Complicated Regular and Chaotic Motions of the Parametrically Excited Pendulum

2005 ◽  
Author(s):  
Eugene I. Butikov
Keyword(s):  
Computer Simulations ◽  
Parametric Resonance ◽  
Inverted Pendulum ◽  
Chaotic Behavior ◽  
Spectral Composition ◽  
Physical Explanation ◽  
Chaotic Motions ◽  
Upper Limit ◽  
Close Relationship ◽  
Subharmonic Resonances

Several new types of regular and chaotic behavior of the parametrically driven pendulum are discovered with the help of computer simulations. A simple physical explanation is suggested to the phenomenon of subharmonic resonances. The boundaries of these resonances in the parameter space and the spectral composition of corresponding stationary oscillations are determined theoretically and verified experimentally. A close relationship between the upper limit of stability of the dynamically stabilized inverted pendulum and parametric resonance of the non-inverted pendulum is established. Most of the newly discovered modes are still waiting a plausible physical explanation.

Lateral Dynamics of Walking-Like Mechanical Systems and Their Chaotic Behavior

2019 ◽  
Vol 29 (09) ◽  
pp. 1930024
Author(s):  
Sergej Čelikovský ◽  
Volodymyr Lynnyk
Keyword(s):  
Stability Region ◽  
Inverted Pendulum ◽  
Chaotic Behavior ◽  
Mechanical Systems ◽  
Small Time ◽  
Time Varying ◽  
External Forcing ◽  
Double Support ◽  
Lateral Dynamics ◽  
Detailed Mathematical Analysis

A detailed mathematical analysis of the two-dimensional hybrid model for the lateral dynamics of walking-like mechanical systems (the so-called hybrid inverted pendulum) is presented in this article. The chaotic behavior, when being externally harmonically perturbed, is demonstrated. Two rather exceptional features are analyzed. Firstly, the unperturbed undamped hybrid inverted pendulum behaves inside a certain stability region periodically and its respective frequencies range from zero (close to the boundary of that stability region) to infinity (close to its double support equilibrium). Secondly, the constant lateral forcing less than a certain threshold does not affect the periodic behavior of the hybrid inverted pendulum and preserves its equilibrium at the origin. The latter is due to the hybrid nature of the equilibrium at the origin, which exists only in the Filippov sense. It is actually a trivial example of the so-called pseudo-equilibrium [Kuznetsov et al., 2003]. Nevertheless, such an observation holds only for constant external forcing and even arbitrary small time-varying external forcing may destabilize the origin. As a matter of fact, one can observe many, possibly even infinitely many, distinct chaotic attractors for a single system when the forcing amplitude does not exceed the mentioned threshold. Moreover, some general properties of the hybrid inverted pendulum are characterized through its topological equivalence to the classical pendulum. Extensive numerical experiments demonstrate the chaotic behavior of the harmonically perturbed hybrid inverted pendulum.

CHAOS IN THE REORIENTATION PROCESS OF A DUAL-SPIN SPACECRAFT WITH TIME-DEPENDENT MOMENTS OF INERTIA

2000 ◽  
Vol 10 (05) ◽  
pp. 997-1018 ◽  
Author(s):  
M. IÑARREA ◽  
V. LANCHARES
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Center Of Mass ◽  
Melnikov Method ◽  
Moments Of Inertia ◽  
Suitable Parameter ◽  
Nutation Angle ◽  
Principal Axes ◽  
Reorientation Process ◽  
Subharmonic Resonances

We study the spin-up dynamics of a dual-spin spacecraft containing one axisymmetric rotor which is parallel to one of the principal axes of the spacecraft. It will be supposed that one of the moments of inertia of the platform is a periodic function of time and that the center of mass of the spacecraft is not modified. Under these assumptions, it is shown that in the absence of external torques and spinning rotors the system possesses chaotic behavior in the sense that it exhibits Smale's horseshoes. We prove this statement by means of the Melnikov method. The presence of chaotic behavior results in a random spin-up operation. This randomness is visualized by means of maps of the initial conditions with final nutation angle close to zero. This phenomenon is well described by a suitable parameter that measures the amount of randomness of the process. Finally, we relate this parameter with the Melnikov function in the absence of the spinning rotor and with the presence of subharmonic resonances.

