Complex dynamics of compliant off-shore structures

1983 ◽  
Vol 387 (1793) ◽  
pp. 407-427 ◽  
Keyword(s):  
Complex Dynamics ◽  
Initial Conditions ◽  
Period Doubling ◽  
Model Studies ◽  
Impact Oscillators ◽  
Starting Conditions ◽  
Extreme Sensitivity ◽  
Subharmonic Resonances ◽  
Sensitivity To Initial Conditions ◽  
Period Doubling Bifurcations

The dynamics of advanced compliant off-shore structures can be extremely complex owing to the inherent nonlinearities. Subharmonic resonances can coexist with stable small-amplitude solutions, the response observed depending solely on the starting conditions of the motion. So care must be taken in digital, analogue and model studies to explore a comprehensive set of initial conditions. ‘Efficient’ automated digital computations could miss an entire subharmonic peak by locking onto a coexisting small-deflexion fundamental solution. Chaotic, non-periodic motions of strange attractors can also arise in well defined deterministic resonance problems. The waveforms of these look like the result of a stochastic process, and because of their extreme sensitivity to initial conditions a statistical description must be sought. Genuinely chaotic solutions can be identified by looking for nearby period-doubling bifurcations, and for the exponential divergence of adjacent starts leading to a loss of correlation. The period-doubling cascades give rise to subharmonics of arbitrarily high order close to the chaotic régimes, so the duration of digital, analogue and laboratory experiments must be long and chosen with care. The resonance of simple bilinear and impact oscillators is used as a vehicle to illustrate these general ideas.

Transitions Through Period Doubling Route to Chaos

1991 ◽  
Author(s):  
R. M. Evan-lwanowski ◽  
Chu-Ho Lu
Keyword(s):  
Period Doubling ◽  
Mirror Image ◽  
Flip Flop ◽  
Physical Systems ◽  
Starting Conditions ◽  
Spatial System ◽  
Stationary Bifurcation ◽  
Extreme Sensitivity ◽  
Route To Chaos ◽  
Parameter Values

Abstract The Duffing driven, damped, “softening” oscillator has been analyzed for transition through period doubling route to chaos. The forcing frequency and amplitude have been varied in time (constant sweep). The stationary 2T, 4T… chaos regions have been determined and used as the starting conditions for nonstationary regimes, consisting of the transition along the Ω(t)=Ω0±α2t,f=const., Ω-line, and along the E-line: Ω(t)=Ω0±α2t;f(t)=f0∓α2t. The results are new, revealing, puzzling and complex. The nonstationary penetration phenomena (delay, memory) has been observed for a single and two-control nonstationary parameters. The rate of penetrations tends to zero with increasing sweeps, delaying thus the nonstationary chaos relative to the stationary chaos by a constant value. A bifurcation discontinuity has been uncovered at the stationary 2T bifurcation: the 2T bifurcation discontinuity drops from the upper branches of (a, Ω) or (a, f) curves to their lower branches. The bifurcation drops occur at the different control parameter values from the response x(t) discontinuities. The stationary bifurcation discontinuities are annihilated in the nonstationary bifurcation cascade to chaos — they reside entirely on the upper or lower nonstationary branches. A puzzling drop (jump) of the chaotic bifurcation bands has been observed for reversed sweeps. Extreme sensitivity of the nonstationary bifurcations to the starting conditions manifests itself in the flip-flop (mirror image) phenomena. The knowledge of the bifurcations allows for accurate reconstruction of the spatial system itself. The results obtained may model mathematically a number of engineering and physical systems.

SYNCHRONIZATION STABILITY ANALYSIS OF THE CHAOTIC RÖSSLER SYSTEM

1996 ◽  
Vol 06 (11) ◽  
pp. 2153-2161 ◽  
Author(s):  
QINGXIAN XIE ◽  
GUANRONG CHEN
Keyword(s):  
Initial Conditions ◽  
Sufficient Conditions ◽  
General Situation ◽  
Wide Range ◽  
Rossler System ◽  
Extreme Sensitivity ◽  
System Synchronization ◽  
Rössler System ◽  
Type Of Stability ◽  
Sensitivity To Initial Conditions

In this paper we show, both analytically and experimentally, that the Rössler system synchronization is either asymptotically stable or orbitally stable within a wide range of the system key parameters. In the meantime, we provide some simple sufficient conditions for synchronization stabilities of the Rössler system in a general situation. Our computer simulation shows that the type of stability of the synchronization is very sensitive to the initial values of the two (drive and response) Rössler systems, especially for higher-periodic synchronizing trajectories, which is believed to be a fundamental characteristic of chaotic synchronization that preserves the extreme sensitivity to initial conditions of chaotic systems.

