scholarly journals Double Grazing Periodic Motions and Bifurcations in a Vibroimpact System with Bilateral Stops

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Qunhong Li ◽  
Limei Wei ◽  
Jieyan Tan ◽  
Jiezhen Xi
Keyword(s):  
Periodic Orbit ◽  
Periodic Motion ◽  
Poincaré Map ◽  
Complex Form ◽  
Original System ◽  
Existence Condition ◽  
Poincare Map ◽  
Periodic Motions ◽  
Continuous Transition ◽  
Vibroimpact System

The double grazing periodic motions and bifurcations are investigated for a two-degree-of-freedom vibroimpact system with symmetrical rigid stops in this paper. From the initial condition and periodicity, existence of the double grazing periodic motion of the system is discussed. Using the existence condition derived, a set of parameter values is found that generates a double grazing periodic motion in the considered system. By extending the discontinuity mapping of one constraint surface to that of two constraint surfaces, the Poincaré map of the vibroimpact system is constructed in the proximity of the grazing point of a double grazing periodic orbit, which has a more complex form than that of the single grazing periodic orbit. The grazing bifurcation of the system is analyzed through the Poincaré map with clearance as a bifurcation parameter. Numerical simulations show that there is a continuous transition from the chaotic band to a period-1 periodic motion, which is confirmed by the numerical simulation of the original system.

scholarly journals Constructing the Second Order Poincaré Map Based on the Hopf-Zero Unfolding Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Gen Ge ◽  
Wang Wei
Keyword(s):  
Homoclinic Orbit ◽  
Poincaré Map ◽  
Complex Form ◽  
Tubular Neighborhood ◽  
Second Order ◽  
Poincare Map ◽  
Third Order ◽  
Normal Form Theory ◽  
Dynamical Behaviors ◽  
Form Theory

We investigate the Shilnikov sense homoclinicity in a 3D system and consider the dynamical behaviors in vicinity of the principal homoclinic orbit emerging from a third order simplified system. It depends on the application of the simplest normal form theory and further evolution of the Hopf-zero singularity unfolding. For the Shilnikov sense homoclinic orbit, the complex form analytic expression is accomplished by using the power series of the manifolds surrounding the saddle-focus equilibrium. Then, the second order Poincaré map in a generally analytical style helps to portrait the double pulse dynamics existing in the tubular neighborhood of the principal homoclinic orbit.

An Estimation Method for Controlling Unstable Periodic Orbit Without Using Poincaré Map

2021 ◽  
Author(s):  
Satoshi Aoki ◽  
Takuji Kousaka ◽  
Shota Uchino ◽  
Daiki Hozumi ◽  
Hiroyuki Asahara
Keyword(s):  
Periodic Orbit ◽  
Poincaré Map ◽  
Estimation Method ◽  
Poincare Map ◽  
Unstable Periodic Orbit

Exponentially stabilizing continuous-time controllers for periodic orbits of hybrid systems: Application to bipedal locomotion with ground height variations

2015 ◽  
Vol 35 (8) ◽  
pp. 977-999 ◽  
Author(s):  
Kaveh Akbari Hamed ◽  
Brian G. Buss ◽  
Jessy W. Grizzle
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Hybrid System ◽  
Continuous Time ◽  
Poincaré Map ◽  
Bipedal Walking ◽  
Poincare Map ◽  
Parameterized Family ◽  
Set Up ◽  
Feedback Laws

This paper presents a systematic approach for the design of continuous-time controllers to robustly and exponentially stabilize periodic orbits of hybrid dynamical systems arising from bipedal walking. A parameterized family of continuous-time controllers is assumed so that (1) a periodic orbit is induced for the hybrid system, and (2) the orbit is invariant under the choice of controller parameters. Properties of the Poincaré map and its first- and second-order derivatives are used to translate the problem of exponential stabilization of the periodic orbit into a set of bilinear matrix inequalities (BMIs). A BMI optimization problem is then set up to tune the parameters of the continuous-time controller so that the Jacobian of the Poincaré map has its eigenvalues in the unit circle. It is also shown how robustness against uncertainty in the switching condition of the hybrid system can be incorporated into the design problem. The power of this approach is illustrated by finding robust and stabilizing continuous-time feedback laws for walking gaits of two underactuated 3D bipedal robots.

scholarly journals Poincaré Map and Periodic Solutions of First-Order Impulsive Differential Equations on Moebius Stripe

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yefeng He ◽  
Yepeng Xing
Keyword(s):  
Periodic Orbit ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Poincare Map ◽  
First Order ◽  
Existence And Stability ◽  
Stability And Bifurcations

This paper is mainly concerned with the existence, stability, and bifurcations of periodic solutions of a certain scalar impulsive differential equations on Moebius stripe. Some sufficient conditions are obtained to ensure the existence and stability of one-side periodic orbit and two-side periodic orbit of impulsive differential equations on Moebius stripe by employing displacement functions. Furthermore, double-periodic bifurcation is also studied by using Poincaré map.

