SQU Journal for Science, 2020, 25(2), 100-106 DOI:10.24200/squjs.vol25iss2pp100-106 Sultan Qaboos University 100 Doubly Periodic Functions and Floquet Theorem Nafya H. Mohammed 1 * and Nazaneen Q.M. Saeed 2 1 Mathematics Department, College of Basic Education, University of Raparin, Kurdistan Region-Iraq. 2 Mathematics Department, College of Education, University of Salahaddin, Kurdistan Region-Iraq. *E-mail: nafya.mohammad@uor.edu.krd ABSTRACT: In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such function must be constant. Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. In this paper, we extend Floquet theorem and another theorem (which is mentioned in [1]) related to it, which are dependent on elliptic functions. Keywords: Meromorphic function; Periodic function; Elliptic function; Floquet Theorem; Fundamental matrix. و مبرهنة فلوكت مضاعفة الدورانالدوال نفيا حميد محمد و نازنين قادر محمد سعيد ، مجالحقيقي بقيمها على حولمثلما يتم تعريف الدالة الدورية لمت. ان باتجاهيندور ذاتدالة مرومورفية يفي التحليل العقدالدالة الناقصة عتبر ت :صلخمال أن تكون تحليلية، انالدورمضاعفة هذه الدالة لال يمكن .األساسي، الذي يتكرر بعد ذلك في الشبكةضالع أللة الناقصة بقيمها على متوازي ايتم تعريف الدا ال الناقصة ألول مرة بواسطة تاريخيا، تم اكتشاف الدو .ثابتة تكونيجب أن دالة كلن فإليوفيل، مبرهنة ، وحسب اكليقيدة م ألنها ستكون عندئذ دالة مع الدوال لعالقتهاهذه ة دراسومن ناحية أخرى تم وستاف جاكوبي؛ جا بواسطة كارل نظري هاللتكامالت الناقصة، وتم تحسينهنريك أبيل كدوال عكسية نيلز بها، ذات صلة[( 1]المذكورة في )ت و مبرهنة أخرى يفلوك عمم في هذا البحث، مبرهنة ن .سمإلا ذا تم إعطائهاناقص، ومن هقطع لقوس السألة طول م .لى الدوال الناقصةتمد عوالتي تع .المصفوفة األساسيةالدالة المرومورفية، الدالة الدورية، الدالة الناقصة، مبرهنة فلوكت، :مفتاحيةالكلمات ال 1. Introduction n our opinion, complex analysis is one of the most beautiful areas of mathematics. It has one of the highest ratios of theorems to definitions (i.e., a very low “entropy”), and many applications to things that seem unrelated to complex numbers. Also, it is a comprehensive subject, which provides every mathematician with helpful data. In this respect and due to the usefulness of this subject, we have chosen elliptic functions to be the focus of our work. We need to give a definition of what an elliptic function is, so we will restrict ourselves to meromorphic functions which are functions having only poles as singularities. A doubly periodic function is a function that has two primitive periods, namely 2𝑤1 and 2𝑤3 with 𝑓(𝑧 + 2𝑚𝑤1 + 2𝑛𝑤3) = 𝑓(𝑧); 𝑚, 𝑛 ∈ ℤ. The set of all points of the form 2𝑚𝑤1 + 2𝑛𝑤3, with 𝑚 and 𝑛 being integers is called the period lattice. An elliptic function is a