scholarly journals Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme

2021 ◽  
Vol 11 (2) ◽  
pp. 741
Author(s):  
Xiangyu Wang ◽  
Zhigang Wu ◽  
Chao Yang
Keyword(s):  
Fixed Points ◽  
State Space ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Initial Conditions ◽  
Time Integration ◽  
Integration Time ◽  
Aeroelastic System ◽  
System States ◽  
Short Integration Time

Time integration is commonly used to obtain accurate system responses, such as the limit cycle oscillations (LCOs) for an aeroelastic system with freeplay. However, the integrations that start with various initial conditions (I.C.s) are usually studied case by case, so only a few system states can possibly be focused on. This paper proposes a state space iterating (SSI) scheme to find LCO solutions using time integration by using another method. First, a large number of arbitrary I.C. cases are used for time integrations, but only a very short integration time is required for each I.C. case. Second, system behaviors are depicted visually through a method that combines a modified Poincaré map and Lorenz map, in which the LCO solutions are found as fixed points via visual inspections. To verify the SSI scheme’s ability to find LCOs, a typical plunge–pitch wing section is established numerically. Time integrations with both the classic scheme and the proposed SSI scheme are carried out. The LCO results of the SSI scheme are well-aligned with those from the classic scheme. The SSI scheme visualizes the patterns of system responses using arbitrary I.C. cases and analyzes the LCO stability, which provides more mathematical insights into an aeroelastic system with freeplay.

CHAOTIC BEHAVIOR AND DYNAMICS OF MAPS USED IN A METHOD OF SCRAMBLING SIGNALS

2010 ◽  
Vol 20 (12) ◽  
pp. 4097-4101
Author(s):  
REZA MAZROOEI-SEBDANI ◽  
MEHDI DEHGHAN
Keyword(s):  
Fixed Points ◽  
Limit Cycles ◽  
Secure Communication ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Chaotic Encryption ◽  
Input And Output ◽  
Sensitive Dependence ◽  
Close Relationship ◽  
Existence And Stability

The close relationship between chaos and cryptography makes chaotic encryption a natural candidate for secure communication and cryptography. In this manuscript, we prove that a class of maps that have been proposed as suitable for scrambling signals possess the property of sensitive dependence on initial conditions (s.d.i.c.) necessary for chaos and cryptography. Our result can also be used for generating other maps with s.d.i.c., through a suitable semiconjugacy between their input and output parts. Using the condition of semiconjugacy we also establish for this class of maps rigorous criteria for the existence and stability of their fixed points and limit cycles.

scholarly journals Limit cycle oscillation control and suppression

1999 ◽  
Vol 103 (1023) ◽  
pp. 257-263 ◽  
Author(s):  
G. Dimitriadis ◽  
J. E. Cooper
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Harmonic Balance Method ◽  
Limit Cycle Oscillation ◽  
Aeroelastic System ◽  
Oscillation Control ◽  
Control Scheme ◽  
Non Linear ◽  
Cycle Oscillation ◽  
The Stability

Abstract The prediction and characterisation of the limit cycle oscillation (LCO) behaviour of non-linear aeroelastic systems has become of great interest recently. However, much of this work has concentrated on determining the existence of LCOs. This paper concentrates on LCO stability. By considering the energy present in different limit cycles, and also using the harmonic balance method, it is shown how the stability of limit cycles can be determined. The analysis is then extended to show that limit cycles can be controlled, or even suppressed, by the use of suitable excitation signals. A basic control scheme is developed to achieve this, and is demonstrated on a simple simulated non-linear aeroelastic system.