HARMONIC BALANCE APPROACH TO PREDICT PERIOD-DOUBLING BIFURCATIONS IN NEARLY SYMMETRIC CNNs

2003 ◽  
Vol 12 (04) ◽  
pp. 435-459 ◽  
Author(s):  
MAURO DI MARCO ◽  
MAURO FORTI ◽  
ALBERTO TESI
Keyword(s):  
Neural Networks ◽  
Computer Simulations ◽  
Harmonic Balance ◽  
Recent Literature ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Third Order ◽  
Small Perturbations ◽  
Basic Issue ◽  
Period Doubling Bifurcations

This paper further investigates a basic issue that has received attention in the recent literature, namely, the robustness of complete stability of standard Cellular Neural Networks (CNNs) with respect to small perturbations of the nominal symmetric interconnections. More specifically, a class of third-order CNNs with a nominal symmetric interconnection matrix is considered, and the Harmonic Balance (HB) method is exploited for addressing the possible existence of period-doubling bifurcations, and complex dynamics, for small perturbations of the nominal interconnections. The main result is that there are indeed parameter sets close to symmetry for which period-doubling bifurcations are predicted by the HB method. Moreover, the predictions are found to be reliable and accurate by means of computer simulations.

scholarly journals A Bifurcation Analysis and Sensitivity Study of Brake Creep Groan

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Shaun Smith ◽  
James Knowles ◽  
Byron Mason ◽  
Sean Biggs
Keyword(s):  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Period Doubling ◽  
Nonlinear Behavior ◽  
Sensitivity Study ◽  
Friction Model ◽  
Model Parameters ◽  
Period Doubling Bifurcations ◽  
Simple Models

Creep groan is the undesirable vibration observed in the brake pad and disc as brakes are applied during low-speed driving. The presence of friction leads to nonlinear behavior even in simple models of this phenomenon. This paper uses tools from bifurcation theory to investigate creep groan behavior in a nonlinear 3-degrees-of-freedom mathematical model. Three areas of operational interest are identified, replicating results from previous studies: region 1 contains repelling equilibria and attracting periodic orbits (creep groan); region 2 contains both attracting equilibria and periodic orbits (creep groan and no creep groan, depending on initial conditions); region 3 contains attracting equilibria (no creep groan). The influence of several friction model parameters on these regions is presented, which identify that the transition between static and dynamic friction regimes has a large influence on the existence of creep groan. Additional investigations discover the presence of several bifurcations previously unknown to exist in this model, including Hopf, torus and period-doubling bifurcations. This insight provides valuable novel information about the nature of creep groan and indicates that complex behavior can be discovered and explored in relatively simple models.

Analytical Reformulation of the Multibody Dynamics of a Satellite

1999 ◽  
Author(s):  
Nestor Sanchez
Keyword(s):  
Initial Conditions ◽  
Period Doubling ◽  
Numerical Approach ◽  
Branch Points ◽  
Extra Effort ◽  
The Everyday ◽  
Symbolic Code ◽  
Three Body ◽  
Near Future ◽  
Period Doubling Bifurcations

Abstract The topic of dynamics has been somehow reshaped by computational power. The areas of computer algebra and symbolics now allow us to deal with a more involved analytical manipulation of equations. At the same time, the everyday increasing power of numerics put into our hands new tools to solve old problems. In this case, we reformulate the problem of the dynamics of a three body multibody system by using symbolic manipulation of the Newtonian equations, to produce a set of differential equations that can be solve with standard codes. This treatment should produce not only the same results as the numerical approach, but it allows us to use the analytical equations to expand the analysis into design, control and stability. The paper shows the process to build the symbolic code using Maple language, or any algebraic manipulator. The proper equations will be derived to solve for the unknowns angles {ψ,ϕ,θ}, in terms of the prescribed quantities {α(t),β(t),γ(t)t}, and initial conditions. This procedure gives a good idea about the nonlinear response of the satellite to the control parameters. The size of the equations obtained is large. However, considering the type of analysis that could be done with a set like this and the capacity of large computers, it will pay off the extra effort. The codes that could be used for further analysis would find folds, branch points, period doubling bifurcations, Hopf bifurcations, torus bifurcations, by changing the parameters of the governing equation. A large number of important applications will develop in this area in the near future.

FOURTH-ORDER NEARLY-SYMMETRIC CNNS EXHIBITING COMPLEX DYNAMICS

2005 ◽  
Vol 15 (05) ◽  
pp. 1579-1587 ◽  
Author(s):  
M. DI MARCO ◽  
M. FORTI ◽  
M. GRAZZINI ◽  
L. PANCIONI
Keyword(s):  
Dynamical System ◽  
Neural Networks ◽  
Fourth Order ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Cellular Neural Networks ◽  
Parameter Family ◽  
Relative Errors ◽  
Period Doubling Bifurcations

In this paper, the possible presence of complex dynamics in nearly-symmetric standard Cellular Neural Networks (CNNs), is investigated. A one-parameter family of fourth-order CNNs is presented, which exhibits a cascade of period-doubling bifurcations leading to the birth of a complex attractor, close to some nominal symmetric CNN. Different from previous work on this topic, the bifurcations and complex dynamics are obtained for small relative errors with respect to the nominal interconnections. The fourth-order CNNs have negative (inhibitory) interconnections between distinct neurons, and are designed by a variant of a technique proposed by Smale to embed a given dynamical system within a competitive dynamical system of larger order.

