Periodic Problems of Difference Equations and Ergodic Theory

Abstract

The necessary and sufficient conditions for solvability of the family of difference equations with periodic boundary condition were obtained using the notion of relative spectrum of linear bounded operator in the Banach space and the ergodic theorem. It is shown that when the condition of existence is satisfied, then such periodic solutions are built using the formula for the generalized inverse operator to the linear limited one.

1. The Problem and The Main Statement

The problem of existence of periodic solutions for classes of equations is well known. Though it is hard to mention all the contributors in a single paper, we would like to mark out well-developed Floke theory [1], which is used in analysis of linear differential equation systems by the means of monodromy matrix. Operator analogy of such theory in noncritical case (when there is single solution) for differential equations in Banach space was developed by Daletskyi and Krein [2].

This paper is dedicated to obtaining analogous conditions for a family of difference equations in Banach space and to building representations of corresponding solutions. The proposed approach allows obtaining solutions for both critical and noncritical cases. Note that this problem can be approached using well-developed pseudoinverse techniques in theory of boundary value problems [3]. In this paper we firstly build a new representation of the pseudoinverse operator based on results of ergodic theory, and then we provide the necessary and sufficient conditions that guarantee the existence of the corresponding solutions.

Let 𝐁-complex Banach space with norm ‖⋅‖ and zero-element 0; ℒ(𝐁)-Banach space of bounded linear operators from 𝐁 to 𝐁. In this paper we consider existence of periodic solutions of the equation𝑥_𝑛+1=𝜆𝐴_𝑛+1𝑥_𝑛+ℎ_𝑛+1,𝑛⩾0, (1.1) with periodicity condition𝑥₀=𝑥_𝑚, (1.2) where 𝐴_𝑛∈ℒ(𝐁), 𝐴_𝑛+𝑚=𝐴_𝑛, for all 𝑛⩾0, 𝜆 is a complex parameter, and {ℎ_𝑛}^∞_𝑛=0 is a sequence in 𝐁. The solution of the corresponding homogeneous equation to (1.1) has the following form [4]: 𝑥_𝑚(𝜆)=Φ(𝑚,𝑛,𝜆)𝑥_𝑛(𝜆),𝑚⩾𝑛, (1.3) where Φ(𝑚,𝑛,𝜆)=𝜆^𝑚−𝑛𝐴_𝑚+1𝐴_𝑚⋯𝐴_𝑛+1,𝑚>𝑛 (1.4) is evolution operator for problem (1.1); Φ(𝑚,𝑚,𝜆)=𝐸, where 𝐸 is identity operator. Let us remark that 𝑈(𝑚,𝜆)=Φ(𝑚,0,𝜆),𝑈(0,𝜆)=𝐸 and 𝑈(𝑘+𝑛,𝜆)=𝑈(𝑘,𝜆)𝑈(𝑛,𝜆). Operator 𝑈(𝑚,𝜆) is traditionally called monodromy operator.

We can represent [4] the solution (1.1) with arbitrary initial condition 𝑥(0,𝜆)=𝑥₀,𝑥₀∈𝐁 in the form 𝑥_𝑘(𝜆)=Φ(𝑘,0,𝜆)𝑥₀+𝑔(𝑘,𝜆), (1.5) where 𝑔(𝑘,𝜆)=^𝑘_𝑖=0Φ(𝑘,𝑖,𝜆)ℎ_𝑖. (1.6) If we substitute this representation in boundary condition (1.2), we obtain operator equation 𝑥₀(𝜆)−𝑥_𝑚(𝜆)=𝑥₀−Φ(𝑚,0,𝜆)𝑥₀−𝑔(𝑚,𝜆)=0. (1.7) According to notations, we get operator equation(𝐸−𝑈(𝑚,𝜆))𝑥₀=𝑔(𝑚,𝜆). (1.8)

Boundary value problem (1.1), (1.2) has periodic solution if and only if operator equation (1.8) is solvable.

