Applied Mathematics and Mechanics (English Edition) ›› 2026, Vol. 47 ›› Issue (3): 509-534.doi: https://doi.org/10.1007/s10483-026-3362-8
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Xinyi HUANG1,2, S. LENCI2, Qingjie CAO1,†(
)
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RSSReceived:2025-10-20
Revised:2026-01-12
Published:2026-03-02
Contact:
Qingjie CAO, E-mail: qingjiecao@hotmail.comSupported by:2010 MSC Number:
Xinyi HUANG, S. LENCI, Qingjie CAO. Singular closed orbits and chaotic behavior of a double-winged quasi-zero-stiffness system. Applied Mathematics and Mechanics (English Edition), 2026, 47(3): 509-534.
Fig. 1
Summary of singular closed orbits and chaotic mechanism of a double-winged QZS system (color online)"
Fig. 2
Dynamical model of a double-winged QZS system"
Fig. 3
Bifurcation diagram of the system (5), where the colored surface is the constraint surface cosθ+cosφ=β, solid lines represent stable equilibria, and dashed lines represent unstable equilibria (color online)"
Fig. 4
Phase portraits of the system (5) on the surface cosθ+cosφ=β: (a) κ=0.1, β=0.6; (b) κ=0.1, β=0.7; (c) κ=0.1, β=1.3; (d) κ=0, β=0.6; (e) κ=0, β=0.7; (f) κ=0, β=1.3 (color online)"
Fig. 5
(a) Bifurcation diagram of the system (15), where solid lines represent stable equilibria, and dashed lines represent unstable equilibria; (b)–(h) phase portraits of the system (15) for β=0.55, βc, 0.74, βh, 0.9, βP, 1.6, respectively, where black lines represent singular closed orbits including homoclinic (homoclinic-like) and heteroclinic (heteroclinic-like) orbits, red lines represent local small periodic orbits, green lines represent small periodic orbits surrounding singular closed orbits, blue lines represent global large periodic orbits, and green dots represent equilibria (color online)"
Fig. 6
Five types of singular closed orbits: (a) discontinuous homoclinic orbit Γ1; (b) homoclinic-like orbit Γe; (c) heteroclinic-like orbit Γh; (d) homoclinic orbit Γc; (e) heteroclinic-like orbit Γg (color online)"
Fig. 7
(a) Chaotic thresholds in the whole parameter domain (β, ω) for r0=2 and ζ=0.2; (b) chaotic thresholds as β varies for ω=1, r0=2, and ζ=0.2 (color online)"
Fig. 8
(a) Chaotic thresholds for β=0.72, r0=2, and ζ=0.2; (b) bifurcation diagram for x (on the Poincaré section) versus f0 when β=0.72, r0=2, ζ=0.2, and ω=1, with threshold values R1(0.72, 0.2, 2, 1)=0.151 and Re(0.72, 0.2, 1)=0.148; (c) bifurcation diagram in detail for local chaos (color online)"
Fig. 9
Basin boundaries of attractors for varying f0 when β=0.72, r0=2, ζ=0.2, and ω=1 (the colored basins correspond to branches with the same colors in the bifurcation diagram): (a) f0=0.14; (b) f0=0.2; (c) f0=0.35; (d) f0=0.5; (e) f0=0.8; (f) f0=1; (g) f0=1.3; (h) f0=2.2; (i) details of Fig. 9(c); (j) details of Fig. 9(i) (color online)"
Fig. 10
Coexisting solutions for β=0.72, r0=2, ζ=0.2, and ω=1 (the colored curves correspond to branches with the same colors in the bifurcation diagram): (a) a pair of smooth period-1 orbits for f0=0.2; (b) a pair of local chaotic attractors for f0=0.5 with the largest Lyapunov exponent 0.019 3; (c) a pair of discontinuous period-1 and period-3 orbits for f0=1.5; (d) a pair of discontinuous period-1 orbits for f0=12; (e) chaotic attractor for f0=15 with the largest Lyapunov exponent 0.132 3; (f) a pair of discontinuous period-1 orbits for f0=18 (color online)"
