Bi-periodicity in an isothermal autocatalytic reaction-diffusion system
Bi-periodicity in an isothermal autocatalytic reaction-diffusion system
复制标题
等温自催化反应扩散系统的双周期性
DOI：
10.1016/0377-0427(89)90155-6
复制
发表时间：
1989
影响因子：
2.4
通讯作者：
C. Kaas
中科院分区：
文献类型：
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作者：
C. Kaas
我们研究了等温条件下的自催化过程，即所谓的Gray-Scott模型，在一个空间维度上进行扩散。我们发现，在对称条件下，我们可以有稳定的平稳解或稳定的周期解，但没有双周期解，尽管存在两对正实部的复共轭特征值。当我们通过使边界条件不相等来打破对称性时，就得到了双周期解。在双参数平面上，双周期区域由具有自交点的Hopf分岔点曲线限定。当第三个参数接近临界值时，该点变形为一个尖点。这个余维数为3的事件可以表述为一个零点问题，我们描述了表述这样一个零点问题的两种不同的方法。采用正交配置法将偏微分方程离散成一个低维的偏微分方程系统，从而减少了计算量。
We have examined an autocatalytic process under isothermal conditions, the so-called Gray-Scott model, with diffusion in one spatial dimension. We have found, that under symmetric conditions we may have stable stationary solutions or stable periodic solutions, but no bi-periodic solutions despite the presence of two pairs of complex conjugated eigenvalues with positive real parts. Bi-periodic solutions are seen, when we break the symmetry by making the boundary conditions unequal. In a two-parameter plane the region of bi-periodicity is bounded by a curve of Hopf bifurcation points with a point of self-intersection. This point is deformed into a cusp point as a third parameter approaches a critical value. This codimension 3 event can be formulated as a zero point problem, and we describe two different methods of formulating such a zero point problem. The computational effort was made small by using an orthogonal collocation method to discretise the PDEs into a low-dimensional system of ODEs.