Following the paper [5], point 𝜆 is called right stable point if monodromy operator satisfies inequality {‖𝑈^𝑛(𝑚,𝜆)‖⩽𝑐,𝑛⩾0}.

Denote 𝜌_𝑁𝑆(𝐸−𝑈(𝑚,𝜆))={𝜆∈ℂ∶𝑅(𝐸−𝑈(𝑚,𝜆))=𝑅(𝐸−𝑈(𝑚,𝜆))} (this set coincides with the set of all 𝜆∈ℂ such that operator 𝐸−𝑈(𝑚,𝜆) is normally solvable). It follows easily that resolvent set 𝜌(𝐸−𝑈(𝑚,𝜆)) of the operator 𝐸−𝑈(𝑚,𝜆) lies in 𝜌_𝑁𝑆(𝐸−𝑈(𝑚,𝜆)).

In the sequel we assume that 𝐁 is reflexive for simplicity [6].

The main result of this paper is contained in Theorem 1.1.

Theorem 1.1. Let 𝜆∈𝜌_𝑁𝑆(𝐸−𝑈(𝑚,𝜆)) be right stability point for (1.1). Then (a) boundary value problem (1.1), (1.2) has solutions if and only if sequence {ℎ_𝑛}_𝑛∈ℤ+,ℎ_𝑛∈𝐁 satisfies condition lim_𝑛→∞∑^𝑛_𝑘=1∑^𝑚_𝑖=0𝑈^𝑘(𝑚,𝜆)Φ(𝑚,𝑖,𝜆)ℎ_𝑖𝑛=0, (1.9)(b)under condition (1.9), solutions of boundary value problem (1.1), (1.2) have the following form: 𝑥_𝑛=𝑈(𝑛,𝜆)lim_𝑘→∞∑^𝑘_𝑚=1𝑈^𝑚(𝑘,𝜆)𝑘𝑐+𝑈(𝑛,𝜆)𝐺(𝑛,𝜆)ℎ_𝑛, (1.10) where 𝑐 is an arbitrary element of Banach space 𝐁, 𝐺(𝑛,𝜆)-generalized Green operator of boundary value (1.1), (1.2), which is defined by equality 𝐺(𝑛,𝜆)ℎ_𝑛=^∞_𝑘=01−𝜇^𝑘∞_𝑙=0𝜇^−𝑙−1𝑈(𝑚,𝜆)−𝑈₀(𝜆)^{𝑙𝑚𝑘+1}_𝑖=0Φ(𝑚,𝑖,𝜆)ℎ_𝑖−𝑈₀(𝜆)^𝑚_𝑖=0Φ(𝑚,𝑖,𝜆)ℎ_𝑖+^𝑛_𝑖=0Φ(𝑛,𝑖,𝜆)ℎ_𝑖. (1.11)

2. Auxiliary Result

Let us formulate and prove a number of auxiliary lemmas, which entail the theorem.

Lemma 2.1. If 𝜆∈𝜌_𝑁𝑆(𝐸−𝑈(𝑚,𝜆)), then boundary value problem (1.1), (1.2) is solvable if and only if sequence ℎ_𝑛 satisfies the condition lim_𝑛→∞∑^𝑛_𝑘=1∑^𝑚_𝑖=0𝑈^𝑘(𝑚,𝜆)Φ(𝑚,𝑖,𝜆)ℎ_𝑖𝑛=0. (2.1)

Proof. From the assumption above it follows that the conditions of statistical ergodic theorem hold [6]. Then 𝑅(𝐸−𝑈(𝑚,𝜆))=𝑥∈𝐁∶lim_𝑛→∞𝑈_𝑛(𝑚,𝜆)𝑥=0,𝑈_𝑛()=∑𝑚,𝜆^𝑛𝑘=1𝑈^𝑘(𝑚,𝜆)𝑛. (2.2) It follows from the equation above that element 𝑔(𝑚,𝜆) lies in value set of the operator 𝐸−𝑈(𝑚,𝜆) if and only if lim_𝑛→∞∑^𝑛_𝑘=1𝑈^𝑘(𝑚,𝜆)𝑛^𝑚_𝑖=0Φ(𝑚,𝑖,𝜆)ℎ_𝑖=0, (2.3) which proves the lemma.