meromorphic function that admits two independent primitive periods. At least one of the two primitive periods of an elliptic function should be complex since the ratio of these two periods should be non-real. I mailto:nafya.mohammad@uor.edu.krd https://en.wikipedia.org/wiki/Complex_analysis https://en.wikipedia.org/wiki/Meromorphic_function https://en.wikipedia.org/wiki/Periodic_function https://en.wikipedia.org/wiki/Fundamental_parallelogram_(complex_analysis) https://en.wikipedia.org/wiki/Lattice_(group) https://en.wikipedia.org/wiki/Doubly_periodic_function https://en.wikipedia.org/wiki/Holomorphic_function https://en.wikipedia.org/wiki/Bounded_function https://en.wikipedia.org/wiki/Entire_function https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis) https://en.wikipedia.org/wiki/Niels_Henrik_Abel https://en.wikipedia.org/wiki/Inverse_function https://en.wikipedia.org/wiki/Elliptic_integral https://en.wikipedia.org/wiki/Elliptic_integral https://en.wikipedia.org/wiki/Carl_Gustav_Jacobi https://en.wikipedia.org/wiki/Arc_length https://en.wikipedia.org/wiki/Ellipse DOUBLY PERIODIC FUNCTIONS AND FLOQUET THEOREM 101 In 1998, Gesztesy and Weikard [2] provided an overview of elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, in which they concentrated on Floquet theorem. Also, Weikard in 2000 [3] dealt with differential equations with meromorphic solutions, which is related to Floquet theorem, while Chouikha [4] paid more attention to properties and developments of elliptic functions, and in particular Jacobi elliptic functions. The main work in this paper is the extension of the Floquet theorem based on elliptic functions. 2. Preliminaries Definition 2.1. A function 𝑓: ℂ ⟶ ℂ∞ with two periods 2𝑤1 and 2𝑤3, the ratio of which is not real, is called ‘doubly periodic’. Definition 2.2. A function that is analytic in the region 𝐷 except for poles in 𝐷, is called ‘meromorphic’ in 𝐷. Definition 2.3. A doubly periodic meromorphic function is called ‘elliptic’. Table (1) contains 12 Jacobi elliptic functions (as examples of elliptic functions) with their periods, zeros, poles, and residues of the functions at the poles. Table 1. Some information on Jacobi elliptic functions. Functions Periods Zeros Poles Residues 𝑐𝑑 (𝑧, 𝑘) 4𝑚𝐾 + 2𝑛𝐾′𝑖 (2𝑚 + 1)𝐾 + 2𝑛𝐾′𝑖 (2𝑚 + 1)𝐾 + (2𝑛 + 1)𝐾′𝑖 (−1)𝑚−1/𝑘 𝑐𝑛 (𝑧, 𝑘) 4𝑚𝐾 + 4𝑛𝐾′𝑖 (2𝑚 + 2𝑛 + 1)𝐾 + 2𝑛𝐾′𝑖 2𝑚𝐾 + (2𝑛 + 1)𝐾′𝑖 (−1)𝑚+𝑛−1𝑖/𝑘 𝑐𝑠 (𝑧, 𝑘) 2𝑚𝐾 + 4𝑛𝐾′𝑖 (2𝑚 + 1)𝐾 + 2𝑛𝐾′𝑖 2𝑚𝐾 + 2𝑛𝐾′𝑖 (−1)𝑛 𝑑𝑐 (𝑧, 𝑘) 4𝑚𝐾 + 2𝑛𝐾′𝑖 (2𝑚 + 1)𝐾 + (2𝑛 + 1)𝐾′𝑖 (2𝑚 + 1)𝐾 + 2𝑛𝐾′𝑖 (−1)𝑚−1 𝑑𝑛 (𝑧, 𝑘) 2𝑚𝐾 + 4𝑛𝐾′𝑖 (2𝑚 + 1)𝐾 + (2𝑛 + 1)𝐾′𝑖 2𝑚𝐾 + (2𝑛 + 1)𝐾′𝑖 (−1)𝑛−1𝑖 𝑑𝑠 (𝑧, 𝑘) 4𝑚𝐾 + 4𝑛𝐾′𝑖 (2𝑚 + 1)𝐾 + (2𝑛 + 1)𝐾′𝑖 2𝑚𝐾 + 2𝑛𝐾′𝑖 (−1)𝑚+𝑛 𝑛𝑐 (𝑧, 𝑘) 4𝑚𝐾 + 4𝑛𝐾′𝑖 2𝑚𝐾 + (2𝑛 + 1)𝐾′𝑖 (2𝑚 + 1)𝐾 + 2𝑛𝐾′𝑖 (−1)𝑚+𝑛−1/𝑘′ 𝑛𝑑 (𝑧, 𝑘) 2𝑚𝐾 + 4𝑛𝐾′𝑖 2𝑚𝐾 + (2𝑛 + 1)𝐾′𝑖 (2𝑚 + 1)𝐾 + (2𝑛 + 1)𝐾′𝑖 (−1)𝑛−1𝑖/𝑘′ 𝑛𝑠 (𝑧, 𝑘) 4𝑚𝐾 + 2𝑛𝐾′𝑖 2𝑚𝐾 + (2𝑛 + 1)𝐾′𝑖 2𝑚𝐾 + 2𝑛𝐾′𝑖 (−1)𝑚 𝑠𝑐 (𝑧, 𝑘) 2𝑚𝐾 + 4𝑛𝐾′𝑖 2𝑚𝐾 + 2𝑛𝐾′𝑖 (2𝑚 + 1)𝐾 + 2𝑛𝐾′𝑖 (−1)𝑛−1/𝑘′ 𝑠𝑑 (𝑧, 𝑘) 4𝑚𝐾 + 4𝑛𝐾′𝑖 2𝑚𝐾 + 2𝑛𝐾′𝑖 (2𝑚 + 1)𝐾 + (2𝑛 + 1)𝐾′𝑖 (−1)𝑚+𝑛−1𝑖/(𝑘. 