Fig. 11
Coexisting solutions for β=0.72, r0=2, ζ=0.2, and ω=1 (the colored curves correspond to branches with the same colors in the bifurcation diagram): (a) chaotic attractor for f0=20.5 with the largest Lyapunov exponent 0.105 2; (b) a pair of discontinuous period-2 orbits for f0=22; (c) chaotic attractor for f0=26 with the largest Lyapunov exponent 0.120 6; (d) discontinuous period-3 orbit for f0=28; (e) chaotic attractor for f0=30 with the largest Lyapunov exponent 0.110 2; (f) a pair of discontinuous period-1 orbits for f0=32 (color online)"
Fig. 12
(a) Chaotic thresholds for β=1, r0=2, and ζ=0.2; (b) bifurcation diagram for x (on the Poincaré section) versus f0 when β=1, r0=2, ζ=0.2, and ω=1, with threshold values Rc(1, 0.2, 1)=0.329 and Rg(1, 0.2, 1)=1.041 (color online)"
Fig. 13
Basin boundaries of attractors for varying f0 when β=1, r0=2, ζ=0.2, and ω=1 (the colored basins correspond to branches with the same colors in the bifurcation diagram): (a) f0=0.1; (b) f0=0.2; (c) f0=0.3; (d) f0=0.4 (color online)"
Fig. 14
Coexisting solutions for β=1, r0=2, ζ=0.2, and ω=1 (the colored curves correspond to branches with the same colors in the bifurcation diagram): (a) a pair of smooth period-2 orbits and a smooth period-5 orbits for f0=0.5; (b) a chaotic saddle converging to a smooth period-3 orbits for f0=1.1; (c) a discontinuous period-1 orbit and a pair of smooth period-3 orbits for f0=1.7; (d) a discontinuous period-1 orbits and a chaotic attractor for f0=2 with the largest Lyapunov exponent 0.130 3; (e) a discontinuous period-1 orbit and a period-11 orbits (in the periodic window) for f0=2.07; (f) a discontinuous period-1 orbit and a chaotic attractor for f0=2.8 with the largest Lyapunov exponent 0.199 6; (g) a discontinuous period-1 orbit and a pair of smooth period-2 orbits for f0=4 (color online)"
Fig. 15
(a) Chaotic thresholds for β=1.45, r0=2, and ζ=0.2; (b) bifurcation diagram for x (on the Poincaré section) versus f0 when β=1.45, r0=2, ζ=0.2, and ω=1, with the threshold value Rg(1.45, 0.2, 1)=3.564 (color online)"
Fig. 16
Basin boundaries of attractors for varying f0 when β=1.45, r0=2, ζ=0.2, and ω=1 (the colored basins correspond to branches with the same colors in the bifurcation diagram): (a) f0=1.4; (b) f0=2; (c) f0=3; (d) f0=4 (color online)"
Fig. 17
Coexisting solutions for β=1.45, r0=2, ζ=0.2, and ω=1 (the colored curves correspond to branches with the same colors in the bifurcation diagram): (a) a discontinuous period-1 orbit, a smooth period-1 orbit, and a smooth period-3 orbit for f0=4; (b) a discontinuous period-1 orbit, a smooth period-1 orbit, and a discontinuous period-3 orbit for f0=8.5; (c) a pair of discontinuous period-1 orbit and a pair of chaotic attractors for f0=16.5; (d) chaotic attractor for f0=19 with the largest Lyapunov exponent 0.190 4; (e) a discontinuous period-3 orbit (in the periodic window) for f0=20.1; (f) a pair of chaotic attractors for f0=21.1 with the largest Lyapunov exponent 0.064 5; (g) chaotic attractor for f0=22 with the largest Lyapunov exponent 0.171 4; (h) a pair of discontinuous period-1 orbits for f0=25; (i) chaotic attractor for f0=30 with the largest Lyapunov exponent 0.302 8 (color online)"
Fig. 18
Chaos and resonant behavior for β=0.8, r0=2, f0=1, ω=1, and ε=0.1: (a) Poincaré section; (b) a pair of resonant orbits with the winding number (1:1); (c) resonant orbit with the winding number (9:9) (color online)"
