The Turing instability is one of the most commonly studied mechanisms leading to pattern formation in reaction–diffusion systems, yet there are still many open questions on the applicability of the Turing mechanism. Although experiments on pattern formation using chemical systems have shown that temperature differences play a role in pattern formation, there is far less theoretical work concerning the interplay between temperature and spatial instabilities. We consider a thermodynamically extended reaction–diffusion system, consisting of a pair of reaction–diffusion equations coupled to an energy equation for temperature, and use this to obtain a natural extension of the Turing instability accounting for temperature. We show that thermal contributions can restrict or enlarge the set of unstable modes possible under the instability, and in some cases may be used to completely shift the set of unstable modes, strongly modifying emergent Turing patterns. Spatial heterogeneity plays a role under several experimentally feasible configurations, and we give particular consideration to scenarios involving thermal gradients, thermodynamics of chemicals transported within a flow, and thermodiffusion. Control of Turing patterns is also an area of active interest, and we also demonstrate how patterns can be modified using time-dependent control of the boundary temperature.

中文翻译：

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温度对图灵模式形成的影响

图灵不稳定性是导致反应扩散系统中形成模式的最常研究的机制之一，但关于图灵机制的适用性仍然存在许多悬而未决的问题。尽管使用化学系统进行图案形成的实验表明温度差异在图案形成中起作用，但关于温度和空间不稳定性之间相互作用的理论工作却少得多。我们考虑一个热力学扩展的反应-扩散系统，由一对反应-扩散方程组成，这些方程与温度的能量方程耦合，并使用它来获得考虑温度的图灵不稳定性的自然扩展。我们表明热贡献可以限制或扩大不稳定模式下可能的不稳定模式集，并且在某些情况下可用于完全改变不稳定模式集，强烈修改涌现的图灵模式。空间异质性在几个实验可行的配置下发挥作用，我们特别考虑了涉及热梯度、流体中传输的化学物质的热力学和热扩散的场景。图灵模式的控制也是一个积极感兴趣的领域，我们还演示了如何使用边界温度的时间相关控制来修改模式。流体中输送的化学物质的热力学和热扩散。图灵模式的控制也是一个积极感兴趣的领域，我们还演示了如何使用边界温度的时间相关控制来修改模式。流体中输送的化学物质的热力学和热扩散。图灵模式的控制也是一个积极感兴趣的领域，我们还演示了如何使用边界温度的时间相关控制来修改模式。