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- In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounded states means crossing a boundary, in the domain of the parameters, where local stability of the stationary states implies global stability (see, e.g. Kalman's conjecture). If a hidden oscillation (or a set of such hidden oscillations filling a compact subset of the phase space of the dynamical system) attracts all nearby oscillations, then it is called a hidden attractor. For a dynamical system with a unique equilibrium point that is globally attractive, the birth of a hidden attractor corresponds to a qualitative change in behaviour from monostability to bi-stability. In the general case, a dynamical system may turn out to be multistable and have coexisting local attractors in the phase space. While trivial attractors, i.e. stable equilibrium points, can be easily found analytically or numerically, the search of periodic and chaotic attractors can turn out to be a challenging problem (see, e.g. the second part of Hilbert's 16th problem). (en)
- 隱藏吸引子（hidden attractor）是动力系统中一種特別的吸引子，系統中不但有穩定的振荡（極限環或混沌吸引子），也存在唯一的穩定平衡點。 在动力系统的分岔理論中，若有不失去平穩集穩定性的有界振荡，會稱為是隱藏振盪（hidden oscillation）。在非線性控制理論中，非時變系統出現狀態有界的隱藏振盪，表示其越過了參數域的邊界，平穩集的局部穩定性也表示全域的穩定性（卡爾曼猜想）。若隱藏振盪（或是动力系统相空間內的某隱藏振盪子集）可以吸引鄰近幾乎所有的振盪，則稱為是隱藏吸引子（hidden attractor）。 針對有單一平衡點，而且平衡點具有全域吸引性的动力系统，隱藏振盪的出現表示其行為特性的改變，由單穩定性變成雙穩定性（bi-stability）。一般來說，动力系统會變成多穩態，同時在相空間中會同時出現局部的吸引子。平凡吸引子（穩定的平衡點）可以用解析或是數值的方式求得。但要找极限环（週期吸引子）或混沌吸引子的困難度就很高了（參考希爾伯特第十六問題）。 (zh)
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- 隱藏吸引子（hidden attractor）是动力系统中一種特別的吸引子，系統中不但有穩定的振荡（極限環或混沌吸引子），也存在唯一的穩定平衡點。 在动力系统的分岔理論中，若有不失去平穩集穩定性的有界振荡，會稱為是隱藏振盪（hidden oscillation）。在非線性控制理論中，非時變系統出現狀態有界的隱藏振盪，表示其越過了參數域的邊界，平穩集的局部穩定性也表示全域的穩定性（卡爾曼猜想）。若隱藏振盪（或是动力系统相空間內的某隱藏振盪子集）可以吸引鄰近幾乎所有的振盪，則稱為是隱藏吸引子（hidden attractor）。 針對有單一平衡點，而且平衡點具有全域吸引性的动力系统，隱藏振盪的出現表示其行為特性的改變，由單穩定性變成雙穩定性（bi-stability）。一般來說，动力系统會變成多穩態，同時在相空間中會同時出現局部的吸引子。平凡吸引子（穩定的平衡點）可以用解析或是數值的方式求得。但要找极限环（週期吸引子）或混沌吸引子的困難度就很高了（參考希爾伯特第十六問題）。 (zh)
- In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounded states means crossing a boundary, in the domain of the parameters, where local stability of the stationary states implies global stability (see, e.g. Kalman's conjecture). If a hidden oscillation (or a set of such hidden oscillations filling a compact subset of the phase space of the dynamical system) attracts all nearby oscillations, then it is called a hidden attractor. For a dynamical system with a unique equilibrium point that is globally attractive, the birth of a hidden attractor corresponds to a qualitative change in behaviour from monost (en)
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- Hidden attractor (en)
- 隱藏吸引子 (zh)
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