Fig. 19
Poincaré sections for coexistence of chaotic attractors, chaotic sea, periodic attractors, quasi-periodic islands and resonant points, when f0=1, ω=1, and ε=0.1: (a) β=1.5, r0=2; (b) β=1.8, r0=2; (c) β=βP, r0=0; (d) detailed structure for Fig. 19(a); (e) and (f) detailed structures for Fig. 19(c) (color online)"
Fig. 20
Periodic and resonant orbits of the system (56) for β=1.5, r0=2, f0=1, ω=1, and ε=0.1: (a) discontinuous periodic orbits with a pair of period-2 orbits and a single period-1 orbit; (b) resonant orbits with the winding numbers (1:1), (3:1), and (7:7); (c) resonant orbits with the winding numbers (5:3) and (13:5); (d) resonant orbit with the winding number (21:7); (e) a pair of resonant orbits with the winding number (4:2); (f) resonant orbit with the winding number (11:5); (g) resonant orbit with the winding number (27:9); (h) a pair of resonant orbits with the winding number (12:4) (color online)"
Fig. 21
SD resonant orbits of the system (56) for β=βP, r0=0, f0=1, ω=1, and ε=0.1: (a) resonant orbits with the winding numbers (1:1) and (9:9); (b) resonant orbits with the winding numbers (7:7) and (5:3); (c) resonant orbits with the winding numbers (3:1) and (15:5); (d) a pair of resonant orbits with the winding number (9:3); (e) resonant orbits with the winding numbers (5:2) and (5:3); (f) resonant orbit with the winding number (11:5); (g) discontinuous resonant orbits with the winding numbers (1:1) and (5:3); (h) a pair of discontinuous resonant orbits with the winding number (5:5) (color online)"
| [1] | CAO, Q., WIERCIGROCH, M., PAVLOVSKAIA, E., GREBOGI, C., and THOMPSON, M. Archetypal oscillator for smooth and discontinuous dynamics. Physical Review E, 74(4), 046218 (2006) |
| [2] | HAN, N. and CAO, Q. Rotating pendulum with smooth and discontinuous dynamics. International Journal of Mechanical Sciences, 127, 91–102 (2017) |
| [3] | HUANG, X. and CAO, Q. The heteroclinic and codimension-4 bifurcations of a triple SD oscillator. Nonlinear Dynamics, 112(7), 5053–5075 (2024) |
| [4] | BUDD, C. and DUX, F. Chattering and related behaviour in impact oscillators. Philosophical Transactions of the Royal Society of London, Series A: Physical and Engineering Sciences, 347, 365–389 (1994) |
| [5] | FAN, J., LI, C., YANG, Z., CHEN, S., CAO, J., and DOU, C. On discontinuous dynamics of a 2-DOF oscillator with an one-sided rigid obstacle. International Journal of Non-Linear Mechanics, 118, 103261 (2020) |
| [6] | DOU, C., FAN, J., and GAO, M. On discontinuous dynamics of a 2-DOF system with bilateral rigid constraints and nonlinear friction. Nonlinear Dynamics, 104(2), 1175–1204 (2021) |
| [7] | DUAN, J., WEI, Z., LI, G., LI, D., and GREBOGI, C. Strange nonchaotic attractors in a class of quasiperiodically forced piecewise smooth systems. Nonlinear Dynamics, 112(14), 12565–12577 (2024) |
| [8] | PENG, R., LI, Q., and ZHANG, W. Chaos analysis of SD oscillator with two-frequency excitation. Nonlinear Dynamics, 112(9), 7649–7677 (2024) |
| [9] | KOU, L., LI, X., and LI, S. Bifurcations of homoclinic torus in a quasiperiodically forced SD oscillator. International Journal of Dynamics and Control, 13(2), 46 (2025) |
| [10] | LI, J., SUN, R., and ZHU, S. Chaotic characteristics of a quadruple-well triple SD oscillator with multiple homoclinic orbits. Nonlinear Dynamics, 113(11), 13807–13835 (2025) |