𝑘′) 𝑠𝑛 (𝑧, 𝑘) 4𝑚𝐾 + 2𝑛𝐾′𝑖 2𝑚𝐾 + 2𝑛𝐾′𝑖 2𝑚𝐾 + (2𝑛 + 1)𝐾′𝑖 (−1)𝑚/𝑘 where 0 < 𝑘 < 1, 𝑘′ = √1 − 𝑘2 , 𝐹 ( 𝜋 2 , 𝑎) = ∫ 1 √(1 − 𝑣2)(1 − 𝑎2𝑣2) 𝑑𝑣 1 0 , 𝐾 = 𝐹 ( 𝜋 2 , 𝑘) and 𝐾′ = 𝐹 ( 𝜋 2 , 𝑘′). More information about elliptic functions is provided in [5]. Definition 2.4 [3, p3]. Two matrices 𝐴, 𝐵 ∈ 𝐸𝑛×𝑛 (where 𝐸 denotes the field of elliptic functions of the same periods) are said to be of the same kind (with respect to 𝐸) if there exists an invertible matrix 𝑇 ∈ 𝐸 𝑛×𝑛 such that 𝐵 = 𝑇−1(𝐴𝑇 − 𝑇′) and 𝑇′ is the derivative of matrix 𝑇. Example 2.5. Two matrices 𝐴 = [ 1 𝑠𝑛 𝑡 0 1 ] and 𝐵 = [ 1 + 2 𝑠𝑛 𝑡 2 𝑠𝑛 𝑡 −2 𝑠𝑛 𝑡 1 − 2 𝑠𝑛 𝑡 ] are of the same kind since there exists an invertible matrix 𝑇 = [ 1 0 2 2 ] such that 𝐵 = 𝑇−1(𝐴𝑇 − 𝑇′). It is obvious that the set 𝐸 is closed under the operations of addition, subtraction, multiplication, division by non- zero divisor, and differentiation [2, p278]. Now, we consider the set 𝑆 of all invertible matrices whose entries are elliptic functions of the same periods (or the entries of the matrices are elements of 𝐸). NAFYA H. MOHAMMED and NAZANEEN Q.M. SAEED 102 Note 2.6. Since the all the operations of addition, subtraction, multiplication and differentiation on 𝑆 depend directly on the operations on 𝐸, and 𝐸 is closed under these operations, we can say that 𝑆 is closed under the all these operations. Theorem 2.7. Any pair of matrices in 𝑆 is of the same kind. Proof. Since the relation “of the same kind” is an equivalence relation [3, p3], then every element of 𝑆 is of the same kind as itself. Thus we prove the theorem for any distinct pair of matrices in 𝑆, and for this purpose we show the elements of 𝑆 by the set {𝑇𝑖 ; 𝑖 ∈ ℕ}. At the first step, we fix 𝐴 = 𝑇1 and choose 𝑇𝑚1 ∈ {𝑇𝑖 ; 𝑖 ∈ ℕ, 𝑖 ≠ 1} arbitrarily. By Note 2.6 and the closedness property of 𝑆, 𝑇𝑚1 −1, 𝑇𝑚1 ′ and 𝐴𝑇𝑚1 are in 𝑆 , and then 𝑇𝑚1 −1(𝐴𝑇𝑚1 − 𝑇𝑚1 ′ ) ∈ 𝑆. We name this element 𝐵𝑚1 (so, 𝐵𝑚1 ∈ 𝑆). Thus for two elements 𝐴 and 𝐵𝑚1 in 𝑆 there exists 𝑇𝑚1 ∈ 𝑆 such that 𝐵𝑚1 = 𝑇𝑚1 −1(𝐴𝑇𝑚1 − 𝑇𝑚1 ′ ). Thus 𝐴 and 𝐵𝑚1 are of the same kind. Again, we choose 𝑇𝑚2 ∈ {𝑇𝑖 ; 𝑖 ∈ ℕ, 𝑖 ≠ 1 𝑎𝑛𝑑 𝑖 ≠ 𝑚1} and, in the same way as above, we can say that 𝑇𝑚2 −1(𝐴𝑇𝑚2 − 𝑇𝑚2 ′ ) ∈ 𝑆. We name this element 𝐵𝑚2 (so, 𝐵𝑚2 ∈ 𝑆). Thus for two elements 𝐴 and 𝐵𝑚2 in 𝑆 there exists 𝑇𝑚2 ∈ 𝑆 such that 𝐵𝑚2 = 𝑇𝑚2 −1(𝐴𝑇𝑚2 − 𝑇𝑚2 ′ ). Thus 𝐴 and 𝐵𝑚2 are of the same kind, and so on. In the second step, we let 𝐴 = 𝑇2 and repeat the previous step. We continue by choosing the elements 𝐴 and 𝑇𝑚𝑖, (𝑚𝑖 ∈ ℕ), such that 𝐴 is fixed and 𝑇𝑚𝑖 is arbitrary, to complete the proof. 