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Previous results show that some oscillators possess finite number of Wada basins. Here we find that a nonlinear oscillator can possess a countable infinity of Wada basins and these Wada basins are connected. Infinite number of coexisting attractors and their Wada basins are investigated by the basin cell theorem and generalized basin cell theorem. Infinite number of Wada basins are systematic, which identical basins structure can be identified in each periodic X-axis coordinate interval. This type of Wada basin boundary can lead to a high level of indeterminacy and an extreme sensitive dependence on initial condition.

Basin cells play a fundamental role in the study of Wada basins having the strange property that every point on the basin boundary is on the boundary of at least three different basins. The main goal of this paper is to present a new test for Wada basins using the concept of generalized basin cells. We show that the basins of the smooth and discontinuous oscillator have no basin cells but generalized basin cells in some given parameters. We analyzed several possible types of generalized basin cells in this oscillator. The results provide verifiable conditions of Wada basin boundaries by searching the generalized basin cells.

A new type of basin catastrophe is investigated in two-dimensional driven oscillators. We identify the organization of multiple coexisting basins of attraction before and after the basin catastrophe. Before the catastrophe, all basins spiral outward toward the attractor at infinity in reverse iteration. After the catastrophe, the systematic basin organization is interrupted and some basins spiral inward toward the unstable attractors. This type of basin catastrophe is called basin reversal. The underlying mechanism for basin reversal is discovered by the reversible manifold of the regular saddles. We further analyze that the origin of the emerging abrupt reversal is related to a limit cycle.

It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A quasiperiodically forced piecewise Logistic system is shown to exhibit many types of routes to the creation of SNAs. We point out that the truncation of border-collision torus-doubling bifurcation can lead to different types of SNAs. We identify and describe the Heagy–Hammel routes, fractalization route and intermittent routes after the two coexisting tori collide at the border and the doubled torus is interrupted in this system. It has been shown that there exist two critical tongue-type regions in the parameter space, where the different mechanisms for the birth of SNAs are investigated. These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents. Different types of SNAs are also characterized by the singular-continuous spectrum, Fourier transform, rational approximations, distribution of finite-time Lyapunov exponents and recurrence analysis.

In driven oscillators, examples are shown that every boundary point of one basin is on the boundary of the two remaining basins and all three boundaries of these basins coincide. When there are more than five basins of attraction, is it possible that every boundary point of one basin is on the boundary of other basins? Is it possible that all basin boundaries coincide? This paper describes some numerical experiments giving evidence of seven Wada basin boundaries and all seven basin boundaries coincide for a driven shallow arch oscillator. The results are verified by the basin cell theory. It suggests that small noises can lead to final state sensitivity and uncertain dynamics in the driven oscillator.

In a recent paper, we examined a driven archetypal oscillator which is derived from an arch bridge model with a sinusoidally varying central load. We showed that this oscillator can possess complex basins of constrained motions in the space of the initial conditions. We here investigate the mechanisms for the metamorphoses of basin boundary with topological complexity as the control parameter (the magnitude of the excitation) varies. The Wada basin boundary metamorphoses include the creation (or disappearance) of some new (or old) basin sets by a series of saddle-node (or inverse saddle-node) bifurcations in the Poincare map. The changes can also be induced by the collision between the attractors and the stable (or unstable) manifold of the regular saddles. The mechanism for Wada basin boundary metamorphoses is also investigated by the homoclinic tangency and rotary homoclinic tangency. The results perhaps can explain a series of jump phenomena in the mechanical oscillator and give a good guide to safe parameter design according to safe basin.

Based on the method of Poincare mapping undercellreference,wedescribebasinsofattractionfor coexisting multi-dimensional tori attractors in a three- degree-of-freedom vibro-impact system. Because the multi-dimensional tori attractors are very rare in the low-dimensional systems, we find that these coexist- ing tori attractors have positive measure basins in the sense of Milnor. The coexisting multi-dimensional tori attractors can be distinguished by the exactly Poincare mappingandLyapunovdimension.Thebasinofattrac- tion can be estimated by the limit cycle in the Poincare section from the viewpoint of engineering.

The basin boundaries with nested structure are investigated in a shallow arch oscillator. Basin organization is complex yet systematic and it is governed by the ordering of heteroclinic and homoclinic connections of regular saddles. The Wada properties are verified for eight basin boundaries, where five basin boundaries are totally Wada basin boundaries for a given set of parameters. The organization of nested basin boundaries is governed by the order of saddle connections. The term “Wada number” is introduced to describe the nested structure. The partially Wada basin boundaries are investigated by the erodent cells and the remnant cells.

The Wada basin boundaries of a switched Henon map have been verified for period-2 switching signals [Zhang in Nonlinear Dyn. 73:2221–2229, 2013]. Based upon the auxiliary dynamical system method, the results are extended to the generally switched systems with time-dependent switching. Under some generic assumptions, some sufficient conditions guaranteeing Wada basin boundaries are presented for the periodic switching signals. The results show that switching signals can give rise to this type of basin structure. It suggests that the unpredictability associated to the Wada property can also occur in the switched system.

Whether Wada basin boundaries can occur typically in dynamical systems other than smooth systems has been an open question. We verify the existence of Wada basin boundaries in a switched Henon map. We combine two basin boundaries (non-Wada property) but when they are alternated in periodic manners the Wada basin boundaries can be created. We give some mathematically rigorous results guaranteeing these emerging Wada basin boundaries by the auxiliary dynamics method. It suggests that switching can also induce the consequence of the existence of a high number of possible final states.

We investigate a new type of codimension two bifurcation and related dynamics in a phase-modulated system with switched strategy. Two curves intersect at a point and are called the crisis–Hopf bifurcation. At the critical crisis–Hopf vertex, a boundary crisis and Hopf bifurcation coincide. Metamorphosis of coexisting attractors can be observed in the vicinity of the vertex. Another novelty is that we discover some sets of measure zero with riddled holes in the neighborhood of the bifurcation point. These sets display some qualitative and quantitative features of riddled basins but they are essentially different from the riddled basin. It may provide a more comprehensive picture for unusual dynamical features.