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.

Subharmonic Resonances and Chaotic Motions of a Bilinear Oscillator

1983 ◽  
Vol 31 (3) ◽  
pp. 207-234 ◽  
Author(s):  
J. M. T. THOMPSON ◽  
A. R. BOKAIAN ◽  
R. GHAFFARI
Keyword(s):  
Chaotic Motions ◽  
Subharmonic Resonances

scholarly journals Traveling Wave Solutions and Chaotic Motions for a Perturbed Nonlinear Schrödinger Equation with Power-Law Nonlinearity and Higher-Order Dispersions

2021 ◽  
Author(s):  
Mati Youssoufa ◽  
Ousmanou Dafounansou ◽  
Camus Gaston Latchio Tiofack ◽  
Alidou Mohamadou
Keyword(s):  
Power Law ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Chaotic Behavior ◽  
Traveling Wave Solutions ◽  
Chaotic Motions ◽  
Auxiliary Equation Method ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Quasiperiodic Route To Chaos

This chapter aims to study and solve the perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano-optical fiber, based upon different methods such as the auxiliary equation method, the Stuart and DiPrima’s stability analysis method, and the bifurcation theory. The existence of the traveling wave solutions is discussed, and their stability properties are investigated through the modulational stability gain spectra. Moreover, the development of the chaotic motions for the systems is pointed out via the bifurcation theory. Taking into account an external periodic perturbation, we have analyzed the chaotic behavior of traveling waves through quasiperiodic route to chaos.

scholarly journals Identification and Machine Learning Prediction of Nonlinear Behavior in a Robotic Arm System

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1445
Author(s):  
Cheng-Chi Wang ◽  
Yong-Quan Zhu
Keyword(s):  
Neural Network ◽  
Lyapunov Exponent ◽  
Chaotic Motion ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Robotic Arm ◽  
Maximum Lyapunov Exponent ◽  
Double Pendulum ◽  
Chaotic Motions ◽  
Robotic Arms

In this study, the subject of investigation was the dynamic double pendulum crank mechanism used in a robotic arm. The arm is driven by a DC motor though the crank system and connected to a fixed side with a mount that includes a single spring and damping. Robotic arms are now widely used in industry, and the requirements for accuracy are stringent. There are many factors that can cause the induction of nonlinear or asymmetric behavior and even excite chaotic motion. In this study, bifurcation diagrams were used to analyze the dynamic response, including stable symmetric orbits and periodic and chaotic motions of the system under different damping and stiffness parameters. Behavior under different parameters was analyzed and verified by phase portraits, the maximum Lyapunov exponent, and Poincaré mapping. Firstly, to distinguish instability in the system, phase portraits and Poincaré maps were used for the identification of individual images, and the maximum Lyapunov exponents were used for prediction. GoogLeNet and ResNet-50 were used for image identification, and the results were compared using a convolutional neural network (CNN). This widens the convolutional layer and expands pooling to reduce network training time and thickening of the image; this deepens the network and strengthens performance. Secondly, the maximum Lyapunov exponent was used as the key index for the indication of chaos. Gaussian process regression (GPR) and the back propagation neural network (BPNN) were used with different amounts of data to quickly predict the maximum Lyapunov exponent under different parameters. The main finding of this study was that chaotic behavior occurs in the robotic arm system and can be more efficiently identified by ResNet-50 than by GoogLeNet; this was especially true for Poincaré map diagnosis. The results of GPR and BPNN model training on the three types of data show that GPR had a smaller error value, and the GPR-21 × 21 model was similar to the BPNN-51 × 51 model in terms of error and determination coefficient, showing that GPR prediction was better than that of BPNN. The results of this study allow the formation of a highly accurate prediction and identification model system for nonlinear and chaotic motion in robotic arms.