scholarly journals The Poincaré Map of Randomly Perturbed Periodic Motion

2013 ◽  
Vol 23 (5) ◽  
pp. 835-861 ◽  
Author(s):  
Pawel Hitczenko ◽  
Georgi S. Medvedev
Keyword(s):  
Periodic Motion ◽  
Poincaré Map ◽  
Poincare Map ◽  
Perturbed Periodic

scholarly journals Periodic Motion and Transition of a Vibro‐Impact System with Multilevel Elastic Constraints

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jie Ding ◽  
Chao Wang ◽  
Wangcai Ding
Keyword(s):  
Periodic Motion ◽  
Simulation Method ◽  
Scientific Basis ◽  
Mapping Method ◽  
Periodic Motions ◽  
Coexistence Of Attractors ◽  
Vibroimpact System ◽  
The Stability ◽  
Elastic Constraints ◽  
Lyapunov Exponent Spectrum

In this paper, a single-degree-of-freedom vibroimpact system with multilevel elastic constraints is taken as the research object. By constructing the Poincaré map of the system and calculating the Lyapunov exponent spectrum of the system, the stability of the system is determined. Using the multiparameter collaborative numerical simulation method, the parameter domains of various periodic motions are determined, and the diversity and transition characteristics of periodic motions are revealed. At the same time, combined with the cell mapping method, the coexistence of attractors induced due to grazing bifurcation, saddle-node bifurcation, and boundary crisis is studied. Finally, the influence of system parameters on periodic motion distribution is analyzed, which provides a scientific basis for system parameter optimization.

Measurement of robustness for biped locomotion using a linearized Poincaré map

Robotica ◽  
1996 ◽  
Vol 14 (3) ◽  
pp. 253-259 ◽  
Author(s):  
M. -Y. Cheng ◽  
C. -S. Lin
Keyword(s):  
Periodic Motion ◽  
Poincaré Map ◽  
Computational Method ◽  
Biped Locomotion ◽  
Poincare Map ◽  
The Past ◽  
Highly Nonlinear ◽  
The Stability ◽  
Linearized Model ◽  
Continuous Time System

SUMMARYMany studies on control of dynamic biped walking have been done in the past two decades. While the biped dynamics is highly nonlinear, the stability analysis, if done, is usually based on a linearized model. The validity of the linearized model may become questionable if the walking involves states that are too far away from the operating point. In this paper, an approach for evaluating the robustness based on the linearized Poincare map is suggested and examined. The Poincare map is a useful tool to investigate the periodic motion of a dynamic system. Using the Poincare“ map, one can study an associated discrete time map instead of studying the continuous time system directly. Investigation of stability of a periodic motion can be reduced to the study of the stability of a fixed point of the Poincaré map. The computational method that results in a measurement for evaluating the robustness of biped locomotion is developed. Our simulation study has verified that the suggested measurement is a good indicator.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

BIFURCATION ANALYSIS OF CURRENT COUPLED BVP OSCILLATORS

2007 ◽  
Vol 17 (03) ◽  
pp. 837-850 ◽  
Author(s):  
SHIGEKI TSUJI ◽  
TETSUSHI UETA ◽  
HIROSHI KAWAKAMI
Keyword(s):  
Periodic Solutions ◽  
Poincaré Map ◽  
Dynamical Behavior ◽  
Coupled Systems ◽  
Poincare Map ◽  
Circuit Implementation ◽  
Coupling Structure ◽  
Bifurcation Phenomena ◽  
Junctional Coupling ◽  
Current Coupling

The Bonhöffer–van der Pol (BVP) oscillator is a simple circuit implementation describing neuronal dynamics. Lately the diffusive coupling structure of neurons attracts much attention since the existence of the gap-junctional coupling has been confirmed in the brain. Such coupling is easily realized by linear resistors for the circuit implementation, however, there are not enough investigations about diffusively coupled BVP oscillators, even a couple of BVP oscillators. We have considered several types of coupling structure between two BVP oscillators, and discussed their dynamical behavior in preceding works. In this paper, we treat a simple structure called current coupling and study their dynamical properties by the bifurcation theory. We investigate various bifurcation phenomena by computing some bifurcation diagrams in two cases, symmetrically and asymmetrically coupled systems. In symmetrically coupled systems, although all internal elements of two oscillators are the same, we obtain in-phase, anti-phase solution and some chaotic attractors. Moreover, we show that two quasi-periodic solutions disappear simultaneously by the homoclinic bifurcation on the Poincaré map, and that a large quasi-periodic solution is generated by the coalescence of these quasi-periodic solutions, but it disappears by the heteroclinic bifurcation on the Poincaré map. In the other case, we confirm the existence a conspicuous chaotic attractor in the laboratory experiments.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

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