Nonlinear interactions of tearing modes in the vicinity of a bifurcation point of codimension two

In this paper, we study effects of the secondary bifurcations in the neighborhood of the primary codimension-two bifurcation point. The twin-well potential Duffing oscillator is considered and the investigations are focused on the new scenario of destruction of the cross-well chaotic attractor. The phenomenon belongs to the category of the subduction scenario and relies on the replacement of the cross-well chaotic attractor by a pair of unsymmetric 2T-periodic attractors. The exploration of a sequence of accompanying bifurcations throws more light on the complex phenomena that may occur in the neighborhood of the primary codimension-two bifurcation point. It shows that in the close vicinity of the point there appears a transition zone in the system parameter plane, the zone which separates the two so-far investigated scenarios of annihilation of the cross-well chaotic attractor.

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Explanation of the onset of bouncing cycles in isotropic rotor dynamics; a grazing bifurcation analysis

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2019.0549 ◽

2020 ◽

Vol 476 (2237) ◽

pp. 20190549

Author(s):

Karin Mora ◽

Alan R. Champneys ◽

Alexander D. Shaw ◽

Michael I. Friswell

Keyword(s):

Bifurcation Analysis ◽

Bifurcation Point ◽

Internal Resonance ◽

Rotor Dynamics ◽

Rotor Speed ◽

Grazing Bifurcation ◽

Codimension Two ◽

Partial Contact ◽

Unbalanced Rotor ◽

Fold Bifurcation

The dynamics associated with bouncing-type partial contact cycles are considered for a 2 degree-of-freedom unbalanced rotor in the rigid-stator limit. Specifically, analytical explanation is provided for a previously proposed criterion for the onset upon increasing the rotor speed Ω of single-bounce-per-period periodic motion, namely internal resonance between forward and backward whirling modes. Focusing on the cases of 2 : 1 and 3 : 2 resonances, detailed numerical results for small rotor damping reveal that stable bouncing periodic orbits, which coexist with non-contacting motion, arise just beyond the resonance speed Ω p : q . The theory of discontinuity maps is used to analyse the problem as a codimension-two degenerate grazing bifurcation in the limit of zero rotor damping and Ω = Ω p : q . An analytic unfolding of the map explains all the features of the bouncing orbits locally. In particular, for non-zero damping ζ , stable bouncing motion bifurcates in the direction of increasing Ω speed in a smooth fold bifurcation point that is at rotor speed O ( ζ ) beyond Ω p : q . The results provide the first analytic explanation of partial-contact bouncing orbits and has implications for prediction and avoidance of unwanted machine vibrations in a number of different industrial settings.

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Spatiotemporal Dynamics in Reaction–Diffusion Neural Networks Near a Turing–Hopf Bifurcation Point

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501542 ◽

2019 ◽

Vol 29 (11) ◽

pp. 1950154 ◽

Cited By ~ 1

Author(s):

Jiazhe Lin ◽

Rui Xu ◽

Xiaohong Tian

Keyword(s):

Neural Networks ◽

Hopf Bifurcation ◽

Normal Form ◽

Bifurcation Point ◽

Reaction Diffusion ◽

Spatiotemporal Dynamics ◽

Normal Form Theory ◽

Hopf Bifurcation Point ◽

Codimension Two ◽

Form Theory

Since the electromagnetic field of neural networks is heterogeneous, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate the existence of Turing–Hopf bifurcation in a reaction–diffusion neural network. By the normal form theory for partial differential equations, we calculate the normal form on the center manifold associated with codimension-two Turing–Hopf bifurcation, which helps us understand and classify the spatiotemporal dynamics close to the Turing–Hopf bifurcation point. Numerical simulations show that the spatiotemporal dynamics in the neighborhood of the bifurcation point can be divided into six cases and spatially inhomogeneous periodic solution appears in one of them.

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Dynamic Behaviors of a Two-Degree-of-Freedom Impact Oscillator with Two-Sided Constraints

Shock and Vibration ◽

10.1155/2021/8854115 ◽

2021 ◽

Vol 2021 ◽

pp. 1-14

Author(s):

Songtao Li ◽

Qunhong Li ◽

Zhongchuan Meng

Keyword(s):

Bifurcation Point ◽

Period Doubling ◽

Harmonic Excitation ◽

Impact Oscillator ◽

Grazing Bifurcation ◽

Constraint System ◽

Codimension Two ◽

Codimension One ◽

Discontinuity Mapping ◽

Elastic Constraints

The dynamic model of a vibroimpact system subjected to harmonic excitation with symmetric elastic constraints is investigated with analytical and numerical methods. The codimension-one bifurcation diagrams with respect to frequency of the excitation are obtained by means of the continuation technique, and the different types of bifurcations are detected, such as grazing bifurcation, saddle-node bifurcation, and period-doubling bifurcation, which predicts the complexity of the system considered. Based on the grazing phenomenon obtained, the zero-time-discontinuity mapping is extended from the single constraint system presented in the literature to the two-sided elastic constraint system discussed in this paper. The Poincare mapping of double grazing periodic motion is derived, and this compound mapping is applied to obtain the existence conditions of codimension-two grazing bifurcation point of the system. According to the deduced theoretical result, the grazing curve and the codimension-two grazing bifurcation points are validated by numerical simulation. Finally, various types of periodic-impact motions near the codimension-two grazing bifurcation point are illustrated through the unfolding diagram and phase diagrams.

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