Polarity balance for attractor self-reproducing.

There are complex chaotic manifolds in practical nonlinear dynamical systems, especially in nonlinear circuits and chemical engineering. Any system attractor has its own geometric and physical properties, such as granularity, orientation, and spatiotemporal distribution. Polarity balance plays an important role in the solution of a dynamical system including symmetrization, attractor merging, and attractor self-reproducing. The absolute value function and the signum function manage and control the polarity balance, strictly regulating the attractor distribution by switching the polarity balances. Attractor self-reproducing is an attractive regime for constructing the desired multistability, where the coexisting attractors at different positions can be extracted by a selected initial value. Polarity balance is the key factor for attractor self-reproducing, where the offset boosting of an attractor needs an available polarity controller to restore the imbalanced polarity.

Doubling the coexisting attractors.

When the offset boosting technique is introduced into a chaotic system for attractor shifting, the number of coexisting attractors in the system can be doubled under the application of the employed absolute-value function. Consequently, the offset booster becomes a doubling parameter determining the distance between the two coexisting attractors, and therefore can polymerize these attractors to become a pseudo-multi-scroll attractor. This paper demonstrates that the attractor doubling operation can be applied to any dimension of the system and can also be nested at any time leading to the geometric growth of the coexisting attractors. Furthermore, various regimes of coexistence can be merged and composed together to reproduce an integrated attractor in the system.