| [11] | HAN, N., LU, P., and LI, Z. An approximate technique to test chaotic region in a rotating pendulum system with bistable characteristics. Nonlinear Dynamics, 104(1), 191–214 (2021) |
| [12] | BANDI, D. and TAMADAPU, G. Hidden dynamics of a self-excited SD oscillator. Nonlinear Dynamics, 113(2), 1063–1082 (2025) |
| [13] | CAO, Q., XIONG, Y., and WIERCIGROCH, M. Resonances of the SD oscillator due to the discontinuous phase. Journal of Applied Analysis and Computation, 1(2), 183–191 (2011) |
| [14] | YUE, X., XU, W., and WANG, L. Stochastic bifurcations in the SD (smooth and discontinuous) oscillator under bounded noise excitation. Science China Physics, Mechanics and Astronomy, 56(5), 1010–1016 (2013) |
| [15] | WANG, Z. and SHANG, H. Multistability and jump in the harmonically excited SD oscillator. Fractal and Fractional, 7(4), 314 (2023) |
| [16] | YANG, Q., ZHOU, B., LI, D., and LI, Y. Multi-attractor coexistence and switching in a smooth and discontinuous oscillator under quasiperiodic excitation. Nonlinear Dynamics, 113, 21845–21861 (2025) |
| [17] | CHEN, Y. and LEUNG, A. Bifurcation and Chaos in Engineering, Springer Science & Business Media, London (1998) |
| [18] | CHEN, H. Global analysis on the discontinuous limit case of a smooth oscillator. International Journal of Bifurcation and Chaos, 26(4), 1650061 (2016) |
| [19] | CHEN, H., LLIBRE, J., and TANG, Y. Global dynamics of a SD oscillator. Nonlinear Dynamics, 91(3), 1755–1777 (2018) |
| [20] | BAHNASY, T., AMER, T., ALMAHALAWY, A., ABOHAMER, M., ABOSHEIAHA, H., and ELAMEER, A. A chaotic behavior and stability analysis on quasi-zero stiffness vibration isolators with multi-control methodologies. Journal of Low Frequency Noise, Vibration and Active Control, 44(2), 1094–1116 (2025) |
| [21] | BANERJEE, P., BALAJI, P., and MURUGAN, S. A dual-function compliant metastructure based on quasi-zero-stiffness for combined vibration isolation and vibration energy harvesting. International Journal of Dynamics and Control, 13(4), 135 (2025) |
| [22] | HU, Y., ZHANG, H., WANG, K., FANG, Y., and MA, C. Analytical analysis of vibration isolation characteristics of quasi-zero stiffness suspension backpack. International Journal of Dynamics and Control, 12, 4387–4397 (2024) |
| [23] | YE, T., NGUYEN, V., and LI, S. Enhancing the vibratory roller’s ride comfort with semi-active seat suspension embedded by quasi-zero stiffness structure. International Journal of Dynamics and Control, 11, 2069–2081 (2023) |
| [24] | ALABUZHEV, P. and RIVIN, E. Vibration Protecting and Measuring Systems with Quasi-Zero Stiffness, Hemisphere Publishing, New York (1989) |
| [25] | BRENNAN, M., ELLIOTT, S., BONELLO, P., and VINCENT, J. The “click" mechanism in dipteran flight: if it exists, then what effect does it have? Journal of Theoretical Biology, 224(2), 205–213 (2003) |
| [26] | ZENG, R., WEN, G., ZHOU, J., and ZHAO, G. Limb-inspired bionic quasi-zero stiffness vibration isolator. Acta Mechanica Sinica, 37(7), 1152–1167 (2021) |
| [27] | HAN, H., SOROKIN, V., TANG, L., and CAO, D. A nonlinear vibration isolator with quasi-zero-stiffness inspired by Miura-origami tube. Nonlinear Dynamics, 105(2), 1313–1325 (2021) |
| [28] | CHEN, Z., CHEN, Z., and WEI, Y. Quasi-zero stiffness-based synchronous vibration isolation and energy harvesting: a comprehensive review. Energies, 15(19), 7066 (2022) |
