1.1 Transitivity and hypercyclicity

First of all, we will recall the basic notions on linear dynamics. The study of hypercyclicity started with the work of Kitai [1, 29]. Beauzamy coined the notion of hypercyclicity for linear operators, see [30, 31]. Further information on the origins of this notion can be found in [5].

This concept is stronger than the ones of supercyclicity or cyclicity, in which one considers the multiples of the elements of the orbit or their linear combinations, respectively, instead of the orbit itself. In this work we will restrict ourselves to hypercyclicity. It consists on the existence of elements with dense orbit. The generalization to semigroups is natural, replacing the discrete orbit of an element by a continuous one.

The separability of X also yields that the hypercyclicity of {T_t}_t≥0 is equivalent to the hypercyclicity of a sequence of operators {T_tn}_n for some discretization of the semigroup given by the sequence (t_n)_n with lim_n t_n = ∞, see for instance [36, Prop. 2.1] and [8, Page 186]. A detailed study of the equivalences between the dynamics of semigroups and the dynamics of their discretizations can be found in [36]. The particular case of autonomous discretizations, that is the existence of hypercyclic operators in the semigroup, requires a special attention. Oxtoby and Ulam obtained, from a clever use of the Baire’s Category Theorem, that there is a G_δ set A ⊂ (0, +∞) such that T_t0 is hypercyclic for every t₀ ∈ A. Costakis and Sambarino proved that such a set can be shared by all the non-trivial operators of the translation semigroup on the complex plane [37]. Those results were later improved in [38], where it is shown that in a hypercylic semigroup {T_t}_t≥0 all the operators T_t0, t₀ > 0 are hypercyclic and they share the set of hypercyclic vectors. This also holds for supercyclic semigroups and their non-trivial operators [39].

However, these results cannot be extended to the frame of semigroups whose index set is a sector, or the whole, complex plane. The translation semigroup considered on a suitable L^p-space may result to be hypercylic, or even mixing, but with no single operator satisfying such property [36, Ex. 4.14 and 4.15]. The situation does not improve if we restrict ourselves to holomorphic semigroups [40].

After the aforementioned works of Birkhoff and Rolewicz, the study of the wild dynamics of semigroups was taken on by Lasota, who considered the existence of turbulent orbits in [21]. In fluid dynamics, turbulence is associated with low momentum diffusion, high momentum convection, and rapid variation of pressure and flow velocity in space and time. The model in which he shows his results was given by the solution C₀-semigroup of

where λ denotes the Reynolds number. On the one hand, if λ < 1 the solutions converge to the laminar solution u ≡ 0. On the other hand, if λ ≥ 2 there are infinitely many turbulent solutions. A solution is strictly turbulent if its closure is a compact non-empty set and it does not contain periodic points. Roughly speaking, they are complicated and irregular. This was quite surprising since turbulence seemed to be tied to very strongly nonlinear partial differential equations of higher order.

Following this line, in [41], Lasota also showed an example of a solution semigroup associated to the following Abstract Cauchy Problem that describes the growth of a population of cells which constantly differentiate (change their properties) in time.

Here t represents the time, x the degree of differentiation which goes from x = 0 (undifferentiated cells) to x = L (mature cells), and c(x, t) the velocity of cell differentiation [42-44]. The usual setting for posing these problems are the spaces of continuous or integrable functions on the interval.

The existence of semigroups is not limited to partial differential operators in function spaces. It is known that every separable infinite-dimensional Banach space admits a hypercyclic semigroup [45]. The proof relies on a construction based on a result on biorthogonal sequences on Banach spaces by Ovsepian and Pelczynski [46] and on an analogous result for single operators [47]. An alternative shorter proof was given in [48]. The generalization to Fréchet spaces different from ω was presented in [49]. In this case, the role of the result of Ovsepian and Pelczynski is played by a more general result on the setting of Fréchet spaces [50].

1.2 Devaney chaos

Godefroy and Shapiro introduced the notion of chaos in the sense of Devaney [51] for linear operators [3]. We recall that x ∈ X is a periodic point for a C₀-semigroup {T_t }_t≥0 if there is some t₀ > 0 such that T_t0 x = x.