Nonlinear Response of Infinitely Long Circular Cylindrical Shells to Subharmonic Radial Loads

1991 ◽  
Vol 58 (4) ◽  
pp. 1033-1041 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Raouf A. Raouf ◽  
Jamal F. Nayfeh
Keyword(s):  
Fixed Points ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Nonlinear Response ◽  
Cylindrical Shells ◽  
Excitation Frequency ◽  
Hopf Bifurcations ◽  
Response Curves ◽  
Modulation Equations ◽  
Circular Cylindrical Shells

The method of multiple scales is used to analyze the nonlinear response of infinitely long, circular cylindrical shells (thin circular rings) in the presence of a two-to-one internal (autoparametric) resonance to a subharmonic excitation of order one-half of the higher mode. Four autonomous first-order ordinary differential equations are derived for the modulation of the amplitudes and phases of the interacting modes. These modulation equations are used to determine the fixed points and their stability. The fixed points correspond to periodic oscillations of the shell, whereas the limit-cycle solutions of the modulation equations correspond to amplitude and phase-modulated oscillations of the shell. The force response curves exhibit saturation, jumps, and Hopf bifurcations. Moreover, the frequency response curves exhibit Hopf bifurcations. For certain parameters and excitation frequencies between the Hopf values, limit-cycle solutions of the modulation equations are found. As the excitation frequency changes, all limit cycles deform and lose stability through either pitchfork or cyclic-fold (saddle-node) bifurcations. Some of these saddlenode bifurcations cause a transition to chaos. The pitchfork bifurcations break the symmetry of the limit cycles.

Propagating Neuronal Discharges in Neocortical Slices: Computational and Experimental Study

1997 ◽  
Vol 78 (3) ◽  
pp. 1199-1211 ◽  
Author(s):  
David Golomb ◽  
Yael Amitai
Keyword(s):  
Experimental Study ◽  
Ampa Receptors ◽  
Potassium Current ◽  
Initial Conditions ◽  
Field Potential ◽  
Integration Time ◽  
Synaptic Coupling ◽  
Discharge Velocity ◽  
Slow Potassium ◽  
Typical Cell

Golomb, David and Yael Amitai. Propagating neuronal discharges in neocortical slices: computational and experimental study. J. Neurophysiol. 78: 1199–1211, 1997. We studied the propagation of paroxysmal discharges in disinhibited neocortical slices by developing and analyzing a model of excitatory regular-spiking neocortical cells with spatially decaying synaptic efficacies and by field potential recording in rat slices. Evoked discharges may propagate both in the model and in the experiment. The model discharge propagates as a traveling pulse with constant velocity and shape. The discharge shape is determined by an interplay between the synaptic driving force and the neuron's intrinsic currents, in particular the slow potassium current. In the model, N-methyl-d-aspartate (NMDA) conductance contributes much less to the discharge velocity than amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) conductance. Blocking NMDA receptors experimentally with 2-amino-5-phosphonovaleric acid (APV) has no significant effect on the discharge velocity. In both model and experiments, propagation occurs for AMPA synaptic coupling g AMPA above a certain threshold, at which the velocity is finite (non-zero). The discharge velocity grows linearly with the g AMPA for g AMPA much above the threshold. In the experiments, blocking AMPA receptors gradually by increasing concentrations of 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX) in the perfusing solution results in a gradual reduction of the discharge velocity until propagation stops altogether, thus confirming the model prediction. When discharges are terminated in the model by the slow potassium current, a network with the same parameter set may display discharges with several forms, which have different velocities and numbers of spikes; initial conditions select the exhibited pattern. When the discharge is also terminated by strong synaptic depression, there is only one discharge form for a particular parameter set; the velocity grows continuously with increased synaptic conductances. No indication for more than one discharge velocity was observed experimentally. If the AMPA decay rate increases while the maximal excitatory postsynaptic conductance (EPSC) a cell receives is kept fixed, the velocity increases by ∼20% until it reaches a saturated value. Therefore the discharge velocity is determined mainly by the cells' integration time of input EPSCs. We conclude, on the basis of both the experiments and the model, that the total amount of excitatory conductance a typical cell receives in a control slice exhibiting paroxysmal discharges is only ∼5 times larger than the excitatory conductance needed for raising the potential of a resting cell above its action potential threshold.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

scholarly journals ON PERIODIC SOLUTIONS OF LIÉNARD TYPE EQUATIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 708-716 ◽  
Author(s):  
Svetlana Atslega ◽  
Felix Sadyrbaev
Keyword(s):  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Type Equation ◽  
Convex Shape ◽  
Period Annulus ◽  
Conservative Equation ◽  
Liénard Type Equation