Consider the following consequences of the assumptions above for further reasoning. Suppose that 𝜆∈𝜌_𝑁𝑆(𝐸−𝑈(𝑚,𝜆)) and 𝜆 is right stable point of the monodromy operator, such that 𝜆 define eigenspace 𝑁(𝐸−𝑈(𝑚,𝜆)), which coincides with the values set of operator 𝑈₀(𝜆)𝑥=lim_𝑛→∞𝑈_𝑛(𝑚,𝜆)𝑥. This operator satisfies the following conditions [6]: (i)𝑈₀(𝜆)=𝑈²₀(𝜆),(ii)𝑈₀(𝜆)=𝑈(𝑚,𝜆)𝑈₀(𝜆),(iii)𝑈₀(𝜆)=𝑈₀(𝜆)𝑈(𝑚,𝜆). (2.4)

Lemma 2.2. Operator 𝐸−𝑈(𝑚,𝜆)+𝑈₀(𝜆)∶𝐁→𝐁 has bounded inverse of the form 𝐸−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹=^∞_𝑘=0𝜇−1^𝑘∞_𝑙=0𝜇^−𝑙−1𝑈(𝑚,𝜆)−𝑈₀(𝜆)^𝑙𝑘+1, (2.5) for all 𝜇>1∶|1−𝜇|<1/‖𝑅_𝜇‖.

Proof. Let us show that Ker(𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆))=0. Indeed, if 𝑥∈Ker(𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)), then 𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)𝑥=0. (2.6) Since (𝐼−𝑈(𝑚,𝜆))𝑥∈Im(𝐼−𝑈(𝑚,𝜆)) and 𝑈₀(𝜆)𝑥∈Ker(𝐼−𝑈(𝑚,𝜆)) [6], subspaces Im(𝐼−𝑈(𝑚,𝜆)) and Ker(𝐼−𝑈(𝑚,𝜆)) intersect only at zero point, and condition (2.6) is satisfied if and only if (𝐼−𝑈(𝑚,𝜆))𝑥=0 and 𝑈₀(𝜆)𝑥=0. This is possible if and only if 𝑥=0. Let us show that Im(𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆))=𝐁. Note [6] 𝐁=Ker(𝐼−𝑈(𝑚,𝜆))⊕Im(𝐼−𝑈(𝑚,𝜆))=Im(𝑈₀(𝜆))⊕𝐼𝑚(𝐼−𝑈(𝑚,𝜆)). It follows from the last decomposition that any element 𝑥∈𝐁 has the form (𝐼−𝑈(𝑚,𝜆))𝑦+𝑈₀(𝜆)𝑧, where 𝑦,𝑧∈𝐁, which proves that Im(𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆))=𝐁. Hence according to the Banach theorem [6] original operator has inverse since it bijectively maps 𝐁 to itself. Therefore point 𝜇=1 is regular [6] for the operator 𝜇𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆). Since powers of the operator 𝑈(𝑚,𝜆) are uniformly bounded and spectral radius 𝑟_{𝑈(𝑚,𝜆)}⩽1 (^𝑛‖𝑈(𝑚,𝜆)^𝑛‖⩽√^𝑛𝑐, then 𝑟_{𝑈(𝑚,𝜆)}=lim_𝑛→∞^𝑛‖𝑈(𝑚,𝜆)^𝑛‖⩽lim√_𝑛→∞^𝑛𝑐=1). It is well known [6] that resolvent set of a bounded operator is open. Number 𝜇=1∈𝜌(𝑈(𝑚,𝜆)−𝑈₀(𝜆)); thus there exist a neighborhood of 𝜇 such that each point from the neighborhood belongs to resolvent set. For any point 𝜇>𝑟_{(𝑈(𝑚,𝜆)−𝑈0(𝜆))} that belongs to the neighborhood there exists a resolvent [6], which has the form of converging in the norm series 𝑅_𝜇∶=𝑅_𝜇𝑈(𝑚,𝜆)−𝑈₀(𝜆)=^∞_𝑙=0𝜇^−𝑙−1𝑈(𝑚,𝜆)−𝑈₀(𝜆)^𝑙. (2.7)
Using the analyticity of the resolvent and well-known identity for points 𝜇>1 such that |1−𝜇|<1/(‖𝑅_𝜇(𝑈(𝑚,𝜆)−𝑈₀(𝜆))‖), we obtain 𝑅₁=^∞_𝑘=0𝜇−1^𝑘𝑅_𝜇^𝑘+1. (2.8) Finally, by substituting the series in the equation above, we get (2.5), which proves the lemma.

