Chaotic Saddles in Wada Basin Boundaries and Their Bifurcations by the Generalized Cell-Mapping Digraph (GCMD) Method

By means of the generalized cell-mapping digraph (GCMD) method, we studybifurcations governing the escape of periodically forced oscillatorsfrom a potential well, in which a chaotic saddle plays an extremelyimportant role. In this paper, we find the chaotic saddle anddemonstrate that it is embedded in a strange fractalbasin boundary which has the Wada property that any point that is on theboundary of that basin is also simultaneously on the boundary of atleast two other basins. The chaotic saddle in the Wada basin boundary,by colliding with a chaotic attractor, leads to a chaotic boundarycrisis with indeterminate outcome. A local saddle-node fold bifurcation,if the saddle of the saddle-node fold is located in tangency with thechaotic saddle in the Wada basin boundary, also results in a strangeglobal phenomenon, namely that the local saddle-node fold bifurcation hasglobally indeterminate outcome. We also investigate the origin andevolution of the chaotic saddle in the Wada basin boundary, particularlyconcentrating on its discontinuous bifurcations (metamorphoses). Wedemonstrate that the chaotic saddle in the Wada basin boundary iscreated by a collision between two chaotic saddles in differentfractal basin boundaries. After a final escape bifurcation, there onlyexists the attractor at infinity and a chaotic saddle with a beautifulpattern is left behind in the phase space.     

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

Common Features of the Onset of the Persistent Chaos in Nonlinear Oscillators: A Phenomenological Approach

Bifurcational precedences of generating the persistent,crossing-the-potential-barrier chaos in nonautonomous, dissipative nonlinearoscillators are examined. The comparative computational study covers thetwin-well oscillator, pendulum with parametric excitation, and pendulumdriven by external harmonic force. The study reveals common features of thesystem response properties and of the bifurcational scenarios, prior to theonset of the chaos. It is pointed out that, in the three considered systems,the chaos is preceded by the two and only two asymmetric periodicattractors, which are simultaneously annihilated via theperiod-doubling-crisis scenario. However, the generating of the chaos is notnecessarily related directly to the escape from a potential well. We alsoshow that, in all three systems, the persistent chaos can be viewed as anirregular combination of the `crossing the potential barrier' and theoscillatory component of motion.     

Effects of Multi Global Bifurcations on Basin Organization,Catastrophes and Final Outcomes in a Driven Nonlinear Oscillator at the 2T-Subharmonic Resonance

The behavior of the escape driven oscillator at the 2T-periodic subharmonic resonance is considered, and the mechanism of generating different fractal patterns of the basins of attraction of coexisting attractors, as well as its effects on the unpredictable asymptotic system behaviors, are the main points of interest. The analysis is based on the numerical study of the sudden qualitative changes of the structure of basin-phase portraits, the changes implied by multi global bifurcations. Attention is focused on two qualitatively different regions of control space: the region prior to the subcritical flip bifurcation, where all three attractors (2T-periodic, T-periodic and the attractor at infinity) coexist, and the region after the bifurcation, where only two attractors (2T-periodic and the attractor at infinity) coexist. In particular, the concept of the global (homoclinic and heteroclinic) bifurcations is extended to the latter region, where the arising flip saddle (instead of the direct saddle) is involved in the events. The possible forms of unpredictable outcomes, which arise in both regions of control parameters, are pointed out.