• Title/Summary/Keyword: Periodic Doubling

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OSCILLATORY THERMAL CONVECTION IN A HORIZONTAL ANNULUS (수평 환형 공간에서의 진동하는 열대류)

  • Yoo Joo-Sik
    • Journal of computational fluids engineering
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    • v.11 no.2 s.33
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    • pp.49-55
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    • 2006
  • This study investigates the oscillatory thermal convection of a fluid with Pr=0.02 in a wide-gap horizontal annulus with constant heat flux inner wall. When Pr=0.02, dual steady-state flows are not found. After the first Hopf bifurcation from a steady to a time-periodic flow, five successive period-doubling bifurcations are recorded before chaos. The power spectrum shows the $period-2^4\;and\;2^5$ flows clearly, and a window of period $3{\times}2^3$ flow is found in the chaotic regime. The approximate value of the Feigenbaum number for the last three period-doubling bifurcations is 4.76. The transition route to chaos of the present simulations is consistent with the period-doubling route of Feigenbaum.

Chaotic Thermal Convection in a Wide-Gap Horizontal Annulus : Pr=0.1 (넓은 수평 환형 공간에서의 혼동 열 대류 : Pr=0.1)

  • 유주식;엄용균
    • Korean Journal of Air-Conditioning and Refrigeration Engineering
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    • v.13 no.2
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    • pp.88-95
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    • 2001
  • Transition to chaotic convection is investigated for natural convection of a fluid with Pr=0.1 in a wide-gap horizontal annuls. The unsteady two-dimensional stream-function-vorticity equation is solved with finite difference method. As the Rayleigh number is increased, the steady 'downward flow' bifurcates to a time-periodic flow with a fundamental frequency, and afterwards a period-doubling bifurcation occurs. As the Rayleigh number is increased further, the chaotic flow regime is reached after a sequence of successive Hopf bifurcation to quasi-periodic and chaotic flow regimes. The route to chaos shows the Ruelle-Takens-Newhouse scenario. The flow of chaotic regime displays complex coalescence and separation of eddies in the side and lower region of the annulus.

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Period doubling of the nonlinear dynamical system of an electrostatically actuated micro-cantilever

  • Chen, Y.M.;Liu, J.K.
    • Smart Structures and Systems
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    • v.14 no.5
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    • pp.743-763
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    • 2014
  • The paper presents an investigation of the nonlinear dynamical system of an electrostatically actuated micro-cantilever by the incremental harmonic balance (IHB) method. An efficient approach is proposed to tackle the difficulty in expanding the nonlinear terms into truncated Fourier series. With the help of this approach, periodic and multi-periodic solutions are obtained by the IHB method. Numerical examples show that the IHB solutions, provided as many as harmonics are taken into account, are in excellent agreement with numerical results. In addition, an iterative algorithm is suggested to accurately determine period doubling bifurcation points. The route to chaos via period doublings starting from the period-1 or period-3 solution are analyzed according to the Floquet and the Feigenbaum theories.

DYNAMICS OF A DISCRETE RATIO-DEPENDENT PREDATOR-PREY SYSTEM INCORPORATING HARVESTING

  • BAEK, HUNKI;HA, JUNSOO;HYUN, DAGYEONG;LEE, SANGMIN;PARK, SUNGJIN;SUH, JEONGWOOK
    • East Asian mathematical journal
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    • v.31 no.5
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    • pp.743-751
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    • 2015
  • In this paper, we consider a discrete ratio-dependent predator-prey system with harvesting effect. In order to investigate dynamical behaviors of this system, first we find out all fixed points of the system and then classify their stabilities by using their Jacobian matrices and local stability method. Next, we display some numerical examples to substantiate theoretical results and finally, we show numerically, by means of a bifurcation diagram, that various dynamical behaviors including cycles, periodic doubling bifurcation and chaotic bands can be existed.

Dynamics and instability of the Karman wake mode induced by periodic forcing

  • Mureithi, Njuki W.
    • Wind and Structures
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    • v.7 no.4
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    • pp.265-280
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    • 2004
  • This paper presents some fundamental results on the dynamics of the periodic Karman wake behind a circular cylinder. The wake is treated like a dynamical system. External forcing is then introduced and its effect investigated. The main result obtained is the following. Perturbation of the wake, by controlled cylinder oscillations in the flow direction at a frequency equal to the Karman vortex shedding frequency, leads to instability of the Karman vortex structure. The resulting wake structure oscillates at half the original Karman vortex shedding frequency. For higher frequency excitation the primary pattern involves symmetry breaking of the initially shed symmetric vortex pairs. The Karman shedding phenomenon can be modeled by a nonlinear oscillator. The symmetrical flow perturbations resulting from the periodic cylinder excitation can also be similarly represented by a nonlinear oscillator. The oscillators represent two flow modes. By considering these two nonlinear oscillators, one having inline shedding symmetry and the other having the Karman wake spatio-temporal symmetry, the possible symmetries of subsequent flow perturbations resulting from the modal interaction are determined. A theoretical analysis based on symmetry (group) theory is presented. The analysis confirms the occurrence of a period-doubling instability, which is responsible for the frequency halving phenomenon observed in the experiments. Finally it is remarked that the present findings have important implications for vortex shedding control. Perturbations in the inflow direction introduce 'control' of the Karman wake by inducing a bifurcation which forces the transfer of energy to a lower frequency which is far from the original Karman frequency.

