Abstract
So far, a single design principle has proved sufficient to generate a whole world of nontrivial flows in three and higher dimensions. The resulting limiting equations appeared rather ‘robust’ in the sense that the limiting parameter \(\varepsilon \) could be increased from zero up to the order of unity (cf. Eq. (2.2)); and also in view of the fact that no more than a single quasi-threshold (an ‘L’ instead of a ‘Z’-shaped slow manifold) with a single quadratic term was required (Eqs. (2.2) and (4.1)).
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Notes
- 1.
- 2.
The original system
$$ \left\{ \begin{array}{l} \dot{x} = -gx + (c - hy) \mathcal{H} \left( \frac{c}{h} - z \right) \\ \dot{y} = mx - ky \\ \dot{z} = ay - bz \end{array} \right. $$accounts for the observed sustained oscillations and includes a specific mechanism by which the pituitary stimulates the thyroid [3]. In this system, x is the concentration of thyrotropin, y is the concentration of activated enzyme, z the concentration of thyroid hormone. b, g and k are loss constants, a, h and m are constants expressing the sensitivity of the glands to simulation or inhibition and c is the rate of production of thyrotropin in the absence of thyroid inhibition.
- 3.
The dimension of the system made of the first equation of system (5.1) with n additional ordinary differential equations is \(n+1\). The corresponding last two equations would be
$$ \left\{ \begin{array}{l} \dot{x}_{n-1} = x_{n-2} - x_{n-1} \\ \dot{x}_{n} = x_{n-1} - x_{n} \, . \end{array} \right. $$ - 4.
1979, personal communication.
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Rössler, O.E., Letellier, C. (2020). The Gluing-Together Principle. In: Chaos. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-44305-4_5
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