Let us introduce some notation first before proving next statement.

Definition 2.3. Operator 𝐿⁻∈ℒ(𝐁) is called generalized inverse for operator 𝐿∈ℒ(𝐁) [3] if the following conditions hold:(1)𝐿𝐿⁻𝐿=𝐿,(2)𝐿⁻𝐿𝐿⁻=𝐿⁻. (2.9)

Lemma 2.4. Operator 𝐸−𝑈(𝑚,𝜆) is generalized inverse and (𝐸−𝑈(𝑚,𝜆))⁻=𝐸−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆), (2.10) or in the form of converging operator series (𝐸−𝑈(𝑚,𝜆))⁻=^∞_𝑘=0𝜇−1^𝑘∞_𝑙=0𝜇^−𝑙−1𝑈(𝑚,𝜆)−𝑈₀(𝜆)^𝑙𝑘+1−𝑈₀(𝜆), (2.11) for all 𝜇>1∶|1−𝜇|<1/‖𝑅_𝜇‖.

Proof. It suffices to check conditions (1) and (2) of the Definition 2.3. We use both representations (2.10), (2.11) and the expression (2.4) for operator 𝑈₀(𝜆). Consider the following product: (𝐼−𝑈(𝑚,𝜆))𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆)((𝜆=()𝐼−𝑈))𝐼−𝑈𝑚,𝜆+𝑈₀(𝜆)−𝑈₀(𝜆)×𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆)((=𝐼−𝑈𝑚,𝜆))𝐼−𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)𝑈₀(𝜆)+𝑈₀(𝜆)²×((=𝐼−𝑈𝑚,𝜆))𝐼−𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹×((=𝐼−𝑈𝑚,𝜆))𝐼−𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)−𝑈₀(𝜆())=𝐼−𝑈𝑚,𝜆+𝑈₀(𝜆)−𝑈₀(𝜆)−𝑈₀(𝜆)+𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝑈₀(𝜆)=𝐼−𝑈(𝑚,𝜆)−𝑈₀(𝜆)+𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝑈₀(𝜆). (2.12) Note that 𝑈₀(𝜆)(𝑈(𝑚,𝜆)−𝑈₀(𝜆))^𝑙=0 for any 𝑙∈ℕ (this directly follows from (2.4) using formula of binominal coefficient). Now, prove that 𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝑈₀(𝜆)=𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹=𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝑈₀(𝜆)=𝑈₀(𝜆). (2.13) Indeed 𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝑈₀(𝜆)=^∞_𝑘=0𝜇−1^𝑘𝑈₀(𝜆)^∞_𝑙=0𝜇^−𝑙−1𝑈(𝑚,𝜆)−𝑈₀(𝜆)^𝑙𝑘+1𝑈₀(𝜆)=^∞_𝑘=0𝜇^−1𝑘+1𝜇−1^𝑘𝑈₀(𝜆)+𝜇−1^𝑘𝑈₀(𝜆)×^∞_𝑙=1𝜇^−𝑙−1𝑈(𝑚,𝜆)−𝑈₀(𝜆)^𝑙𝑘+1𝑈₀(𝜆)=^+∞_𝑘=0𝜇^−𝑘−1𝜇−1^𝑘𝑈₀(𝜆)=1𝜇^+∞_𝑘=0𝜇−1𝜇^𝑘𝑈₀(𝜆)=1𝜇1𝑈1−𝜇−1/𝜇⁰(𝜆)=𝑈₀(𝜆). (2.14) Thus 𝐼−𝑈(𝑚,𝜆)−𝑈₀(𝜆)+𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝑈₀(𝜆)=𝐼−𝑈(𝑚,𝜆). (2.15) We have that the operator 𝐼−𝑈(𝑚,𝜆) satisfies condition (1) of the Definition 2.3. Let us check condition (2) 𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆)(𝐼−𝑈(𝑚,𝜆))𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆)=𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)−𝑈₀(𝜆)×𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆)=𝐼−𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)−𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝑈₀(𝜆)+𝑈₀(𝜆)²×𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆)=𝐼−𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆)=𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝑈₀(𝜆)𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆)+𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹𝑈₀(𝜆)=𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆)−𝑈₀(𝜆)+𝑈₀(𝜆)=𝐼−𝑈(𝑚,𝜆)+𝑈₀(𝜆)⁻¹−𝑈₀(𝜆). (2.16)