3. Extension of Floquet theorem Remark 3.1. In [3] it has been mentioned that, in the basic work of Floquet, the independent variable is complex, and the entries of the matrix of the coefficients are analytic functions, and that if these coefficients are not so, then the only possible singularities are isolated singularities. Thus, if we want to extend the Floquet theorem the poles of the entries of the matrix of the coefficients do not affect the extension, because when establishing the theorem, Floquet took it into consideration that some of the functions might have isolated singularities and we extend this theorem depending on the periods of the matrix of the coefficients, and assume that the matrix of the coefficients belongs to 𝑆. In other words, in this paper the entries of the matrix of the coefficients are meromorphic and doubly periodic functions. Now, let 𝑋1(𝑡), ⋯ , 𝑋𝑛(𝑡) be 𝑛 solutions of the linear homogenous system 𝑋′ = 𝐴(𝑡) 𝑋 (1) and 𝑋(𝑡) = [[𝑋1(𝑡)] ⋯ [𝑋𝑛(𝑡)]], so 𝑋(𝑡) is an 𝑛 × 𝑛 matrix solution of (1). If 𝑋1(𝑡), ⋯ , 𝑋𝑛(𝑡) are linearly independent, then 𝑋(𝑡) is non-singular and is called a fundamental matrix. Theorem 3.2. Consider the linear homogenous system (1), where 𝐴(𝑡) ∈ 𝑆. If 𝑊(𝑡) is a fundamental matrix of system (1) such that 𝑊(𝑡0) = 𝐼 (where 𝐼 represents the identity matrix), then: i. 𝑊(𝑡 + 2𝑚𝑤1 + 2𝑛𝑤3) are also fundamental matrices of (1), ∀ 𝑚, 𝑛 ∈ ℤ. ii. Corresponding to every such 𝑊(𝑡) there exist an invertible periodic matrix 𝑃(𝑡) of period 2𝑚𝑤1 + 2𝑛𝑤3 and a constant matrix 𝑅 such that 𝑊(𝑡) = 𝑃(𝑡)𝑒𝑡𝑅. Proof. At the beginning, we mention that our proof will be based on using mathematical double induction. We divide the proof of the theorem into two parts: i. First we prove the theorem for fundamental periods of 𝐴(𝑡). Case 1: If 𝑚 = 1, 𝑛 = 0, then similar to the proof of the Floquet theorem in [6], 𝑊(𝑡 + 2𝑤1) is a fundamental matrix of (1) and there exist an invertible matrix 𝐶0 and a constant matrix 𝑅0 such that 𝐶0 = 𝑊(2𝑤1) = 𝑒 2𝑤1𝑅0 , and we define the matrix 𝑃0(𝑡) by 𝑃0(𝑡) = 𝑊(𝑡)𝑒 −𝑡𝑅0 then it is clear that 𝑃0(𝑡) is periodic of period 2𝑤1 and invertible. So 𝑊(𝑡) = 𝑃0(𝑡)𝑒 𝑡𝑅0 is a fundamental matrix of (1). Case 2: If 𝑚 = 0, 𝑛 = 1, in the same way 𝑊(𝑡 + 2𝑤3) is a fundamental matrix of (1) and there exist 𝐶0 ∗ and 𝑅0 ∗ such that 𝐶0 ∗ = 𝑊(2𝑤3) = 𝑒 2𝑤3𝑅0 ∗ and we define the matrix 𝑃0 ∗(𝑡) = 𝑊(𝑡)𝑒−𝑡𝑅0 ∗ . Clearly 𝑃0 ∗(𝑡) is periodic