Stability and Bifurcation Analysis of a Forced Cylinder Wake

2005 ◽  
Author(s):  
N. W. Mureithi ◽  
M. Rodriguez
Keyword(s):  
Bifurcation Analysis ◽  
Interaction Mechanism ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Mode Interaction ◽  
Cylinder Wake ◽  
Modal Interaction ◽  
Amplitude Equations ◽  
Vortex Merging ◽  
Interaction Amplitude

We present a study on the dynamics of a cylinder wake subjected to forced excitation. Williams et al. (1992) discovered that the spatial symmetry of the excitation plays a crucial role in determining the resulting wake dynamics. Reflection-symmetric forcing was found to affect the Karman wake much more strongly compared to Z2(κ, π) asymmetric forcing. For low forcing amplitudes, the existence of a nonlinear mode interaction mechanism was postulated to explain the observed “beating” phenomenon observed in the wake. Previous work by the authors (Mureithi et al. 2002, 2003) presented general forms of the modal interaction amplitude equations governing the dynamics of the periodically forced wake. In the present work, numerical CFD computations of the forced cylinder wake are presented. It is shown that the experimentally observed wake bifurcations can be reproduced by numerical simulations with reasonable accuracy. The CFD computations show that the forced wake first looses reflection symmetry followed by a bifurcation associated with vortex merging as the forcing amplitude is increased. A bifurcation analysis of a simplified amplitude equation shows that these two transitions are due to a pitchfork bifurcation and a period-doubling bifurcation of mixed mode solutions.

Dynamics of a Cylinder Wake Under Controlled Excitation

2003 ◽  
Author(s):  
Njuki W. Mureithi ◽  
Syuki Goda ◽  
Tomomichi Nakamura
Keyword(s):  
Frequency Ratio ◽  
Period Doubling ◽  
Experimental Tests ◽  
The Other ◽  
Mode Interaction ◽  
Theoretic Approach ◽  
Cylinder Wake ◽  
Strong Response ◽  
Asymmetric Mode ◽  
Other Hand

This paper presents some results of experimental tests as well as a group theoretic analysis of a 2D cylinder wake under forced excitation. The response of the Karman wake (K mode) to external perturbations is studied. Reflection-symmetric (S mode) perturbations and asymmetric (K1 mode) perturbations are considered. The perturbations are generated by mechanically oscillating the test cylinder. Tests were done in a small wind tunnel. Depending on the excitation to Karman shedding frequency ratio, mode locked states, in the form of spatio-temporally fixed patterns, could be observed. Harmonic asymmetric (mode K1=K) forcing at the Karman frequency strongly enhanced the Karman mode. Superharmonic forcing (with mode K1 ≠ K) had little effect on the Karman mode K. However, a detuning effect was observed. On the other hand, subharmonic (1/2, 1/3) K1 mode forcing significantly affected the K mode, with strong response at K1 mode harmonics. Subharmonic S mode excitation had a damping effect on the K mode. On the other hand harmonic and superharmonic forcing triggered a period-doubling instability, destroying the original K mode. Using a group theoretic approach, the general amplitude equations governing the interaction of the S/K1 modes with the K mode have been derived. A qualitative analysis of the equations helps explains some of the experimental results. For K1/K mode interactions, the symmetrical ‘compatibility’, via common subgroups, explains the strong resonances observed experimentally for 1/1 and 1/3 frequency ratios. For a frequency ratio 1/2, it is shown that K1 and K mode symmetries are incompatible; the two modes do not have a common symmetry subgroup. Consequently, traveling wave solutions, induced by total symmetry breaking, rather than standard steady state modes are expected to be more likely to occur. For S/K mode interaction, an earlier result is reiterated; thus, the Karman mode is shown, theoretically, to be destroyed via a period-doubling instability. This effect occurs for S mode frequencies as high as 3 times the Karman frequency.

Second Order Averaging Study of an Autoparametric System

1993 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj ◽  
Patricia Davies
Keyword(s):  
Dynamical Behavior ◽  
Period Doubling ◽  
Nonlinear Effects ◽  
Single Mode ◽  
Pitchfork Bifurcation ◽  
Second Order ◽  
System Response ◽  
Response Curves ◽  
Vibratory System ◽  
First Order

Abstract The autoparametric vibratory system consisting of a primary spring-mass-dashpot system coupled with a damped simple pendulum serves as an useful example of two degree-of-freedom nonlinear systems that exhibit complex dynamic behavior. It exhibits 1:2 internal resonance and amplitude modulated chaos under harmonic forcing conditions. First-order averaging studies of this system using AUTO and KAOS have yielded useful information about the amplitude dynamics of this system. Response curves of the system indicate saturation and the pitchfork bifurcation sets are found to be symmetric. The period-doubling route to chaotic solutions is observed. However questions about the range of the small parameter ε (a function of the forcing amplitude) for which the solutions are valid cannot be answered by a first-order study. Some observed dynamical behavior, like saturation, may not persist when higher-order nonlinear effects are taken into account. Second-order averaging of the system, using Mathematica (Maeder, 1991; Wolfram, 1991) is undertaken to address these questions. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of ε appearing in the averaged equations. The dynamics of the system is found to be very sensitive to damping, with extremely complicated behavior arising for low values of damping. For large ε second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response.

scholarly journals Assessing the Impacts of Competition and Dispersal on a Multiple Interactions Type Model

2021 ◽  
Vol 29 (3) ◽  
Author(s):  
Murtala Bello Aliyu ◽  
Mohd Hafiz Mohd ◽  
Mohd Salmi Md. Noorani
Keyword(s):  
Bifurcation Analysis ◽  
Species Coexistence ◽  
Real Life ◽  
Period Doubling ◽  
Type Model ◽  
Coexistence Mechanisms ◽  
Multiple Species ◽  
The Stability ◽  
Transcritical Bifurcations ◽  
Multiple Interactions

