Period-bubbling transition to chaos in thermo-viscoelastic fluid systems. (English) Zbl 1446.34065

Summary: We report a 6D nonlinear dynamical system for thermo-viscoelastic fluid by selecting higher modes of infinite Fourier series of flow quantities. This nonlinear system demonstrates overstable convective motion and some organized structures such as period-bubbling and Arnold tongue-like structures. Studies reveal that the stability of the conduction state does not alter for the new 6D system in comparison with the lowest order 4D system of R. E. Khayat [“Fluid elasticity and the transition to chaos in thermal convection”, Phys. Rev. E51, 380–399 (1995; doi.org/10.1103/PhysRevE.51.380)]. However, the stabilities of the convective state have some differences. The onset of unsteady convection in the 6D system is delayed for weak elasticity of the fluid. There exists a critical range of fluid elasticity where the 4D system exhibits subcritical Hopf bifurcation while the 6D system shows supercritical Hopf bifurcation, which ensures the increase of the domain of stability. In this range, catastrophic route to chaos occurs in the 4D system, whereas the 6D system exhibits intermittent onset of chaos. Comparing the two-parameter dependent dynamics for the two systems, the chaotic zones enclosed by periodic regions are suppressed in the 6D system, so the flow behaviors become more predictable. Owing to interacting thermal buoyancy and fluid elasticity, both the models exhibit period-bubbling transition to chaos, but the period-bubbling cascade in the 6D model occurs at lower Rayleigh number than the 4D model. The convergence rate of the period-bubbling process slows down compared to usual period-doubling and approaches the square root of the Feigenbaum constant asymptotically.


34C60 Qualitative investigation and simulation of ordinary differential equation models
76A10 Viscoelastic fluids
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
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[1] Bogdanov, R. I. [1975] “ Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues,” Funct. Anal. Appl.9, 144-145. · Zbl 0447.58009
[2] Brand, H. R. & Zielinska, B. J. A. [1986] “ Tricritical codimension-two point near the onset of convection in viscoelastic liquids,” Phys. Rev. Lett.57, 3167-3170.
[3] Carr, J. [1981] Applications of Center Manifold Theory (Springer-Verlag, NY). · Zbl 0464.58001
[4] Curry, J. H. [1978] “ A generalized Lorenz system,” Commun. Math. Phys.60, 193-204. · Zbl 0387.76052
[5] Dawson, S. P., Grebogi, C., Yorke, J. A., Kan, I. & Koçak, H. [1992] “ Antimonotonicity: Inevitable reversals of period-doubling cascades,” Phys. Lett. A162, 249-254.
[6] Eckmann, J. P. & Ruelle, D. [1985] “ Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys.57, 617-656. · Zbl 0989.37516
[7] Feigenbaum, M. J. [1978] “ Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys.19, 25-52. · Zbl 0509.58037
[8] Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, NY). · Zbl 0515.34001
[9] Kengne, J., Negou, A. N. & Tchiotsop, D. [2017] “ Antimonotonicity, chaos and multiple attractors in a novel autonomous memristor-based jerk circuit,” Nonlin. Dyn.88, 2589-2608.
[10] Khayat, R. E. [1995] “ Fluid elasticity and the transition to chaos in thermal convection,” Phys. Rev. E51, 380-399.
[11] Kocarev, L., Halle, K. S., Eckert, K. & Chua, L. O. [1993] “ Experimental observation of antimonotonicity in Chua’s circuit,” Int. J. Bifurcation and Chaos3, 1051-1055. · Zbl 0894.58061
[12] Kyprianidis, I. M., Stouboulos, I. N., Haralabidis, P. & Bountis, T. [2000] “ Antimonotonicity and chaotic dynamics in a fourth-order autonomous nonlinear electric circuit,” Int. J. Bifurcation and Chaos10, 1903-1915.
[13] Layek, G. C. [2015] An Introduction to Dynamical Systems and Chaos (Springer, India). · Zbl 1354.34001
[14] Layek, G. C. & Pati, N. C. [2019] “ Organized structures of two bidirectionally coupled logistic maps,” Chaos29, 093104. · Zbl 1423.37048
[15] Lepers, C., Legrand, J. & Glorieux, P. [1991] “ Experimental investigation of the collision of Feigenbaum cascades in lasers,” Phys. Rev. A43, 2573-2575.
[16] Lorenz, E. N. [1963] “ Deterministic nonperiodic flow,” J. Atmos. Sci.20, 130-141. · Zbl 1417.37129
[17] Oppo, G. L. & Politi, A. [1984] “ Collision of Feigenbaum cascades,” Phys. Rev. A30, 435-441.
[18] Parlitz, U. & Lauterborn, W. [1985] “ Superstructure in the bifurcation set of the Duffing equation \(ẍ+dẋ+x+ x^3=f\cos(\omegat)\),” Phys. Lett. A107, 351-355.
[19] Pomeau, Y. & Manneville, P. [1980] “ Intermittent transition to turbulence in dissipative dynamical systems,” Commun. Math. Phys.74, 189-197.
[20] Shen, B. W. [2014] “ Nonlinear feedback in a five-dimensional Lorenz model,” J. Atmos. Sci.71, 1701-1723.
[21] Shen, B. W. [2015] “ Nonlinear feedback in a six-dimensional Lorenz model: Impact of an additional heating term,” Nonlin. Process. Geophys.22, 749-764.
[22] Shen, B. W. [2019] “ Aggregated negative feedback in a generalized Lorenz model,” Int. J. Bifurcation and Chaos29, 1950037. · Zbl 1414.34011
[23] Takens, F. [1974] “ Singularities of vector fields,” Publ. Math. IHES43, 47-100. · Zbl 0279.58009
[24] Vandermeer, J. [1997] “ Period ‘bubbling’ in simple ecological models: Pattern and chaos formation in a quartic model,” Ecol. Model.95, 311-317.
[25] Vest, C. M. & Arpaci, V. S. [1969] “ Overstability of a viscoelastic fluid layer heated from below,” J. Fluid Mech.36, 613-623. · Zbl 0175.24201
[26] Zhang, S., Zeng, Y., Li, Z. & Zhou, C. [2018] “ Hidden extreme multistability, antimonotonicity and offset boosting control in a novel fractional-order hyperchaotic system without equilibrium,” Int. J. Bifurcation and Chaos28, 1850167-1-18. · Zbl 1406.34047
[27] Zhou, L., Wang, C., Zhang, X. & Yao, W. [2018] “ Various attractors, coexisting attractors and antimonotonicity in a simple fourth-order memristive twin-T oscillator,” Int. J. Bifurcation and Chaos28, 1850050-1-18. · Zbl 1391.34084
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