of period 2𝑤3 and invertible. So 𝑊(𝑡) = 𝑃0 ∗(𝑡)𝑒𝑡𝑅0 ∗ . DOUBLY PERIODIC FUNCTIONS AND FLOQUET THEOREM 103 ii. In this part we prove the theorem for any period of 𝐴(𝑡). Case 1. If 𝑚 = 𝑛, suppose 𝑚 = 𝑛 ≠ 0. Now, for 𝑚 = 𝑛 = 1, 𝑊(𝑡 + 2𝑚𝑤1 + 2𝑛𝑤3) = 𝑊(𝑡 + 2𝑤1 + 2𝑤3) and since 𝐴(𝑡) ∈ 𝑆, 𝑊′(𝑡 + 2𝑤1 + 2𝑤3) = 𝐴(𝑡) 𝑊(𝑡 + 2𝑤1 + 2𝑤3). Then 𝑊(𝑡 + 2𝑤1 + 2𝑤3) is a matrix solution of (1) and, since it is invertible, 𝑊(𝑡 + 2𝑤1 + 2𝑤3) is the fundamental matrix of (1). Therefore, there exists an invertible matrix 𝐶1 such that 𝑊(𝑡 + 2𝑤1 + 2𝑤3) = 𝑊(𝑡) 𝐶1; By taking 𝑡 = 𝑡0 = 0, 𝐶1 = 𝑊(2𝑤1 + 2𝑤3). So, there exists a matrix 𝑅1 such that 𝐶1 = 𝑒 (2𝑤1+2𝑤3)𝑅1, see [6, P.139]. We define the matrix 𝑃1(𝑡) = 𝑊(𝑡)𝑒 −𝑡𝑅1. In this way we easily show that 𝑃1(𝑡) is periodic of period 2𝑤1 + 2𝑤3 and invertible. So 𝑊(𝑡) = 𝑃1(𝑡)𝑒 𝑡𝑅1 . Suppose that the theorem is true for = 𝑛 = 𝑘 . This means 𝑊(𝑡 + 2𝑘𝑤1 + 2𝑘𝑤3) is a fundamental matrix of (1) and there exists an invertible matrix 𝐶𝑘 = 𝑊(2𝑘𝑤1 + 2𝑘𝑤3) and a constant matrix 𝑅𝑘 such that 𝐶𝑘 = 𝑒 (2𝑘𝑤1+2𝑘𝑤3)𝑅𝑘 and 𝑊(𝑡) = 𝑃𝑘(𝑡)𝑒 𝑡𝑅𝑘 where 𝑃𝑘(𝑡)is invertible and periodic of period 2𝑘𝑤1 + 2𝑘𝑤3. We want to prove that it is true for 𝑚 = 𝑛 = 𝑘 + 1. Now, 𝑊′(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) = 𝐴(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) . 𝑊(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) = 𝐴((𝑡 + 2𝑘𝑤1 + 2𝑘𝑤3) + (2𝑤1 + 2𝑤3)) . 𝑊(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) = 𝐴(𝑡 + 2𝑘𝑤1 + 2𝑘𝑤3) . 𝑊(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) = 𝐴(𝑡) . 𝑊(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) So, 𝑊(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) is a matrix solution of (1) and also an invertible matrix, then it is the fundamental matrix of (1). Since 𝑊(𝑡) and 𝑊(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) are both fundamental matrices of (1) we must find 𝐶𝑘+1, in which 𝑊(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) = 𝑊(𝑡). 𝐶𝑘+1. For 𝑡 = 𝑡0 = 0, 𝐶𝑘+1 = 𝑊(2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) = 𝑊(2𝑘𝑤1 + (2𝑤1 + 2𝑘𝑤3 + 2𝑤3)) = 𝑊(2𝑘𝑤1)𝑊(2𝑘𝑤3 + (2𝑤1 + 2𝑤3)) ⋮ = 𝑊(2𝑘𝑤1)𝑊(2𝑘𝑤3)𝑊(2𝑤1)𝑊(2𝑤3) = 𝐶𝑘 ∙ 𝐶1. Hence we have found an invertible matrix 𝐶𝑘+1, and for this invertible matrix there exists a matrix 𝑅𝑘+1 such that 𝐶𝑘+1 = 𝑒 (2(𝑘+1)𝑤1+2(𝑘+1)𝑤3)𝑅𝑘+1. We define a matrix 𝑃𝑘+1(𝑡) by 𝑃𝑘+1(𝑡) = 𝑊(𝑡)𝑒 −𝑡𝑅𝑘+1 . 𝑃𝑘+1(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3) = 𝑊(𝑡 + 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3). 𝑒 −(𝑡+2(𝑘+1)𝑤1+2(𝑘+1)𝑤3)𝑅𝑘+1 = 𝑊(𝑡). 𝑊(2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3). 𝑒 −𝑡 𝑅𝑘+1 . 𝑒−(2(𝑘+1)𝑤1+2(𝑘+1)𝑤3)𝑅𝑘+1 = 𝑊(𝑡). 