A sharp condition for the chaotic behaviour of a size structured cell population

S. EL Mourchid^{1}, A. Rhandi^{2}, H. Vogt^{3}, and J. Voigt^{3}

Abstract

We show that the condition 0 ≤ β ≤ \frac{1}{2 ln 2} is necessary for the chaoticity of the solution of the cell population model

(0.1)  \{ \begin{matrix}\frac{∂u(t,x)}{∂t} = −\frac{∂(xu(t,x))}{∂x} + γu(t, x) − βu(t, x) + 4βu(t, 2x)χ_{(0, \frac{1}{2} )}(x), \\ u(0, ·) = f ∈ L^{1}(0, 1). \end{matrix}

(If γ − 3β > 0, then this condition is known to be sufficient.) The analysis depends on solving a forward delay equation.

1 Preliminaries

In [2] we studied the chaotic behaviour of the above simplified model of a size structured cell population. We supposed that γ −3β > 0 and we proved that the evolution associated with (0.1) is chaotic if 0 ≤ β ≤ \frac{1}{2 ln 2}, and is subchaotic if β > \frac{1}{2 ln 2} (see [2; Theorem 2.3]). In the present note we show that in fact the evolution is not chaotic for β > \frac{1}{2 ln 2} (i.e., the space of subchaoticity is a proper subspace).

The following result due to Desch, Schappacher and Webb will be used to rule out chaoticity in our main result.

Theorem 1.1. ([1; Theorem 3.3]) If a C_{0}-semigroup T (·) generated by A in a Banach space X is hypercyclic, then the adjoint A^{∗} of A and the dual semigroup have the following properties:

1Department of Mathematics, Faculty of Science Semlalia, Cadi Ayyad University, B.P. 2390, 40000, Marrakesh, Morocco.

e-mail: s.elmourchid@ucam.ac.ma

2DIIMA, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy.

e-mail: rhandi@diima.unisa.it

3Fachrichtung Mathematik, Technische Universität Dresden, 01062 Dresden, Germany.

e-mail: hendrhugo@egon ik.vogt@tu-darmstadt resden.de e-mail: juergen.voigt@tu-dresden.de

(i) If φ ∈ X^{∗}, φ ≠ 0, then the orbit (T^{∗}(t)φ : t ≥ 0) is unbounded,

(ii) the point spectrum of A^{∗} is empty.

Remark 1.2. On a separable Banach space, a chaotic C_{0}-semigroup is automatically hypercyclic (see [1; Definition 2.1 and Theorem 2.2]; see also [2; Definition 1.2] for “chaotic”).

2 Main result

From [2] we know that, by the change of variable x = e^{−y} (y > 0), one transforms equation (0.1) into

(2.1) \{ \begin{matrix}\frac{∂v(t,y)}{∂t} = e^{y} \frac{∂(e^{−y}u(t,y))}{∂y} + (γ − β)v(t, y) + 4βv(t, y − ln 2)χ_{(ln 2,∞)}(y) =: Av(t, y), \\ v(0, ·) = g ∈ L^{1}((0, ∞), e^{−y}dy), \end{matrix}

with g(y) = f(e^{−y}) and D(A) = \{v ∈ L^{1}((0, ∞), e^{−y}dy); \frac{∂v}{∂y} ∈ L^{1}((0, ∞), e^{−y}dy)\}. By a standard perturbation argument one sees that A generates a C_{0}-semigroup T (·) on L^{1}((0, ∞), e^{−y}dy) and hence equation (2.1) (or (0.1)) is well-posed.

The following is the main result of this paper.

Theorem 2.1. If β > \frac{1}{2 ln 2}, then the C_{0}-semigroup associated with (2.1) is not hypercyclic, and therefore not chaotic.

Remark 2.2. Note that the hypothesis γ − 3β > 0 of [2; Theorem 2.3] is not needed in Theorem 2.1.

Proof of Theorem 2.1. In view of Theorem 1.1 and Remark 1.2 it is sufficient to show that the point spectrum of A^{∗} is not empty.

A standard computation shows that A^{∗} is given by

A^{∗}φ(y) = −φ′(y) + (γ − β)φ(y) + 2βφ(y + ln 2) (y ≥ 0),

with D(A^{∗}) = {φ ∈ W_{∞}^{1}(0, ∞); φ(0) = 0}.

