A solvable nonlinear autonomous recursion of arbitrary order

2021/11/18 by F. Calogero, Calogero, Francesco, Farrin Payandeh +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2111.09973

openalex publication_date 2021/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The initial-values problem of the following nonlinear autonomous recursion of order p , z (s + p) = c product of [z (s + l)]^a_l ; with p an arbitrary positive integer, z (s) the dependent variable (possibly a complex number), s the independent variable (a non negative integer), c an arbitrarily assigned, possibly complex, number, and the p exponents a_l arbitrarily assigned integers (positive, negative or vanishing, so that the right-hand side of the recursion be univalent)|is solvable by algebraic operations, involving the solution of a system of linear algebraic equations (generally explicitly solvable) and of a single polynomial equation of degree p (hence explicitly solvable for p = 1; 2; 3; 4 ).

Citations

• Two classes of explicitly solvable sextic equations
• Explicitly Solvable Systems of First-Order Difference Equations With\n Homogeneous Polynomial Right-Hand Sides

Related

• Solvable nonlinear systems of 2 recursions displaying interesting evolutions
• Some properties of the simple nonlinear recursion y(ℓ + 1) = [1-y(ℓ)]^p with p an arbitrary positive integer
• Simple systems of nonlinear recursions whose evolution is rather easily ascertained
• Simple recursions displaying interesting evolutions
• Interesting system of 3 first-order recursions