 0 is the zero solution, then the semigroup {T (t)}_{t≥0} is chaotic in each of these spaces. But any element of l^{r} is necessarily bounded, thus the statement follows from the unique solvability of AΦ = 0 in l^{∞}.

3. The birth process

We can consider the ”opposite” process to the one considered above, that is related to birth (if the former is related to death), or to heating with creation [PA]. The rate equations are

(8)  \begin{aligned}\frac{df_{0}}{dt} & = −α_{0}f_{0} , \\ \frac{df_{n}}{dt} & = −α_{n}f_{n} + β_{n−1}f_{n−1} ,  n ∈ ℕ . \end{aligned}

We adopt the same assumptions on the sequences {α_{n}}_{n∈ℕ_{0}} and {β_{n}}_{n∈ℕ_{0}} as in the previous section. Then it is clear that the matrix

K^{T} = ( \begin{matrix}−α_{0} & 0 & 0 & 0 & … \\ β_{0} & −α_{1} & 0 & 0 & … \\ 0 & β_{1} & −α_{2} & 0 & … \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \end{matrix} )

defines a bounded operator K_{0}^{T} in c_{0}, and K_{p}^{T} in l^{p}, 1 ≤ p ≤ ∞, respectively. Moreover, the transpose matrix K defines a bounded operator K_{1} in l^{1} and, for p ≠ ∞, K_{r} in l^{r}, where 1/p + 1/r = 1. Define the semigroup generated by K_{p}^{T}, p ∈ {0} ∪ [1, ∞], in the respective space, by {S_{p}(t)}_{t≥0}. Let us also recall that a semigroup is chaotic if it is hypercyclic (topologically transitive) and has a dense set of periodic points. Then we have

Theorem 4. The semigroup {S_{p}(t)}_{t≥0} is not hypercyclic for any p ∈ 0 ∪ [1, ∞).

Proof. First we note that since the generators are bounded, the semigroups are uniformly continuous and therefore the adjoint of the generator generates the adjoint semigroup. Thus {S_{p}^{∗}(t)}_{t≥0} = {T_{r}(t)}_{t≥0} for p ∈ [1, ∞), 1/p + 1/r = 1 and {S_{0}^{∗}(t)}_{t≥0} = {T_{1}(t)}_{t≥0}. Using Theorem 2 we know that if {S_{p}(t)}_{t≥0} was hypercyclic, then all the orbits if the adjoint semigroup would be unbounded. Let first p ≠ 1. Then we know from Theorem 3 that {T_{r}(t)}_{t≥0} is chaotic so, in particular, it has periodic orbits which cannot be unbounded. Thus in this case {S_{p}(t)}_{t≥0} is not hypercyclic.

If p = 1, then the adjoint semigroup acts in l^{∞} and we do not have any result about its chaoticity. However, since clearly K_{0} is the restriction of K_{∞} to the closed invariant subspace c_{0} ⊂ l^{∞}, we obtain that {T_{0}(t)}_{t≥0} is a restriction of {T_{∞}(t)}_{t≥0}, ([EN], p. 43). From the proof of Theorem 3 we know that there exists a nonempty subspace X_{0} ⊂ c_{0} (as \overline{X}_{0} = l^{1} ⊂ c_{0}) such that for x ∈ X_{0} we have lim_{t→∞} ∥T_{0}(t)x∥_{∞} = 0. Since X_{0} ⊂ c_{0} ⊂ l^{∞}, for x ∈ X_{0} we get ∥T_{∞}(t)x∥_{∞} = ∥T_{0}(t)x∥_{∞} so that the trajectories of {T_{∞}(t)}_{t≥0} starting from X_{0} tend to 0 as t → ∞. This shows that {S_{1}(t)}_{t≥0} cannot be hypercyclic.

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