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One way to rigorously analyze such systems is by studying the stability of the system far from equilibrium. Close to equilibrium, one can show the existence of a Lyapunov function which ensures that the entropy tends to a stable maximum. Fluctuations are damped in the neighborhood of the fixed point and a macroscopic description suffices. However, far from equilibrium stability is no longer a universal property and can be broken. In chemical systems, this occurs with the presence of autocatalytic reactions, such as in the example of the Brusselator. If the system is driven beyond a certain threshold, oscillations are no longer damped out, but may be amplified. Mathematically, this corresponds to a Hopf bifurcation where increasing one of the parameters beyond a certain value leads to limit cycle behavior. If spatial effects are taken into account through a reaction-diffusion equation, long-range correlations and spatially ordered patterns arise, such as in the case of the Belousov–Zhabotinsky reaction. Systems with such dynamic states of matter that arise as the result of irreversible processes are dissipative structures.

One way to rigorously analyze such systems is by studying the stability of the system far from equilibrium. Close to equilibrium, one can show the existence of a Lyapunov function which ensures that the entropy tends to a stable maximum. Fluctuations are damped in the neighborhood of the fixed point and a macroscopic description suffices. However, far from equilibrium stability is no longer a universal property and can be broken. In chemical systems, this occurs with the presence of autocatalytic reactions, such as in the example of the Brusselator. If the system is driven beyond a certain threshold, oscillations are no longer damped out, but may be amplified. Mathematically, this corresponds to a Hopf bifurcation where increasing one of the parameters beyond a certain value leads to limit cycle behavior. If spatial effects are taken into account through a reaction-diffusion equation, long-range correlations and spatially ordered patterns arise, such as in the case of the Belousov–Zhabotinsky reaction. Systems with such dynamic states of matter that arise as the result of irreversible processes are dissipative structures.

严格分析这类系统的一种方法是研究系统远离平衡点的稳定性。接近平衡点时，我们可以证明存在一个'''李亚普诺夫函数 Lyapunov function'''，它确保熵趋于稳定的最大值。波动在固定点附近被阻尼，宏观描述就足够了。然而，远离平衡的稳定性不再是一个普遍的性质，可以被打破。在化学系统中，这发生在自催化反应的存在时，例如在布鲁塞尔子的例子中。如果系统驱动超过一定的阈值，振荡不再阻尼，但可能被放大。数学上，这相当于一个霍普夫分岔，其中一个参数的增加超过某个值会导致极限环行为。如果通过反应扩散方程考虑空间效应，就会产生长程相关性和空间有序图案，例如 Belousov-Zhabotinsky 反应。具有这种不可逆过程所产生的动态物质状态的系统是耗散结构。

严格分析此类系统的一种方法是研究远离平衡的系统的稳定性。接近平衡点时，我们可以证明存在'''<font color="#FFD700">李亚普诺夫函数 Lyapunov function</font>'''，它确保熵趋于稳定的最大值。波动在固定点附近被阻尼，宏观描述就已足够。然而，远离平衡的稳定性不再是一个普遍的性质，并且可以被打破。在化学系统中，这是在存在自催化反应的情况下发生的，例如在'''<font color="#FFD700">布鲁塞尔模型Brusselator</font>'''中。如果系统被驱动超过一定的阈值，振荡不再被阻尼，而是可能被放大。从数学上讲，这相当于一个'''<font color="#FFD700">霍普夫分岔Hopf bifurcation</font>'''，其中一个参数增加超过一定的值会导致'''<font color="#FFD700">极限环行为limit cycle behavior</font>'''。如果通过反应扩散方程来考虑空间效应，就会产生长程关联和'''<font color="#FFD700">空间有序模式spatially ordered patterns</font>'''，例如'''<font color="#FFD700">BZ反应Belousov–Zhabotinsky reaction</font>'''。由于不可逆过程而产生的具有这种物质动态状态的系统是耗散结构。