Chaotic Behavior in the Double Pendulum Under Parametric Resonance

2016 ◽  
Author(s):  
Rafael H. Avanço ◽  
Helio A. Navarro ◽  
Airton Nabarrete ◽  
José M. Balthazar ◽  
Angelo Marcelo Tusset
Keyword(s):  
Friction Coefficient ◽  
Parametric Resonance ◽  
Chaotic Behavior ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Double Pendulum ◽  
Bifurcation Diagrams ◽  
Harmonic Motion ◽  
Parametric Pendulum

In literature, the classic parametrically excited pendulum is vastly studied. It consists of a pendulum vertically displaced with a harmonic motion in the support while it oscillates. The chaos in this mechanism may appear depending on the frequency and amplitude of excitation in superharmonic and subharmonic resonance. The double pendulum is also well analyzed in literature, but not under parametric excitation. Therefore, this is the novelty in the present paper. The present analysis considers a double pendulum under a harmonic excitation following the same idea performed previously for a single pendulum. The results are obtained based on methods, such as, phase portraits, Poincaré sections and bifurcation diagrams. The 0–1 tests analyze the presence of chaos while the parameters are varied. The dimensionless parameters take into account the excitation frequency and amplitude as mentioned for the classic parametric pendulum. In this case, we have the particular characteristic that the two pendulums have the same length, the same mass and the same friction coefficient in the joints. The types of motion observed include fixed points, oscillations, rotations and chaos. Results also demonstrated that there was a self-synchronization between these pendulums in ideal excitation.

scholarly journals Harmonic Balance Method for Chaotic Dynamics in Fractional-Order Rössler Toroidal System

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Huijian Zhu
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Harmonic Balance ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Chaotic Motions ◽  
Behavior Based

This paper deals with the problem of determining the conditions under which fractional order Rössler toroidal system can give rise to chaotic behavior. Based on the harmonic balance method, four detailed steps are presented for predicting the existence and the location of chaotic motions. Numerical simulations are performed to verify the theoretical analysis by straightforward computations.

scholarly journals Chaos Control of a Fractional-Order Financial System

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
Mohammed Salah Abd-Elouahab ◽  
Nasr-Eddine Hamri ◽  
Junwei Wang
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Financial System ◽  
Linearization Method ◽  
Nonlinear Feedback Control ◽  
Nonlinear Feedback ◽  
Chaotic Motions ◽  
Control Scheme ◽  
Fractional System

Fractional-order financial system introduced by W.-C. Chen (2008) displays chaotic motions at order less than 3. In this paper we have extended the nonlinear feedback control in ODE systems to fractional-order systems, in order to eliminate the chaotic behavior. The results are proved analytically by applying the Lyapunov linearization method and stability condition for fractional system. Moreover numerical simulations are shown to verify the effectiveness of the proposed control scheme.

scholarly journals Physical explanation for the galaxy distribution on the (λR, ε) and (V/σ, ε) diagrams or for the limit on orbital anisotropy

2020 ◽  
Vol 500 (1) ◽  
pp. L27-L31
Author(s):  
Bitao Wang ◽  
Michele Cappellari ◽  
Yingjie Peng
Keyword(s):  
Early Type ◽  
Equilibrium Solutions ◽  
Physical Explanation ◽  
Anisotropic Models ◽  
Galaxy Distribution ◽  
Upper Limit ◽  
The Galaxy ◽  
Radial Anisotropy ◽  
Axisymmetric Equilibrium ◽  
Variable Anisotropy

ABSTRACT In the (λR, ε) and (V/σ, ε) diagrams for characterizing dynamical states, the fast-rotator galaxies (both early type and spirals) are distributed within a well-defined leaf-shaped envelope. This was explained as due to an upper limit to the orbital anisotropy increasing with galaxy intrinsic flattening. However, a physical explanation for this empirical trend was missing. Here, we construct Jeans Anisotropic Models (JAM), with either cylindrically or spherically aligned velocity ellipsoid (two extreme assumptions), and each with either spatially constant or variable anisotropy. We use JAM to build mock samples of axisymmetric galaxies, assuming on average an oblate shape for the velocity ellipsoid (as required to reproduce the rotation of real galaxies), and limiting the radial anisotropy β to the range allowed by physical solutions. We find that all four mock samples naturally predict the observed galaxy distribution on the (λR, ε) and (V/σ, ε) diagrams, without further assumptions. Given the similarity of the results from quite different models, we conclude that the empirical anisotropy upper limit in real galaxies, and the corresponding observed distributions in the (λR, ε) and (V/σ, ε) diagrams, are due to the lack of physical axisymmetric equilibrium solutions at high β anisotropy when the velocity ellipsoid is close to oblate.

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