How Does State Space Definition Influence the Measure of Chaotic Behavior?

This work presents the analysis of nonlinear aeroelastic time series from wing vibrations due to airflow separation during wind tunnel experiments. Surrogate data method is used to justify the application of nonlinear time series analysis to the aeroelastic system, after rejecting the chance for nonstationarity. The singular value decomposition (SVD) approach is used to reconstruct the state space, reducing noise from the aeroelastic time series. Direct analysis of reconstructed trajectories in the state space and the determination of Poincaré sections have been employed to investigate complex dynamics and chaotic patterns. With the reconstructed state spaces, qualitative analyses may be done, and the attractors evolutions with parametric variation are presented. Overall results reveal complex system dynamics associated with highly separated flow effects together with nonlinear coupling between aeroelastic modes. Bifurcations to the nonlinear aeroelastic system are observed for two investigations, that is, considering oscillations-induced aeroelastic evolutions with varying freestream speed, and aeroelastic evolutions at constant freestream speed and varying oscillations. Finally, Lyapunov exponent calculation is proceeded in order to infer on chaotic behavior. Poincaré mappings also suggest bifurcations and chaos, reinforced by the attainment of maximum positive Lyapunov exponents.

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Damage Assessment Using Hyperchaotic Excitation and State-Space Geometry Changes

ASME 2010 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, Volume 2 ◽

10.1115/smasis2010-3705 ◽

2010 ◽

Author(s):

Shahab Torkamani ◽

Eric A. Butcher ◽

Michael D. Todd ◽

Gyuhae Park

Keyword(s):

State Space ◽

Damage Assessment ◽

Chaotic Behavior ◽

High Sensitivity ◽

Space Geometry ◽

Monitoring Applications ◽

Increased Sensitivity ◽

Interrogation Technique ◽

Structural Excitation ◽

Higher Sensitivity

The current study explores the use of a steady-state hyperchaotic signal to probe a system for state-space geometry changes used for structural health monitoring applications. This is an extension of a previous chaotic interrogation technique approach that exploited the intrinsic high sensitivity of chaotic systems to subtle changes of the parameters. The enhanced technique proposed in this paper explores a novel structural excitation, namely a hyperchaotic excitation, which exhibits a chaotic behavior wherein at least two Lyapunov exponents are positive due to stretching of the phase space in multiple directions. A feature called average local attractor variance ratio (ALAVR), which is based on attractor geometry, is used to compare the geometry of a baseline attractor to a test attractor. The enhanced technique is applied to analytically and experimentally analyze the response of an 8-degree-of-freedom system to the hyperchaotic excitation for the sake of damage assessment. A comparison between the results obtained from current hyperchaotic excitation vs. a chaotic excitation highlights the higher sensitivity of the hyperchaotic excitation. Thus, hyperchaotic interrogation can be used as an alternative damage assessment technique when increased sensitivity to small amounts of damage is required.

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ON THE GENERATION OF CHAOTIC BEATS IN SIMPLE NONAUTONOMOUS CIRCUITS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127405013216 ◽

2005 ◽

Vol 15 (07) ◽

pp. 2247-2256 ◽

Cited By ~ 7

Author(s):

DONATO CAFAGNA ◽

GIUSEPPE GRASSI

Keyword(s):

State Space ◽

Time Domain ◽

Chaotic Behavior ◽

Second Order ◽

Nonlinear Circuit ◽

Early Results ◽

Chaotic Beats ◽

Sinusoidal Signals

In this Letter, attention is focused on the dynamics of a second-order nonlinear circuit driven by two sinusoidal signals. The early results reported herein show that the application of signals with slightly different frequencies enables the phenomenon of chaotic beats to be generated. In particular, the beats dynamics are analyzed both in time-domain and state-space, confirming the chaotic behavior of the proposed circuit.

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STATE SPACE RECONSTRUCTION USING EXTENDED STATE OBSERVERS TO CONTROL CHAOS IN A NONLINEAR PENDULUM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127405014490 ◽

2005 ◽

Vol 15 (12) ◽

pp. 4051-4063 ◽

Cited By ~ 4

Author(s):

FRANCISCO HEITOR I. PEREIRA-PINTO ◽

ARMANDO M. FERREIRA ◽

MARCELO A. SAVI

Keyword(s):

State Space ◽

Chaos Control ◽

Control Method ◽

Chaotic Behavior ◽

Unstable Periodic Orbits ◽

Extended State ◽

State Space Reconstruction ◽

State Observers ◽

Nonlinear Pendulum ◽

Delay Coordinates

Chaos control may be understood as the use of tiny perturbations for the stabilization of unstable periodic orbits embedded in a chaotic attractor. Since chaos may occur in many natural processes, the idea that chaotic behavior may be controlled by small perturbations of some physical parameter allows this kind of behavior to be desirable in different applications. In general, it is not necessary to have a mathematical model to achieve the control goal since all control parameters may be resolved from time series analysis. Therefore, state space reconstruction is an important task related to chaos control. This contribution analyzes chaos control performed using a semi-continuous method based on OGY approach and proposes the use of extended state observers in order to perform state space reconstruction. The use of extended state observers allows a direct application of the control method. Comparing with the delay coordinates method, extended state observers avoids the calculation of parametric changes related to delayed Poincaré sections that influence the system dynamics. The proposed procedure is applied in the control of chaos in a nonlinear pendulum, showing that it may be used to control chaos in mechanical systems.

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