Bifurcation and number of subharmonic solutions of a 2n-dimensional non-autonomous system and its application

Quan, Tingting; Li, Jing; Zhang, Wei; Sun, Min

doi:10.1007/s11071-019-05192-2

Cited by 3 publications

(2 citation statements)

References 23 publications

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“…Define a global cross section Σ = s 1 , n 1 , s 2 , n 2 , t | t = 0 in the phase space 2 × 3 , and construct the m 0 -th iteration of Poincaré map P m 0 ∶ Σ → Σ as s 10 , n 10 , s 20 , n 20 → s 1 (m 0 T), n 1 (m 0 T), s 2 (m 0 T), n 2 (m 0 T) , where s i0 = s i (0), n i0 = n i (0) (i = 1, 2) are initial values. A fixed point of P m 0 corresponds to a periodic solution of system (11) T will be obtained as follows [32].…”

Section: Multiple Periodic Vibration Analysismentioning

confidence: 99%

Multiple Periodic Vibrations of Auxetic Honeycomb Sandwich Plate with 1:2 Internal Resonance

Zhu

Qiao

et al. 2022

J Nonlinear Math Phys

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In this paper, we focus on the multiple periodic vibration behaviors of an auxetic honeycomb sandwich plate subjected to in-plane and transverse excitations. Nonlinear equation of motion for the plate is derived based on the third-order shear deformation theory and von Kármán type nonlinear geometric assumptions. The Melnikov method is extended to detect the bifurcation and multiple periodic vibrations of the plate under 1:2 internal resonance. The effects of transverse excitation on nonlinear vibration behaviors are discussed in detail. Evolution laws and waveforms of multiple periodic vibrations are obtained to analyze the energy transfer process between the first two order modes. Even quite small transverse excitation can cause periodic vibration in the system, and there can be at most three periodic orbits in certain bifurcation regions. The periodic orbits are classified into two families by tracing their sources. The study provides the possibility for the classification study on generation mechanism of system complexity and energy transfers between different modes.

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Section: Multiple Periodic Vibration Analysismentioning

confidence: 99%

Multiple Periodic Vibrations of Auxetic Honeycomb Sandwich Plate with 1:2 Internal Resonance

Zhu

Qiao

et al. 2022

J Nonlinear Math Phys

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“…Meanwhile, the research on periodic solutions and bifurcations of high-dimensional complex nonlinear dynamic systems closely related to Hilbert's 16th problem is of great significance in solving practical scientific problems [26]. The nonlinear dynamics have been greatly permeated into various systems, such as the isochronous center system and Hamilton system from theoretical analysis and numerical simulation [27,28]. Barreira et al demonstrated upper bounds for the number of periodic solutions generated by the bifurcation of a 1:N resonance center system using average theory [29].…”

Section: Introductionmentioning

confidence: 99%

Periodic Solutions and Stability Analysis for Two-Coupled-Oscillator Structure in Optics of Chiral Molecules

Chen

Zhu

2022

Mathematics

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Chirality is an indispensable geometric property in the world that has become invariably interlocked with life. The main goal of this paper is to study the nonlinear dynamic behavior and periodic vibration characteristic of a two-coupled-oscillator model in the optics of chiral molecules. We systematically discuss the stability and local dynamic behavior of the system with two pairs of identical conjugate complex eigenvalues. In particular, the existence and number of periodic solutions are investigated by establishing the curvilinear coordinate and constructing a Poincaré map to improve the Melnikov function. Then, we verify the accuracy of the theoretical analysis by numerical simulations, and take a comprehensive look at the nonlinear response of multiple periodic motion under certain conditions. The results might be of important significance for the vibration control, safety stability and design optimization for chiral molecules.

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Bifurcation of Multiple Periodic Solutions for a Class of Nonlinear Dynamical Systems in $$(m+4)$$-Dimension

Quan,

Li,

Sun

et al. 2024

J Nonlinear Math Phys

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In this paper, we introduce a curvilinear coordinate transformation to study the bifurcation of periodic solutions from a 2-degree-of-freedom Hamiltonian system, when it is perturbed in $${\textbf{R}}^{m+4}$$ R m + 4 , where m represents any positive integer. The extended Melnikov function is obtained by constructing a Poincaré map on the curvilinear coordinate frame of the trajectory of the unperturbed system. Then the criteria for bifurcation of periodic solutions of these Hamiltonian systems under isochronous and non-isochronous conditions are obtained. As for its application, we study the number of periodic solutions of a composite piezoelectric cantilever rectangular plate system whose averaged equation can be transformed into a $$(2+4)$$ ( 2 + 4 ) -dimensional dynamical system. Furthermore, under the two resonance conditions of 1:1 and 1:2, we obtain the periodic solution numbers of this system with the variation of parametric excitation coefficient $$p_1.$$ p 1 .

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Bifurcation and number of subharmonic solutions of a 2n-dimensional non-autonomous system and its application

Cited by 3 publications

References 23 publications

Multiple Periodic Vibrations of Auxetic Honeycomb Sandwich Plate with 1:2 Internal Resonance

Multiple Periodic Vibrations of Auxetic Honeycomb Sandwich Plate with 1:2 Internal Resonance

Periodic Solutions and Stability Analysis for Two-Coupled-Oscillator Structure in Optics of Chiral Molecules

Bifurcation of Multiple Periodic Solutions for a Class of Nonlinear Dynamical Systems in $$(m+4)$$-Dimension

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