In [53], following an ergodic-theoretical approach, Rudnicki showed the existence of invariant measures having strong analytic and mixing properties of the solution semigroup of equation (5). He also showed the existence of Devaney chaos, see also [54-56]. An updated revision of these results can be found in [57, 58].

The dynamics associated to this equation has been also considered in other spaces: in Hölder spaces of continuous functions [59-61] and Orlicz spaces [62], and in Sobolev spaces of type W^1,p, 1 ≤ p < ∞ [63]. The reader can find more information about the study of the dynamics of semigroups induced by semiflows in [64].

1.3 Mixing & weakly mixing properties

The notion of transitivity can be strengthened in some ways. The weak mixing property was considered in the setting of linear dynamics by Herrero [65].

1.4 Frequent hypercyclicity

With the goal of studying linear transformations from the point of view of ergodic theory, Bayart and Grivaux introduced the notion of frequent hypercyclicity in order to quantify the frequency with which an orbit meets open sets [73]. This notion was extended to semigroups in [74]. For this purpose we recall the notion of lower density of a set of real numbers: Let B⊆R0+ , we define the lower density of BasDens_(B)=lim inft→∞μ(B∩[0,t])t and the upper density as Dens¯(B)=lim supt→∞μ(B∩[0,t])t , where μ denotes the Lebesgue measure on R0+ .

1.5 Li-Yorke chaos

In the celebrated paper of Li and Yorke [76], they introduced the concept of scrambled set. In this flavor, this notion was studied in linear dynamics in [77] and [78].

1.6 Distributional chaos

The notion of distributional chaos was inspired by the work of Schweizer and Smítal [80]. It was incorporated to the setting of linear dynamics in [81, 82], and thoroughly studied in [83]. The corresponding version for C₀-semigroups was given in [84]. In this notion we require a scrambled set S such that the orbits of any couple of distinct points are closed enough or at least at some distance, measured in terms of upper densities. We say that a semigroup {T_t }_t≥0 is distributionally chaotic if there is an uncountable set Γ ⊆ X, δ > 0, so that for each ɛ > 0 and pair x, y ∈ Γ of distinct points

See also [85, 86] where distributionally chaotic semigroups defined on Fédchet spaces are characterized.

Inspired by the concept of irregular vectors, a new notion of distributionally irregular vector was given in [77] for single operators, later generalized for C₀-semigroups:

1.7 The specification property

The specification property was introduced by Bowen in [87] in order to indicate that there exist periodic points that can approximate prescribed parts of the orbits of certain elements. It is a very strong property, and it was stated for compact metric spaces. In the context of linear dynamics, the corresponding concept was given in [88], and thoroughly studied in [89] for single operators. A recent adaptation to C₀-semigroups was provided:

2.1 The role of the infinitesimal generator

Godefroy and Shapiro reformulated the HC in terms of the abundance of vectors of an operator [3], also known as the Eigenvalue Criterion, see also [96, Th. 3.7] and [97]. For hypercyclicity, it was necessary to have “plenty of” eigenvalues of modulus greater than 1 and smaller than 1 (and for Devaney chaos to have additionally many of them with modulus equal to 1).

For C₀-semigroups, especially for those associated to solutions of linear PDEs and infinite systems of linear ODEs, it turns out to be necessary to have at our disposal dynamical criteria that can be expressed in terms of the infinitesimal generator. In this subsection X will be a complex Banach space.

Kalmes showed that all the nontrivial operators of such semigroups are in fact Devaney chaotic [98, Th. 2.1]. This criterion has presented several different formulations along the time, in order to relax the hypothesis, as far as possible [8, 94, 99-102]. Banasiak and Moszynski [103] reformulated it in order to find subspaces where hypercyclicity and chaos hold (sub-hypercyclicity and sub-chaoticity). A fractional version of criterion 2.7 for semigroups was provided in [104]. As an illustration of its usefulness, we recall the following result. The Desch-Schappacher-Webb Criterion is very strong since it also implies the Semigroup Specification Property [90], which is the strongest dynamical property mentioned in this paper.