| [29] | HUANG, X. and CAO, Q. Complicated dynamical behaviors of a geometrical oscillator with a mass parameter. International Journal of Bifurcation and Chaos, 33(12), 2330031 (2023) |
| [30] | STUART, A. and FLOATER, M. On the computation of blow-up. European Journal of Applied Mathematics, 1(1), 47–71 (1990) |
| [31] | GALAKTIONOV, V. and VAZQUEZ, J. The problem of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 8(2), 399–434 (2002) |
| [32] | CHO, C. On the computation of the numerical blow-up time. Japan Journal of Industrial and Applied Mathematics, 30(2), 331–349 (2013) |
| [33] | POLYANIN, A. and SHINGAREVA, I. Nonlinear problems with blow-up solutions: numerical integration based on differential and nonlocal transformations, and differential constraints. Applied Mathematics and Computation, 336, 107–137 (2018) |
| [34] | MERLE, F., RAPHAEL, P., RODNIANSKI, I., and SZEFTEL, J. On blow up for the energy super critical defocusing nonlinear Schrodinger equations. Inventiones Mathematicae, 227(1), 247–413 (2022) |
| [35] | FUJIE, K. and SENBA, T. Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions. Nonlinear Analysis, 222, 112987 (2022) |
| [36] | JI, J., LUO, Q., and YE, K. Vibration control based metamaterials and origami structures: a state-of-the-art review. Mechanical Systems and Signal Processing, 161, 107945 (2021) |
| [37] | ZHANG, Z., TIAN, R., ZHANG, X., WEI, F., and YANG, X. A novel butterfly-shaped auxetic structure with negative Poisson’s ratio and enhanced stiffness. Journal of Materials Science, 56(25), 14139–14156 (2021) |
| [38] | RAO, E. and FU, T. Acoustic radiation response of functionally graded sandwich plates cored by butterfly-shaped honeycombs with negative Poisson’s ratio. Journal of Mechanical Science and Technology, 38(5), 2321–2333 (2024) |
| [39] | WEI, F., TIAN, R., ZHANG, X., HAO, H., and WANG, M. Geometric nonlinear analysis of dielectric layer based on concave paper-cut structure with zero Poisson’s ratio. Smart Materials and Structures, 32(8), 085031 (2023) |
| [40] | TIAN, R., GUAN, H., LU, X., ZHANG, X., HAO, H. FENG, W., and ZHANG, G. Dynamic crushing behavior and energy absorption of hybrid auxetic metamaterial inspired by Islamic motif art. Applied Mathematics and Mechanics (English Edition), 44(3), 345–362 (2023) https://doi.org/10.1007/s10483-023-2962-9 |
| [41] | LU, Y., LI, H., SHEN, C., ZHOU, Y., YAO, Y., and SUN, G. Novel subcell-augmented re-entrant honeycomb with simultaneous stiffness and auxeticity enhancement. Thin-Walled Structures, 215, 113517 (2025) |
| [42] | CHEN, T., PAULY, M., and REIS, P. A reprogrammable mechanical metamaterial with stable memory. nature, 589(7842), 386–390 (2021) |
| [43] | HUANG, X. and CAO, Q. A novel nonlinear oscillator consisting torsional springs and rigid rods. International Journal of Non-Linear Mechanics, 161, 104684 (2024) |
| [44] | HUANG, X., CAO, Q., and LENCI, S. The complicated behaviours of a novel model of smooth and discontinuous dynamics with quasi-zero-stiffness property. Nonlinear Dynamics, 112, 20879–20902 (2024) |
| [45] | XU, W., FENG, J., and RONG, H. Melnikov’s method for a general nonlinear vibro-impact oscillator. Nonlinear Analysis: Theory, Methods & Applications, 71(1-2), 418–426 (2009) |
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