stability and bifurcations
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2021 ◽  
Vol 2070 (1) ◽  
pp. 012068
A George Maria Selvam ◽  
R Janagaraj ◽  
S Britto Jacob ◽  
D Vignesh

Abstract In ecology, by refuge an organism attains protection from predation by hiding in an area where it is unreachable or cannot simply be found. In population dynamics, once refuges are available, both prey-predator populations are expressively greater and meaningfully extra species can be sustained in the region. This examine the stability of a discrete predator prey model incorporating with constant prey refuge. Existence results and the stability conditions of the system are analyzed by obtaining fixed points and Jacobian matrix. The chaotic behavior of the system is discussed with bifurcation diagrams. Numerical experiments are simulated for the better understanding of the qualitative behavior of the considered model. Mathematics Subject Classification. [2010] : 37C25, 39A28, 39A30, 92D25.

2021 ◽  
Vol 31 (12) ◽  
pp. 2150186
Siyuan Xing ◽  
Albert C. J. Luo

This paper studies the dynamics and bifurcations of a vibration-assisted, regenerative, nonlinear turning-tool system using an implicit mapping method. Machine vibration has been studied for a century for the improvement of machine accuracy and metal removal rate. In fact, this problem is unsolved yet. This is because such dynamical systems are involved in nonlinearity, discontinuity and time-delay. Thus, a comprehensive understanding of nonlinear machining dynamics with time-delay is indispensable. In this paper, period-[Formula: see text] motions in the turning machine-tool system are studied through specific mapping structures, and the corresponding stability and bifurcations of the period-[Formula: see text] motion are determined through the eigenvalue analysis. The analytical bifurcation scenarios for two sets of sequential period-[Formula: see text] motions in a turning-tool system are presented. Numerical simulations of period-[Formula: see text] motions are carried out to verify the prediction of periodic motions. The complex dynamics of vibration-assisted machining with strong nonlinearity are presented, which can provide a good overview for nonlinear dynamics of machine-tool systems.

2021 ◽  
Vol 31 (11) ◽  
pp. 2130032
William Duncan ◽  
Tomas Gedeon

In this paper, we study equilibria of differential equation models for networks. When interactions between nodes are taken to be piecewise constant, an efficient combinatorial analysis can be used to characterize the equilibria. When the piecewise constant functions are replaced with piecewise linear functions, the equilibria are preserved as long as the piecewise linear functions are sufficiently steep. Therefore the combinatorial analysis can be leveraged to understand a broader class of interactions. To better understand how broad this class is, we explicitly characterize how steep the piecewise linear functions must be for the correspondence between equilibria to hold. To do so, we analyze the steady state and Hopf bifurcations which cause a change in the number or stability of equilibria as slopes are decreased. Additionally, we show how to choose a subset of parameters so that the correspondence between equilibria holds for the smallest possible slopes when the remaining parameters are fixed.

2021 ◽  

Abstract In this work, a single degree of freedom system consisting of a mass and a Pneumatic Artificial Muscle (PAM) subjected to time varying pressure inside the muscle is considered. The system is subjected to hard excitation and the governing equation of motion is found to be that of a nonlinear forced and parametrically excited system under super- and sub-harmonic resonance conditions. The solution of the nonlinear governing equation of motion is obtained using the method of multiple scales (MMS). The time and frequency response, phase portraits and basin of attraction have been plotted to study the system response along with the stability and bifurcations. Further, the different muscle parameters have been evaluated by performing experiments which are further used for numerically evaluating the system response using the theoretically obtained closed form equations. The responses obtained from the experiments are found to be in good agreement with those obtained from the method of multiple scales. With the help of examples, the procedure to obtain the safe operating range of different system parameters have been illustrated.

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