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(x, x') so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS

2007 ◽  
Vol 17 (03) ◽  
pp. 953-963 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Cellular Neural Networks ◽  
Lyapunov Spectrum ◽  
Regular Array ◽  
Poincaré Maps ◽  
One Dimensional ◽  
Poincare Maps ◽  
New Family

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

A novel methodology to explore the periodic gait of a biped walker under uncertainty using a machine learning algorithm

Robotica ◽  
2021 ◽  
pp. 1-16
Author(s):  
Namjung Kim ◽  
Bongwon Jeong ◽  
Kiwon Park
Keyword(s):  
Limit Cycle ◽  
Systematic Approach ◽  
Learning Algorithm ◽  
Initial Conditions ◽  
Angular Displacement ◽  
External Disturbance ◽  
Rehabilitation Robotics ◽  
Stable Region ◽  
Gait Rehabilitation ◽  
External Disturbances

Abstract In this paper, we present a systematic approach to improve the understanding of stability and robustness of stability against the external disturbances of a passive biped walker. First, a multi-objective, multi-modal particle swarm optimization (MOMM-PSO) algorithm was employed to suggest the appropriate initial conditions for a given biped walker model to be stable. The MOMM-PSO with ring topology and special crowding distance (SCD) used in this study can find multiple local minima under multiple objective functions by limiting each agent’s search area properly without determining a large number of parameters. Second, the robustness of stability under external disturbances was studied, considering an impact in the angular displacement sampled from the probabilistic distribution. The proposed systematic approach based on MOMM-PSO can find multiple initial conditions that lead the biped walker in the periodic gait, which could not be found by heuristic approaches in previous literature. In addition, the results from the proposed study showed that the robustness of stability might change depending on the location on a limit cycle where immediate angular displacement perturbation occurs. The observations of this study imply that the symmetry of the stable region about the limit cycle will break depending on the accelerating direction of inertia. We believe that the systematic approach developed in this study significantly increased the efficiency of finding the appropriate initial conditions of a given biped walker and the understanding of robustness of stability under the unexpected external disturbance. Furthermore, a novel methodology proposed for biped walkers in the present study may expand our understanding of human locomotion, which in turn may suggest clinical strategies for gait rehabilitation and help develop gait rehabilitation robotics.

scholarly journals Phase-Coupled Oscillations in the Brain: Nonlinear Phenomena in Cellular Signalling

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Vikas Rai ◽  
Sreenivasan Rajamoni Nadar ◽  
Riaz A. Khan
Keyword(s):  
Limit Cycles ◽  
Initial Conditions ◽  
Phase Relationship ◽  
Oscillatory System ◽  
Calcium Currents ◽  
Nonlinear Phenomena ◽  
Inhibitory Interneurons ◽  
Stable Limit ◽  
Principal Cells ◽  
Coupled Oscillations

We report the existence of phase-coupled oscillations in a model neural system. The model consists of a group of excitatory principal cells in interaction with local inhibitory interneurons. The voltages across the membranes of excitatory cells are governed primarily by calcium and potassium ion conductivities. The number of potassium channels open at any given instant changes in accordance with a deterministic law. The time scale of this change is set by a constant which depends on midpoint potentials at which potassium and calcium currents are half-activated. The growth of mean membrane potential of excitatory principal cells is controlled by that of the inhibitory interneurons. Nonlinear oscillatory system associated with these limit cycles starting from two different initial conditions maintain a definite phase relationship. The phase-coupled oscillations in electrical activity of the neuronal cells carry together amplitude, phase, and time information for cellular signaling. This mechanism supports an energy efficient way of information processing in the central nervous system. The information content is encoded as persistent periodic oscillations represented by stable limit cycles in the phase space.

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

2008 ◽  
Vol 18 (10) ◽  
pp. 3013-3027 ◽  
Author(s):  
MAOAN HAN ◽  
JIAO JIANG ◽  
HUAIPING ZHU
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Unperturbed System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Cubic System ◽  
Limit Cycle Bifurcation ◽  
Almost All ◽  
Nilpotent Center ◽  
Computing Method

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

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