𝑒−𝑡 𝑅𝑘+1 . So, 𝑃𝑘+1(𝑡) is periodic of period 2(𝑘 + 1)𝑤1 + 2(𝑘 + 1)𝑤3 and it is invertible. Hence 𝑊(𝑡) = 𝑃𝑘+1(𝑡). 𝑒 𝑡 𝑅𝑘+1 ; and the theorem is true for all 𝑚, 𝑛 ∈ ℕ; 𝑚 = 𝑛. Case 2: If 𝑚 ≠ 𝑛. a. We fix 𝑚 = 𝑎; 𝑎 ∈ ℕ; and prove the theorem for 𝑛 = 1, 2, 3, ⋯ by mathematical induction. For 𝑛 = 1, similar to case 1, we can easily show that 𝑊(𝑡 + 2𝑎𝑤1 + 2𝑤3) is a fundamental matrix of (1), and we can find the invertible matrix 𝐶1 ∗ = 𝑊(2𝑎𝑤1 + 2𝑤3), and for this matrix there exists a matrix 𝑅1 ∗ such that 𝐶1 ∗ = 𝑒(2𝑎𝑤1+2𝑤3)𝑅1 ∗ . We define a matrix 𝑃1 ∗(𝑡) by 𝑃1 ∗(𝑡) = 𝑊(𝑡)𝑒−𝑡𝑅1 ∗ . We can also show that it is periodic of period 2𝑎𝑤1 + 2𝑤3, and is an invertible matrix. Then 𝑊(𝑡) = 𝑃1 ∗(𝑡)𝑒𝑡𝑅1 ∗ . Suppose the theorem is true when 𝑛 = 𝑘. That means 𝑊(𝑡 + 2𝑎𝑤1 + 2𝑘𝑤3) is the fundamental matrix of (1) and there exist 𝐶𝑘 ∗ = 𝑊(2𝑎𝑤1 + 2𝑘𝑤3) and 𝑅𝑘 ∗ such that 𝐶𝑘 ∗ = 𝑒(2𝑎𝑤1+2𝑘𝑤3)𝑅𝑘 ∗ and 𝑃𝑘 ∗(𝑡) = 𝑊(𝑡)𝑒−𝑡 𝑅𝑘 ∗ which is invertible and periodic of period 2𝑎𝑤1 + 2𝑘𝑤3. NAFYA H. MOHAMMED and NAZANEEN Q.M. SAEED 104 For 𝑛 = 𝑘 + 1 we can easily show that 𝑊(𝑡 + 2𝑎𝑤1 + 2(𝑘 + 1)𝑤3) is a fundamental matrix of (1) and find the invertible matrix 𝐶𝑘+1 ∗ = 𝐶𝑘 ∗ . 𝐶0 ∗ = 𝑊(2𝑎𝑤1 + 2𝑘𝑤3). 𝑊(2𝑤3) and for this invertible matrix there exists a matrix 𝑅𝑘+1 ∗ such that 𝐶𝑘+1 ∗ = 𝑒(2𝑎𝑤1+2(𝑘+1)𝑤3)𝑅𝑘+1 ∗ . We define the matrix 𝑃𝑘+1 ∗ (𝑡) by 𝑃𝑘+1 ∗ (𝑡) = 𝑊(𝑡)𝑒−𝑡𝑅𝑘+1 ∗ and show that it is periodic of period 2𝑎𝑤1 + 2(𝑘 + 1)𝑤3 and is an invertible matrix. Then 𝑊(𝑡) = 𝑃𝑘+1 ∗ (𝑡)𝑒𝑡𝑅𝑘+1 ∗ . Again, we fix 𝑚 = 𝑎 + 1 and repeat the previous steps b. We fix 𝑛 = 𝑏; 𝑏 ∈ ℕ and prove the theorem for 𝑚 = 1, 2, 3, ⋯ by mathematical induction. Hence the theorem is true for all 𝑚, 𝑛 ∈ ℕ. Example 3.3. Consider the linear homogenous system { 𝑥1 ′ = 𝑥1 + 𝑠𝑛 𝑡 𝑥2 𝑥2 ′ = 𝑥2 ; the fundamental matrix of this system is 𝑊(𝑡) = [ 𝑒𝑡 1 𝑘 𝑒𝑡(−𝑙𝑛(𝑑𝑛 𝑡 + 𝑘 𝑐𝑛 𝑡) + ln (1 + 𝑘)) 0 𝑒𝑡 ]. Note that 𝑊(0) = 𝐼 , and then, by the above theorem, 𝑊(𝑡 + 4𝐾 + 8𝐾′𝑖) is also the fundamental matrix of the system where 𝐾 = ∫ 𝑑𝜃 √1−𝑘2 𝑠𝑖𝑛2𝜃 ; 𝜋 2 0 |𝑘| < 1 , 𝐾′ = ∫ 𝑑𝜃 √1−𝑘′ 2 𝑠𝑖𝑛2𝜃 𝜋 2 0 ; 𝑘′ = √1 − 𝑘2. Also for 𝑊(𝑡) we can find a constant invertible matrix 𝑅 = [ 1 0 0 1 ] and a doubly periodic matrix 𝑃(𝑡) of periods 4𝐾 + 8𝐾′𝑖 such that 𝑃(𝑡) = [1 −1 𝑘 ln (√1 − 𝑘2𝑠𝑛2𝑡 + 𝑘√1 − 𝑠𝑛2𝑡 + 1 𝑘 ln(1 + 𝑘)) 0 1 ] and 𝑊(𝑡) = 𝑃(𝑡)𝑒𝑡𝑅. Note 3.4. We have only explained the case for the extension of the Floquet theorem when 𝑚, 𝑛 ∈ ℕ. However it is clear that this extension is true for all 𝑚, 𝑛 ∈ ℤ, and we can show this by considering −2𝑤1 and −2𝑤3 as the fundamental periods of 𝐴(𝑡). Thus the proof of the extended theorem by depending on this note is completed. 