Concerning distributional chaos, some criteria for C₀-semigroups were provided in [84], inspired by the ones given for single operators in [83]. We recall the following useful one:

This theorem can be rephrased in terms of the infinitesimal generator of the semigroup, in order to try to use directly some of its spectral properties, see [77, Cor. 31] for the discrete case and [105, Th. 3.7] for the continuous version.

3.1 The translation semigroup of Lρp,1≤p<∞

Rolewicz analyzed the dynamics of the translation semigroup in the setting of weighted spaces of p-integrable functions Lρp(R),1≤p≤∞ , with ρ(s) := a^−s, a > 1 [19]. As we will see, the linear dynamics of the translation semigroup has permitted to express the dynamics in terms of the behaviour of the weight function ρ.

Desch et al. showed that the translation semigroup {τ_t}_t≥0 is hypercyclic in Lρp(R) if, and only if,

Devaney chaos can also be characterized, but in this case the condition is stated in terms of the integrability of ρ. In [109] it is proved that {τ_t}_t≥0 is Devaney chaotic on Lρp(R) if, and only if,

Moreover, this condition results to be equivalent to the existence of a single periodic point.

The mixing property for the translation semigroup on Lρp is, somehow, in the middle of hypercyclicity and Devaney chaos. This can be seen in the corresponding characterization provided by [70]. {τ_t}_t≥0 is mixing on Lρp(R) if, and only if

The translation semigroup is frequently hypercyclic on Lρp(R) if and only if ∫−∞+∞ρ(t)dt<∞ , as it was completely characterized in [94], see also [110] for a generalization of this result.

Barrachina and Peris [111] showed that {τ_t}_t≥0 is (densely) distributionally chaotic on Lρp(R0+) if we can find a measurable subset A⊆R0+ such that Dens¯(A)=1and∫Aρ(s)ds<∞ . Sufficient conditions for Li-Yorke chaos were provided in [112]. The interrelations between Devaney chaos and distributional chaos for this semigroup were studied in [105]: There are examples of a hypercyclic and distributionally chaotic translations that are not Devaney chaotic. This fact can be compared with the one provided in [111, Ex. 2] of a distributionally chaotic translation semigroup that is neither hypercyclic nor Devaney chaotic. In [90] we get that the translation semigroup has the Semigroup Specification Property on Lρp(R) if, and only if, ∫−∞+∞ρ(s)ds<∞ , which is in fact the condition for Devaney chaos.

3.2 The water hammer equations

A direct application of some of the aforementioned results can be seen when studying the solution of the water hammer equations [113]. A water hammer is a pressure wave that occurs, accidentally or intentionally, in a filled liquid pipeline when a tap is suddenly closed, or a pump starts or stops, or when a valve closes or opens. A water hammer wave propagates through pipes reflecting on features and boundaries. This phenomenon is governed by a pair of coupled quasi-linear partial differential equations of first order, the dynamic and continuity equations. A detailed derivation of the governing water hammer equations can be found in Chaudry [114]. Here we provide a representation of the classical solution of the linear water hammer equations with the help of the translation semigroup [113, Cor. 4.2].

where Q(x, t) represents the discharge, H(x, t) denotes the piezometric head at the centerline of the conduit above the specified datum, f is the friction factor (which is assumed to be constant), g is the acceleration due to gravity, υ is the fluid wave velocity, and 𝒜 and 𝒟 are the the cross-sectional area and the diameter of a conduit, respectively. The parameters 𝒜 and 𝒟, are characteristics of the conduit system and are time invariant, but may be functions of x.

This strategy is similar to the case in which the dynamics of the cosine family is analyzed [115, 116] to describe the orbit in terms of a forward and backward orbit. Here, we present how the discharge and the piezometric head evolve from a given initial condition Q(x, 0) = ϕ(x) and H(x, 0) = φ(x).

for every x ∈ ℝ, t ≥ 0 and initial conditions (Q(0), H(0)) = (φ, ϕ) ∈ X × X.