Analysis of Nonlinear Behavior in Love Model with External Force (외력을 가진 사랑 모델에서 비선형 거동 해석)

  • Huang, Lyni-Un;Bae, Young-Chul
    • The Journal of the Korea institute of electronic communication sciences
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    • v.10 no.7
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    • pp.845-850
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    • 2015
  • Love which is one of the emotional of mankind, has been studied in sociology and psychology as a matter of great concern. Through such a research, the researchers have provided the basic mathematical model for love model, we cannot find nonlinear characteristics through the basic love model. Therefore, in this paper, in order to find nonlinear behaviors in the basic love model, we apply external force to the basic love model. Then we confirm the existence of nonlinear behaviors through time series and phase portrait. We also confirm that this nonlinear behaviors have the periodic doubling, chaotic phenomena and periodic process which are very similar to typical chaotic occurrence phenomena.

The chaotic motion analysis by hardware implementation of Bonhoeffer Van der Pol oscillation model (Bonhoeffer Van der Pol 오실레이터 모델의 하드웨어 구현에 의한 카오스 운동 해석)

  • Bae, Yeong-Cheol;Seo, Sam-Mun;Im, Hwa-Yeong
    • The Transactions of the Korea Information Processing Society
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    • v.3 no.4
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    • pp.877-882
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    • 1996
  • The effects of periodic and chaotic behaviour in the Bonhoeffer-Van der Pol (BVP) oscillation of the nerve membrane driven by a periodic stimulating current A1 coswtare investigated through hardware implementation.For hardware implementation of the BVP model. real element values were escaled with computer simulation results to determine the parameter real value.As the parameter A1 varied in the range 0 to 1.3, the BVP model showed an ordinary and reversed period-doubling cascade and a chaotic state. At the low driving amplitude ofa1 the period-doubling showed and at the high driving amplitude of A1 the chaotic state occured. To analyse the BVP model for chaotic behaviour Phase Plane, Time series are used to verify that properties.

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Bifurcation to Chaotic Thermal Convection in a Horizontal Annulus (수평 환형 공간에서의 혼돈 열대류로의 분기)

  • Yoo, Joo-Sik;Kim, Yong-Jin
    • Transactions of the Korean Society of Mechanical Engineers B
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    • v.24 no.9
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    • pp.1210-1218
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    • 2000
  • Thermal convection in a horizontal annulus is considered, and the bifurcation phenomena of flows from time-periodic to chaotic convection are numerically investigated. The unsteady two-dimensional streamfunction-vorticity equation is solved with finite difference method. As Rayleigh number is increased, the steady flow bifurcates to a time-periodic flow with a fundamental frequency, and afterwards a period-tripling bifurcation occurs with further increase of the Rayleigh number. Chaotic convection is established after a period-doubling bifurcation. A periodic convection with period 4 appears after the first chaotic convection. At still higher Rayleigh numbers, chaotic flows reappear.

On the Chaotic Vibrations of Thin Beams by a Bifurcation Mode (분기 모우드를 활용한 얇은 빔의 혼돈 역학에 관한 연구)

  • 이영섭;주재만;박철희
    • Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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    • 1995.04a
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    • pp.121-128
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    • 1995
  • The results are summarized as what follows: 1) The modeling of thin beams, which is a continuous system, into a two DOF system yields satisfactory results for the chaotic vibrations. 2) The concept of "natural forcing function" derived from the eigenfunction of the bifurcation mode is very useful for the natural responses of the system. 3) Among the perturbation techniques, HBM is a good estimate for the response when the geometry of motion is known. 4) It is known that there exist periodic solutions of coupled mode response for somewhat large damping and forcing amplitude, as well as weak damping and forcing. 5) The route-to-chaos related with lateral instability in thin beams is composed of period-doubling and quasiperiodic process and finally follows discontinuous period-doubling process. 6) The chaotic vibrations are verified by using Poincare maps, FFT's, time responses, trajectories in the configuration space, and the very powerful technique Lyapunov characteristics exponents.exponents.

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Nonlinear Torsional Oscillations of a System incorporating a Hooke's Joint : 2-DOF Model (훅조인트로 연결된 축계의 비선형 비틀림 진동의 분기해석 :2-자유도계 모델)

  • 장서일
    • Transactions of the Korean Society for Noise and Vibration Engineering
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    • v.13 no.4
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    • pp.317-322
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    • 2003
  • Torsional oscillations of a system incorporating a Hooke's joint are investigated by adopting a nonlinear 2-degree-of-freedom model. Linear and Van der Pol transformations are applied to obtain the equations of motion to which the method of averaging can be readily applied. Various subharmonic and combination resonances are identified with the conditions of their occurrences. Applying the method of averaging leads to the reduced amplitude- and phase-equations of motion, of which constant and periodic solutions are obtained numerically. The periodic solution which emerges from Hopf bifurcation point experiences period doubling bifurcation leading to infinite solution rather than chaotic solution.