Bifurcation Behavior of a Supercavitating Vehicle

2006 ◽  
Author(s):  
Guojian Lin ◽  
Balakumar Balachandran ◽  
Eyad H. Abed
Keyword(s):  
Complex Dynamics ◽  
Numerical Study ◽  
Period Doubling ◽  
Linear Feedback ◽  
Hopf Bifurcations ◽  
Linear Feedback Control ◽  
Chaotic Motions ◽  
Supercavitating Vehicle ◽  
Bifurcation Behavior ◽  
Period Doubling Bifurcations

In this effort, a numerical study of the bifurcation behavior of a supercavitating vehicle is conducted. The nonsmoothness of this system is due to the planing force acting on the vehicle. With a focus on dive-plane dynamics, bifurcations with respect to a quasi-static variation of the cavitation number are studied. The system is found to exhibit rich and complex dynamics including nonsmooth bifurcations such as the grazing bifurcation and smooth bifurcations such as Hopf bifurcations, cyclic-fold bifurcations, and period-doubling bifurcations, chaotic attractors, transient chaotic motions, and crises. The tailslap phenomenon of the supercavitating vehicle is identified as a consequence of the Hopf bifurcation followed by a grazing event. It is shown that the occurrence of these bifurcations can be delayed or triggered earlier by using dynamic linear feedback control aided by washout filters.

Control Lorenz System with Vibration Estimation with Averaging Method

2010 ◽  
Vol 43 ◽  
pp. 36-39
Author(s):  
Chun Zhou
Keyword(s):  
Chaotic Systems ◽  
Inverted Pendulum ◽  
Lorenz System ◽  
Stable Equilibrium ◽  
Averaging Method ◽  
Control Method ◽  
Initial Conditions ◽  
Vibrational Control ◽  
Extreme Sensitivity ◽  
Sensitivity To Initial Conditions

The vibrational control theory stems from the well-known of stabilization of the upper unstable equilibrium position of the inverted pendulum having suspension point vibration along the vertical line with amplitude as small as desired and a frequency reason high. Chaotic phenomena have been found in many nonlinear systems including continuous time and discrete time. The chaotic systems are characterized by their extreme sensitivity to initial conditions, nonperiodic and boundary. The trajectories start even from close initial states will diverge from each other at an exponential rate as time goes. The vibrational control method was applied to Lorenz system. The effect of the control can be estimated with the APAZ method. It was showed that vibrational control brought the controlled Lorenz system to stable equilibrium with appropriate parameters. Numerical simulation demonstrated validity of the proposed method.

scholarly journals The Formation and Early Evolution of the Greensburg, Kansas, Tornadic Supercell on 4 May 2007

2009 ◽  
Vol 24 (4) ◽  
pp. 899-920 ◽  
Author(s):  
Howard B. Bluestein
Keyword(s):  
Doppler Radar ◽  
Initial Conditions ◽  
Radar Data ◽  
Early Evolution ◽  
Convective Storm ◽  
Extreme Sensitivity ◽  
The Right ◽  
Subsequent Splitting ◽  
Operational Data ◽  
Sensitivity To Initial Conditions

Abstract During the evening of 4 May 2007, a large, powerful tornado devastated Greensburg, Kansas. The synoptic and mesoscale environments of the parent supercell that spawned this and other tornadoes are described from operational data. The formation and early evolution of this long-track supercell, within the context of its larger-scale environment, are documented on the basis of Weather Surveillance Radar-1988 Doppler (WSR-88D) data and mobile Doppler radar data. The storm produced tornadoes cyclically for about 30 min before producing a large, long-lived tornado. It is shown that in order to have forecasted the severe weather locations and times accurately, it would have been necessary to have predicted 1) the localized formation of an isolated convective storm near/east of a dryline, 2) the subsequent splitting and resplitting of the storm several times, 3) the growth of a new storm along the right-rear flank of an existing storm, and 4) the transition from the cyclic production of small tornadoes to the production of one, large, long-track tornado. It is therefore suggested that both extreme sensitivity to initial conditions associated with storm formation and the uncertainty of storm behavior made it unusually difficult to forecast this event accurately.

Subharmonic and Chaotic Motions of Compliant Offshore Structures and Articulated Mooring Towers

1984 ◽  
Vol 106 (2) ◽  
pp. 191-198 ◽  
Author(s):  
J. M. T. Thompson ◽  
A. R. Bokaian ◽  
R. Ghaffari
Keyword(s):  
Nonlinear Dynamics ◽  
Initial Conditions ◽  
Offshore Structures ◽  
High Order ◽  
Chaotic Motions ◽  
Laboratory Simulations ◽  
Starting Conditions ◽  
Subharmonic Resonances

Compliant offshore structures have complex nonlinear dynamics. Subharmonic resonances can co-exist with small fundamental motions. To prevent a simulation missing an entire subharmonic peak, a comprehensive set of initial conditions must be explored. Chaotic nonperiodic motions can also arise in deterministic problems. These are extremely sensitive to starting conditions and require a statistical description. Chaotic responses and their adjacent subharmonics of arbitrarily high order mean that the duration of digital, analogue and laboratory simulations must be chosen with considerable care. Resonances of articulated mooring towers, encountered in tank tests, are used to illustrate these general concepts.

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