Kalmes studied the semigroup induced by semiflows in [117]. He characterized hypercyclicity and mixing property for cosine operator functions generated by second order partial differential operators on space of integrable functions and continuous functions. Similar results for the case of cosine operator functions generated from shifts have been given by Chang and Chen in [118]. In addition, Chen also considered chaos in the sense of Devaney [116], giving a characterization of chaotic cosine operator functions generated by weighted translations on the Lebesgue space L^p(G), where G is a locally compact group.

3.3 The quasi-linear Lasota equation

Hung has recently studied the hypercyclicity and Devaney chaos for the quasi-linear Lasota equation by reducing their dynamics to the one of the translation semigroup [119]. He considered the equation

where k(x) is a bounded function on the interval and c(x) satisfies

The particular case c(x) = −1 and k(x) = 0 yields the translation semigroup.

3.4 The conjugation lemma

The dynamics of the translation semigroup has many applications, since the study of the behaviour of the solution semigroups associated to many partial differential equations and infinite systems of linear ordinary differential equations, can be reduced to the analysis of the translation semigroup in certain spaces. This is due to an application of the conjugation lemma, which can be formulated as follows: Given {S_t}_t≥0 a C₀-semigroup on a Banach space Y and {T_t}_t≥0 on a Banach space X. Then {T_t}_t≥0 is called conjugate to {S_t}_t≥0 if there exists a homeomorphism Φ : Y → X such that T_t ∘ Φ = Φ ∘ S_t for all t ≥ 0. Dynamical properties such as frequent hypercyclicity, hypercyclicity, mixing, weak mixing and Devaney chaos for a C₀-semigroup are preserved under conjugacy.

Using the conjugation lemma we are going to revisit the dynamics of solution semigroups to some Lasota type equation. Further information on these examples can be found in [54, 105].

4.1 Cell structured models

Howard [123] studied the linear dynamics of the solution semigroup of a size structured model for describing cell growth that can be found in [124, 125].

In particular, the idea of cells of size zero is considered for describing an abnormality in the division process within our population resulting in a accumulation of cells of various size including a population of non-functional “dwarf” cells. The presence of such “dwarf” cells is seen in the blood disorder Alpha-thalassemia, a genetic disease associated with sickle cell anemia [126].

If η(x) := v(x) − µ(x) − β(x) − 1 and there exists some η₀ ∈ ℝ such that −1 < η₀ ≤ η(x) for all 0 ≤ x ≤ 1, then the solution semigroup of (25) is hypercyclic [123, Prop. 5.3]. This model was also considered by El Mourchid et al. [127, 128].

4.2 Birth-and-death models

Protopopescu and Azmy introduced kinetic models that describe “death” process in the study of linear dynamics [23].

Such a model can be used for describing chemical reactions, biological process, or a generalized automaton rule. From the biological point of view, this represents a population of neoplastic cells divided into subpopulations characterized by different levels of cellular resistance to antineoplastic drugs [129]. Here, particles at level n are absorbed at rate α > 0 and particles of level n + 1 are re-emited at rate β > 0 to level n. Particles with internal energy n = 0 are absorbed and cease to exist. The solution semigroup generated by the associated linear operator is Devaney chaotic provided that β > α ≥ 0 [23] and [130, Ex. 2.1].

The study of the dynamics of the general case in which the rates α and β are different for every stage was started in [131] and continued in [132, 133]:

The associated Abstract Cauchy Problem can be restated as

with

The natural setting for considering this problem is the space of summable sequences ℓ₁. The most general result that ensures its chaotic dynamics is the following [8, Ch. 7].

5.1 Solution semigroups associated to second order PDEs

The classical Fourier heat equation is not the right governing equation to model heat transfer processes in which extremely short periods of time or extreme temperature gradients are involved. In these situations, in order to obtain the adequate temperature profiles, we need to use the hyperbolic heat transfer equation (HHTE).

This last equation predicts a finite speed of heat conduction and assumes a wavy character of the heat transfer, contrary to the FHTE.

This equation can be reduced to a first order differential equation as follows,

which permits to express its solution in terms of semigroups.

5.2 Semigroups associated to PDEs of order greater than or equal to 2

Following a similar approach as described in the previous section, one can analyze the chaotic behavior of different linear partial differential equations that describe physical phenomena that include an underlying propagation process. We gather here some of these examples.