Solution of the System of Two Coupled First-Order ODEs with Second-Degree Polynomial Right-Hand Sides

IF 0.9 3区 数学 Q3 MATHEMATICS, APPLIED
Francesco Calogero, Farrin Payandeh
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引用次数: 3

Abstract

The explicit solution \(x_{n}\left (t\right ) ,\) n = 1,2, of the initial-values problem is exhibited of a subclass of the autonomous system of 2 coupled first-order ODEs with second-degree polynomial right-hand sides, hence featuring 12 a priori arbitrary (time-independent) coefficients:

$$ \dot{x}_{n}=c_{n1}\left( x_{1}\right)^{2}+c_{n2}x_{1}x_{2}+c_{n3}\left( x_{2}\right)^{2}+c_{n4}x_{1}+c_{n5}x_{2}+c_{n6}~,~~~n=1,2~. $$

The solution is explicitly provided if the 12 coefficients cnj (n = 1,2; j = 1,2,3,4,5,6) are expressed by explicitly provided formulas in terms of 10 a priori arbitrary parameters; the inverse problem to express these 10 parameters in terms of the 12 coefficients cnj is also explicitly solved, but it is found to imply—as it were, a posteriori—that the 12 coefficients cnj must then satisfy 4 algebraic constraints, which are explicitly exhibited. Special subcases are also identified the general solutions of which are completely periodic with a period independent of the initial data (“isochrony”), or are characterized by additional restrictions on the coefficients cnj which identify particularly interesting models.

右手边为二次多项式的两个耦合一阶ode系统的解
对于右手边为二阶多项式的2个耦合一阶ode自治系统的一个子集,具有12个先验的任意(时间无关的)系数,给出了初值问题的显式解\(x_{n}\left (t\right ) ,\) n = 1,2: $$ \dot{x}_{n}=c_{n1}\left( x_{1}\right)^{2}+c_{n2}x_{1}x_{2}+c_{n3}\left( x_{2}\right)^{2}+c_{n4}x_{1}+c_{n5}x_{2}+c_{n6}~,~~~n=1,2~. $$如果12个系数cnj (n = 1,2;J = 1,2,3,4,5,6)用明确提供的公式表示为10个先验任意参数;用12个系数CNJ来表示这10个参数的逆问题也被显式地解决了,但它被发现意味着——就像它是一个后验——12个系数CNJ必须满足4个代数约束,这些约束被显式地展示出来。还确定了特殊子情况的一般解是完全周期性的,其周期与初始数据无关(“等时性”),或者通过对系数cnj的附加限制来表征,这些限制可以识别出特别有趣的模型。
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来源期刊
Mathematical Physics, Analysis and Geometry
Mathematical Physics, Analysis and Geometry 数学-物理:数学物理
CiteScore
2.10
自引率
0.00%
发文量
26
审稿时长
>12 weeks
期刊介绍: MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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