3. Proof of Theorem 1.1

According to general theory of linear equations solvability [3], we obtain that the problem (1.1), (1.2) is solvable for sets {ℎ_𝑛}_𝑛∈ℤ₊ that satisfy the condition 𝑈₀(𝜆)𝑔(𝑚,𝜆)=0. (3.1) This condition along with Lemma 2.1 is equivalent to represantion (a) of the Theorem 1.1.

Under such a condition, all solutions of the problem (1.1), (1.2) have the form 𝑥_𝑛=𝑈(𝑛,𝜆)𝑈₀(𝜆)𝑐+𝑈(𝑛,𝜆)(𝐼−𝑈(𝑚,𝜆))⁻𝑔()()()𝑈𝑚,𝜆+𝑔𝑛,𝜆=𝑈𝑛,𝜆₀(𝜆)𝑐+𝑈(𝑛,𝜆)^∞_𝑘=0𝜇−1^𝑘∞_𝑙=0𝜇^−𝑙−1𝑈(𝑚,𝜆)−𝑈₀(𝜆)^𝑙𝑘+1𝑔()()𝑈𝑚,𝜆−𝑈𝑛,𝜆₀(𝜆)𝑔(𝑚,𝜆)+𝑔(𝑛,𝜆), (3.2) which along with notations introduced is equivalent to representation (b) of the theorem.

4. Comments and Examples

Remark 4.1. Suppose 𝐵 is Hilbert space, in such case we can show that formulas (2.10), (2.11) give us the representation for the Moore-Penrose pseudoinverse [7, 8] for 𝐸−𝑈(𝑚,𝜆) with 𝑈₀(𝜆) being self-adjoint operator (orthogonal projector) [6].

Remark 4.2. Supposing 𝐴_𝑘⁻¹∈ℒ(𝐁)∈𝐿(𝐵) exist for all 𝑘=0,𝑚−1, then the following equation holds: Φ(𝑘,𝑖,𝜆)=𝑈(𝑘,𝜆)𝑈⁻¹(𝑖,𝜆),𝑘>𝑖. This allows representing the solutions of (1.1), (1.2) using only the family of operators 𝑈(𝑛,𝜆) and their inverse.

Let us illustrate the statements proved above on example of two-dimensional systems.