Multiple interactions (e.g., mutualist-resource-competitor-exploiter interactions) type models are known to exhibit oscillatory behaviour as a result of their complexity. This large-amplitude oscillation often de-stabilises multispecies communities and increases the chances of species extinction. What mechanisms help species in a complex ecological system to persist? Some studies show that dispersal can stabilise an ecological community and permit multi-species coexistence. However, previous empirical and theoretical studies often focused on one- or two-species systems, and in real life, we have more than two-species coexisting together in nature. Here, we employ a (four-species) multiple interactions type model to investigate how competition interacts with other biotic factors and dispersal to shape multi-species communities. Our results reveal that dispersal has (de-)stabilising effects on the formation of multi-species communities, and this phenomenon shapes coexistence mechanisms of interacting species. These contrasting effects of dispersal can best be illustrated through its combined influences with the competition. To do this, we employ numerical simulation and bifurcation analysis techniques to track the stable and unstable attractors of the system. Results show the presence of Hopf bifurcations, transcritical bifurcations, period-doubling bifurcations and limit point bifurcations of cycles as we vary the competitive strength in the system. Furthermore, our bifurcation analysis findings show that stable coexistence of multiple species is possible for some threshold values of ecologically-relevant parameters in this complex system. Overall, we discover that the stability and coexistence mechanisms of multiple species depend greatly on the interplay between competition, other biotic components and dispersal in multi-species ecological systems.

scholarly journals Transition to Periodic Behaviour of Flow Past a Circular Cylinder under the Action of Fluidic Actuation in the Transitional Regime

Energies ◽  
2021 ◽  
Vol 14 (16) ◽  
pp. 5069
Author(s):  
Wasim Sarwar ◽  
Fernando Mellibovsky ◽  
Md. Mahbub Alam ◽  
Farhan Zafar
Keyword(s):  
Circular Cylinder ◽  
Period Doubling ◽  
Underlying Mechanism ◽  
Time Dependent ◽  
Cylinder Wake ◽  
Transitional Regime ◽  
Periodic Behaviour ◽  
Vortex Shedding Frequency ◽  
Fluidic Actuation ◽  
Periodic Dynamics

This study focuses on the numerical investigation of the underlying mechanism of transition from chaotic to periodic dynamics of circular cylinder wake under the action of time-dependent fluidic actuation at the Reynolds number = 2000. The forcing is realized by blowing and suction from the slits located at ±90∘ on the top and bottom surfaces of the cylinder. The inverse period-doubling cascade is the underlying physical mechanism underpinning the wake transition from mild chaos to perfectly periodic dynamics in the spanwise-independent, time-dependent forcing at twice the natural vortex-shedding frequency.

Fold-Pitchfork Bifurcation, Arnold Tongues and Multiple Chaotic Attractors in a Minimal Network of Three Sigmoidal Neurons

2018 ◽  
Vol 28 (10) ◽  
pp. 1850123 ◽  
Author(s):  
Yo Horikawa ◽  
Hiroyuki Kitajima ◽  
Haruna Matsushita
Keyword(s):  
Periodic Solutions ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Quasiperiodic Solution ◽  
Chaotic Attractors ◽  
Dimensional Parameter ◽  
Chaotic Neural Networks ◽  
Minimal Network ◽  
Arnold Tongues ◽  
Period Doubling Bifurcations

Bifurcations and chaos in a network of three identical sigmoidal neurons are examined. The network consists of a two-neuron oscillator of the Wilson–Cowan type and an additional third neuron, which has a simpler structure than chaotic neural networks in the previous studies. A codimension-two fold-pitchfork bifurcation connecting two periodic solutions exists, which is accompanied by the Neimark–Sacker bifurcation. A stable quasiperiodic solution is generated and Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation in a two-dimensional parameter space. The merging, splitting and crossing of the Arnold tongues are observed. Further, multiple chaotic attractors are generated through cascades of period-doubling bifurcations of periodic solutions in the Arnold tongues. The chaotic attractors grow and are destroyed through crises. Transient chaos and crisis-induced intermittency due to the crises are also observed. These quasiperiodic solutions and chaotic attractors are robust to small asymmetry in the output function of neurons.

PERIOD-DOUBLING SCENARIO WITHOUT FLIP BIFURCATIONS IN A ONE-DIMENSIONAL MAP

2005 ◽  
Vol 15 (04) ◽  
pp. 1267-1284 ◽  
Author(s):  
V. AVRUTIN ◽  
M. SCHANZ
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Symmetry Breaking ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Period Doubling Bifurcation ◽  
Coexisting Attractors ◽  
One Dimensional ◽  
Poincaré Return Map ◽  
Smooth Dynamical System

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

Instability caused by the coupling between non-resonant shape oscillation modes of a charged conducting drop

1997 ◽  
Vol 333 ◽  
pp. 1-21 ◽  
Author(s):  
Z. C. FENG
Keyword(s):  
Liquid Drop ◽  
Multiple Scale ◽  
High Mode ◽  
Mode Number ◽  
Nonlinear Coupling ◽  
Modal Interaction ◽  
Amplitude Equations ◽  
Oscillation Modes ◽  
Shape Oscillation ◽  
Shape Perturbation

By examining the modal interaction between two non-resonant shape oscillation modes of a charged liquid drop, we have identified a new route to instability via nonlinear coupling. We present numerical simulation results to show that when shape perturbation of a high-mode number Legendre mode is applied to the drop, the prolate–oblate mode of the drop may grow unbounded. Using multiple-scale analysis, we derive amplitude equations for the high-mode-number shape mode and the prolate–oblate mode to show the nonlinear coupling between the two modes.

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