A function φ is an eigenfunction of A^{∗}, associated with the eigenvalue λ, if and only if φ is a bounded C^{1}-function satisfying

(2.2)  \{ \begin{matrix}φ′(y) = ωφ(y) + 2βφ(y + ln 2) (y ≥ 0), \\ φ(0) = 0, \end{matrix}

with ω := γ − β − λ.

By setting h(y) := e^{−(ω ln 2)y}φ(y ln 2), c := 2β ln 2 e^{ω ln 2}, we transform problem (2.2) into a problem for a “forward delay equation”,

(2.3)  \{ \begin{matrix}h′(y) = c h(y + 1) (y ≥ 0), \\ h(0) = 0. \end{matrix}

The function h(y) := ye^{y} is a solution of (2.3), for c = e^{−1}. For the corresponding solution φ of (2.2) we obtain

φ(y ln 2) = e^{(ω ln 2)y}h(y) = ye^{(ω ln 2+1)y}.

From c = e^{−1} and 2β ln 2 > 1 we obtain that ω ln 2 + 1 = ln \frac{c}{2β ln 2} + 1 = − ln(2β ln 2) < 0. We conclude that φ is bounded, and therefore φ is an eigenfunction of A^{∗}.

Combining Theorem 2.1 and [2; Theorem 2.3], we obtain the following conclusion.

Corollary 2.3. Assume that γ − 3β > 0. Then the C_{0}-semigroup associated with (2.1) is chaotic if and only if 0 ≤ β ≤ \frac{1}{2 ln 2}.

Appendix. On solutions of h′(y) = c h(y + 1)

In this appendix we present further solutions of problem (2.3), and we explain how we found the above solution.

We start with the observation that for any λ ∈ ℂ the function g_{λ}(y) := e^{λy} satisfies

g′_{λ}(y) = λe^{−λ}g_{λ}(y + 1).

If λ_{1}, λ_{2} ∈ ℂ, λ_{1} ≠ λ_{2}, are such that λ_{1}e^{−λ_{1}} = λ_{2}e^{−λ_{2}}, then obviously h := g_{λ_{1}} − g_{λ_{2}} is a non-trivial solution of (2.3).

Let ξ < 1. We are looking for η > 0 such that

(ξ + iη)e^{−(ξ+iη)} = (ξ − iη)e^{−(ξ−iη)},

i.e., (ξ + iη)e^{−(ξ+iη)} is real, 0 = Im ((ξ + iη)e^{−(ξ+iη)}) = η cos η − ξ sin η, ξ = \frac{η cos η}{sin η}. The function η ↦ \frac{η cos η}{sin η} maps the interval [0, π) continuously and strictly decreasingly onto (−∞, 1]. Therefore there exists a unique η (=: η_{ξ}) ∈ (0, π) such that ξ = \frac{η cos η}{sin η}. Moreover, η_{ξ} depends continuously and strictly decreasingly on ξ, and η_{ξ} → 0 as ξ → 1. For a pair (ξ, η) of this kind one concludes that

h(y) := \frac{1}{2i}(e^{(ξ+iη)y} − e^{(ξ−iη)y}) = e^{ξy} sin ηy

is a solution of (2.3), for c = c_{ξ} := (ξ + iη)e^{−(ξ+iη)} = e^{−ξ}(ξ cos η + η sin η) = e^{−ξ} \frac{η}{sin η}. Using the function h above we obtain that

h_{ξ}(y) := e^{ξy} \frac{sin η_{ξ}y}{η_{ξ}}

is a solution of (2.3), for c = c_{ξ}. Taking the pointwise limit one obtains

h(y) := lim_{ξ→1} h_{ξ}(y) = ye^{y},

and it is easy to check that h satisfies (2.3) for c = e^{−1} = lim_{ξ→1} c_{ξ}.

References

[1] W. Desch, W. Schappacher, and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems 17 (1997), 793–819.

[2] S. EL Mourchid, G. Metafune, A. Rhandi, and J. Voigt, On the chaotic behaviour of size structured cell populations, J. Math. Anal. Appl. 339 (2008), 918-924.