(1) Consider equation 𝑥_𝑛+1=𝜆𝐴_𝑛+1𝑥_𝑛+ℎ_𝑛+1,𝑛⩾0 (4.1) with periodicity condition 𝑥₃=𝑥₀, (4.2) where 𝑥_𝑛=(𝑥¹_𝑛,𝑥²_𝑛)^𝑇,𝑥¹_𝑛,𝑥²_𝑛∈ℝ, ℎ_𝑛=((3√3𝑟)/4𝜋,0)^𝑇,𝐴_𝑛=−12−√32√32−12,∀𝑛⩾0. (4.3) It is easy to see that𝑥₃=𝜆³𝑥₀+𝑔(3,𝜆), (4.4) where 𝑔()=√3,𝜆−33𝑟𝜆−3√3𝑟𝜆²+6√3𝑟,8𝜋9𝑟𝜆−9𝑟𝜆²8𝜋^𝑇. (4.5) Then the following hold for all 𝑘⩾0𝑈(3𝑘+1,𝜆)=𝜆^3𝑘+1𝐴₂,𝑈(3𝑘+2,𝜆)=𝜆^3𝑘+2−12√32√32−12,𝑈(3𝑘+3,𝜆)=𝜆^3𝑘+3𝐸. (4.6) By substituting periodicity condition (4.2) into (4.4) we obtain an equation depending on 𝑥₀∶1−𝜆³𝑥₀=𝑔(3,𝜆). (4.7) Consider the case when 𝜆=1. In such case (4.7) turns into 0𝑥₀=(0,0)^𝑇 which holds for arbitrary initial vector 𝑥₀∈ℝ². Obviously 𝑈^𝑛(1,1)=𝑈(𝑛,1) and 𝑈₀(1)=𝐸. According to Theorem 1.1, all periodic solutions of (4.1) have the form𝑥¹_𝑛𝑐₁,𝑐₂𝑥²_𝑛𝑐₁,𝑐₂=cos2𝜋3𝑛sin2𝜋3𝑛−sin2𝜋3𝑛cos2𝜋3𝑛𝑐₁𝑐₂+3𝑟2𝜋sin2𝜋3𝑛0, (4.8) for all 𝑐⃗=(𝑐₁,𝑐₂)^𝑇∈ℝ².

(2) We can search for periodic solutions of any period 𝑤 in previous problem. They have common view𝑥_𝑛𝑐₁,𝑐₂,𝑤,𝑟=cos2𝜋𝑤𝑛sin2𝜋𝑤𝑛−sin2𝜋𝑤𝑛cos2𝜋𝑤𝑛𝑐₁𝑐₂+𝑟𝑤2𝜋sin2𝜋𝑤𝑛0, (4.9) where 𝑐₁,𝑐₂,𝑤,𝑟 are parameters.

To illustrate complexity of the set we did the following.

Recall that the length of vector 𝑥_𝑛 is ℓ𝑥_𝑛=(𝑥¹_𝑛)²+(𝑥²_𝑛)². System (4.9) was implemented using the Wolfram Mathematica 7 framework. 𝑥-axis corresponds to time, while 𝑦-axis corresponds to the length of the vector. The length of the vector was calculated in the integer moments of time 𝑛. The points obtained in such way were connected in a piecewise linear way. The results obtained for particular values of the parameters are depicted on the following figures.

We can see how the trajectory of vector length densely fills rectangle or turns into a line (Figures 3 and 4). Figures 1, 2, 5, and 6 demonstrate that the trajectory can fill structured sets. The structure depicted on Figure 1 resembles fractal.

Figure 1 

Figure 2 

Figure 3 

Figure 4 

Figure 5 

Figure 6 

This allows us to conclude that behavior of the system is rather complex; it can undergo unpredictable changes with the slightest variations of a single parameter. We must admit that effects described need further theoretical investigation.

Acknowledgment

A. A. Boichuk was supported by the Grant 1/0090/09 of the Grant Agency of Slovak Republic (VEGA) and by the Project APVV-0700-07 of the Slovak Research and Development Agency.