4. Another relative to Floquet theorem The Halphen theorem is another relative of the Floquet theorem and expresses the fact that, if in a homogeneous linear system of differential equations the matrix of the coefficients are rational functions that are bounded at infinity and if also the general solution is meromorphic, then a fundamental matrix of solutions exists such that its elements are in the form 𝑅(𝑥) exp (𝜆𝑥), in which 𝑅 is a rational function and 𝜆 is a special complex number. Due to the closeness of the Halphen theorem to the Floquet theorem, the rest of the article presents a version of the Halphen theorem as a relative of Floquet theorem. In this version, the entries of the matrix of the coefficients of system (1) are bounded at a bounded region which is suitably large but contains a finite number of parallelograms. Definition 4.1. For any elliptic function 𝑓 on ℂ with two fundamental periods 2𝑤1 and 2𝑤3, we define the function 𝑓 ∘ by 𝑓∘(𝑡) = 𝑓 ( 𝑤 (4𝑚𝐾 + 2𝑛𝐾′𝑖)𝑖 log 𝑡) ; 𝑤 = 2𝑚𝑤1 + 2𝑛𝑤3 , which is a meromorphic function on ℂ − {0}. Remark 4.2. Since the entries of 𝐴(𝑡) are elliptic functions of two periods 2𝑤1 and 2𝑤3, then the 𝑧 −plane will be divided into an infinite number of parallelograms and period strips by these two periods, in such a way that each two non-parallel period strips will intersect each other in one parallelogram. Let 𝐿1 and 𝐿2 be two period strips which intersect each other in the period parallelogram denoted by ∆. Figure 1. Period parallelogram generated by the intersection of two non-parallel period strips. DOUBLY PERIODIC FUNCTIONS AND FLOQUET THEOREM 105 Definition 4.3. We define a bounded region 𝐿 by taking the period strip 𝐿2 which is suitably large but contains a finite number of parallelograms. Theorem 4.4. Suppose that all entries of the matrix of the coefficients 𝐴(𝑡), of system (1) are bounded at 𝐿. If system (1) has only meromorphic solutions, then there exists a constant (𝑛 × 𝑛) matrix 𝐽 in Jordan normal form and an (𝑛 × 𝑛) matrix 𝑅∘ whose entries are rational functions over ℂ, such that the following statements hold: i. Suppose that there are non-negative integers 𝑣1, ⋯ , 𝑣𝑟−1 such that 𝜆, 𝜆 + 𝑖𝑣1, ⋯ , 𝜆 + 𝑖𝑣𝑟−1 are all the eigenvalues of 𝐴∘(0) which are equal to 𝜆 modulo . Then 𝜆 is an eigenvalue of 𝐽 with algebraic multiplicity 𝑟. ii. System (1) has a fundamental matrix given by 𝑋(𝑡) = [𝑅∘(exp(𝑖(4𝑚𝐾 + 2𝑛𝐾′𝑖)𝑡/𝑤)) ∙ 𝑒𝑥𝑝((4𝑚𝑘 + 2𝑛𝑘′𝑖)𝐽𝑡/𝑤)]. (2) Conversely, suppose that 𝑅∘ is an invertible (𝑛 × 𝑛) matrix whose entries are meromorphic functions and 𝐽 is a constant (𝑛 × 𝑛) matrix. Then 𝑋(𝑡) as in the equation (2) is a fundamental matrix of system (1) where 𝐴(𝑡) ∈ 𝑆 and is of the same kind as a matrix whose entries are bounded at 𝐿. Proof. We define the function 𝑓∘ as in definition 4.1. On the other hand, in [1] it was mentioned that if 𝑓 is a doubly periodic function, then 𝑓 does not have finitely many poles in the period strip, and hence does not have definite limits at the ends of the period strip, and consequently we cannot say 𝑓∘ is a rational function. So, to deal with this, we define the bounded region 𝐿 as in definition 4.3. Now, the theorem can be proved by taking = 4𝑚𝐾 + 2𝑛𝐾′𝑖 , and the rest of the proof is similar to the proof of the theorem 1 in [3]. To avoid our repeating the technical steps of the proof and for better understanding, it is necessary that the reader to refer to [3]. Example 4.5. This example explains the converse of the above theorem. Consider the linear homogenous system 𝑥1 ′ = 𝑥1 + √1 − 𝑠𝑛 2(𝑓(𝑡)) ∙ √1 − 𝑘2𝑠𝑛2(𝑓(𝑡)) ∙ 𝑥2, 𝑥2 ′ = 𝑥2 (3) and let 𝑅0 = [ 1 𝑠𝑛 (𝑓(𝑡)) 0 1 ] (where 𝑓(𝑡) = 1 𝑖 log 𝑒𝑖𝑡) be an invertible matrix whose entries are meromorphic functions and 𝐽 = [ 1 0 0 1 ] be a constant matrix, then by the above theorem 𝑅0 × 𝑒𝑡𝐽 = [ 1 𝑠𝑛 (𝑓(𝑡)) 0 1 ] × [ 𝑒𝑡 0 0 𝑒𝑡 ] = [ 𝑒𝑡 𝑒𝑡𝑠𝑛 (𝑓(𝑡)) 0 𝑒𝑡 ] is the fundamental matrix of system (3) and it is clear that the matrix of the coefficients of the system 𝐴(𝑡) = [1 √1 − 𝑠𝑛 2(𝑓(𝑡)) ∙ √1 − 𝑘2𝑠𝑛2(𝑓(𝑡)) 0 1 ] is of the same kind as matrix 𝐵 that is bounded at 𝐿 and 𝐵 = [ 3 . √1 − 𝑠𝑛2(𝑓(𝑡)) √1 − 𝑘2𝑠𝑛2(𝑓(𝑡)) + 1 − 3 2 + 9 2 √1 − 𝑠𝑛2(𝑓(𝑡)) √1 − 𝑘2𝑠𝑛2(𝑓(𝑡)) + 1 + 3 2 −2 ∙ √1 − 𝑠𝑛2(𝑓(𝑡)) √1 − 𝑘2𝑠𝑛2(𝑓(𝑡)) 1 − 3 ∙ √1 − 𝑠𝑛2(𝑓(𝑡)) √1 − 𝑘2𝑠𝑛2(𝑓(𝑡)) ] . 5. Conclusion In this work, doubly periodic functions are introduced generally and some Jacobi elliptic functions are specifically illustrated. The concepts of matrices of the same kind and additionally doubly periodic functions were applied for the extension of Floquet theorem. Furthermore, there is a detailed description of any pair of matrices, the entries of which are elliptic functions of the same periods, which are of the same kind. In addition it has been proved that if 𝑊(𝑡) is a fundamental matrix of system (1), then there exists an invertible doubly periodic matrix 𝑃(𝑡) and a constant matrix 𝑅 such that 𝑊(𝑡) = 𝑃(𝑡)𝑒𝑡𝑅. Finally, another theorem that is related to the theorem of Floquet is presented, with an example to explain it. Conflict of interest The authors declare no conflict of interest. Acknowledgment The authors thank the anonymous reviewers for their valuable suggestions, helpful comments, and constructive criticisms for improving the manuscript during the process of preparing this article for publication. NAFYA H. MOHAMMED and NAZANEEN Q.M. SAEED 106 References 1. Mohammad, N.H. 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