scholarly journals STUDYING THE BASIN OF CONVERGENCE OF METHODS FOR COMPUTING PERIODIC ORBITS

2011 ◽  
Vol 21 (08) ◽  
pp. 2079-2106 ◽  
Author(s):  
MICHAEL G. EPITROPAKIS ◽  
MICHAEL N. VRAHATIS
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Newton's Method ◽  
Poincaré Map ◽  
Specific Problem ◽  
Poincare Map ◽  
Robust Methods ◽  
Surface Of Section ◽  
Cubic Equation ◽  
Nonlinear Mappings

Starting from the well-known Newton's fractal which is formed by the basin of convergence of Newton's method applied to a cubic equation in one variable in the field ℂ, we were able to find methods for which the corresponding basins of convergence do not exhibit a fractal-like structure. Using this approach we are able to distinguish reliable and robust methods for tackling a specific problem. Also, our approach is illustrated here for methods for computing periodic orbits of nonlinear mappings as well as for fixed points of the Poincaré map on a surface of section.

Poincaré Map-Based Continuation of Periodic Orbits in Dynamic Discontinuous and Hysteretic Systems

1999 ◽  
Author(s):  
Walter Lacarbonara ◽  
Fabrizio Vestroni ◽  
Danilo Capecchi
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Poincaré Map ◽  
Period Doubling ◽  
Path Following ◽  
Return Time ◽  
Poincare Map ◽  
Response Curves ◽  
Hysteretic Systems

Abstract A numerical algorithm is proposed to compute variation of periodic solutions and their codimension-one bifurcations in discontinuous and hysteretic systems in the relevant control parameter space. For dynamic systems with discontinuities and hysteresis, some components of the associated vector fields are nondifferentiable. Therefore, one cannot resort on classical numerical tools based on the evaluation of the Jacobian of the vector field for path-following analyses. Using the pertinent state space, periodic orbits are sought as the fixed points of a Poincaré map based on an appropriate return time. The Jacobian of the map is computed numerically via either a forward or a central finite-difference scheme and a Newton-Raphson procedure is used to determine the fixed points. The continuation scheme is a pseudo-arclength algorithm based on arclength parameterization. The eigenvalues of the Jacobian of the map — Floquet multipliers — are computed to ascertain the stability of the periodic orbits and the associated bifurcations. The procedure is used to construct frequency-response curves of a bilinear, a Masing-type, and a Bouc-Wen oscillator in the primary and superharmonic frequency ranges for various excitation levels. The proposed numerical strategy proves to be very effective in capturing a rich class of solutions and bifurcations — including jump phenomena, pitchfork (symmetry-breaking), and period-doubling.

Computing the Number of Fixed Joints on Poincare Map in Nonlinear Mathieu Equation

1993 ◽  
Author(s):  
Hisato Fujisaka ◽  
Chikara Sato
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Numerical Method ◽  
Topological Degree ◽  
Poincaré Map ◽  
Mathieu Equation ◽  
Time Varying ◽  
Poincare Map ◽  
Newton's Iteration ◽  
Combined Use

Abstract A numerical method is presented to compute the number of fixed points of Poincare maps in ordinary differential equations including time varying equations. The method’s fundamental is to construct a map whose topological degree equals to the number of fixed points of a Poincare map on a given domain of Poincare section. Consequently, the computation procedure is simply computing the topological degree of the map. The combined use of this method and Newton’s iteration gives the locations of all the fixed points in the domain.

VALIDATED STUDY OF THE EXISTENCE OF SHORT CYCLES FOR CHAOTIC SYSTEMS USING SYMBOLIC DYNAMICS AND INTERVAL TOOLS

2011 ◽  
Vol 21 (02) ◽  
pp. 551-563 ◽  
Author(s):  
ZBIGNIEW GALIAS ◽  
WARWICK TUCKER
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Poincaré Map ◽  
Lorenz System ◽  
Interval Methods ◽  
Theoretical Argument ◽  
Poincare Map ◽  
Distinct Symbol ◽  
The Lorenz System ◽  
Symbol Sequences

We show that, for a certain class of systems, the problem of establishing the existence of periodic orbits can be successfully studied by a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate positions of periodic points, and the existence of periodic orbits in a neighborhood of these approximations is proved using an interval operator. As an example, the Lorenz system is studied; a theoretical argument is used to prove that each periodic orbit has a distinct symbol sequence. All periodic orbits with the period p ≤ 16 of the Poincaré map associated with the Lorenz system are found. Estimates of the topological entropy of the Poincaré map and the flow, based on the number and flow-times of short periodic orbits, are calculated. Finally, we establish the existence of several long periodic orbits with specific symbol sequences.

Dynamic Output Controllers for Exponential Stabilization of Periodic Orbits for Multidomain Hybrid Models of Robotic Locomotion

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
Kaveh Akbari Hamed ◽  
Bita Safaee ◽  
Robert D. Gregg
Keyword(s):  
Periodic Orbits ◽  
Output Feedback ◽  
Poincaré Map ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Dynamic Output Feedback ◽  
Poincare Map ◽  
Feedback Controllers ◽  
Robotic Locomotion ◽  
Dynamic Output

Abstract The primary goal of this paper is to develop an analytical framework to systematically design dynamic output feedback controllers that exponentially stabilize multidomain periodic orbits for hybrid dynamical models of robotic locomotion. We present a class of parameterized dynamic output feedback controllers such that (1) a multidomain periodic orbit is induced for the closed-loop system and (2) the orbit is invariant under the change of the controller parameters. The properties of the Poincaré map are investigated to show that the Jacobian linearization of the Poincaré map around the fixed point takes a triangular form. This demonstrates the nonlinear separation principle for hybrid periodic orbits. We then employ an iterative algorithm based on a sequence of optimization problems involving bilinear matrix inequalities to tune the controller parameters. A set of sufficient conditions for the convergence of the algorithm to stabilizing parameters is presented. Full-state stability and stability modulo yaw under dynamic output feedback control are addressed. The power of the analytical approach is ultimately demonstrated through designing a nonlinear dynamic output feedback controller for walking of a three-dimensional (3D) humanoid robot with 18 state variables and 325 controller parameters.

Stabilizing periodic orbits of chaotic systems using fuzzy control of Poincaré map

2008 ◽  
Vol 36 (3) ◽  
pp. 682-693 ◽  
Author(s):  
Mohammad Bonakdar ◽  
Mostafa Samadi ◽  
Hassan Salarieh ◽  
Aria Alasty
Keyword(s):  
Fuzzy Control ◽  
Periodic Orbits ◽  
Chaotic Systems ◽  
Poincaré Map ◽  
Poincare Map

scholarly journals Dynamics of Symmetric Dynamical Systems with Delayed Switching

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1111-1140 ◽  
Author(s):  
J. Sieber ◽  
P. Kowalczyk ◽  
S.J. Hogan ◽  
M. Di Bernardo
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Poincaré Map ◽  
Invariant Tori ◽  
Poincare Map ◽  
Single Degree Of Freedom ◽  
Finite Dimensional ◽  
Symmetric Periodic Orbits ◽  
Full Reflection

We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value, so-called event collisions occur. This paper classifies and analyzes event collisions, a special type of discontinuity-induced bifurcations, for periodic orbits. Our focus is on event collisions of symmetric periodic orbits in systems with full reflection symmetry, a symmetry that is prevalent in applications. We derive an implicit expression for the Poincaré map near the colliding periodic orbit. The Poincaré map is piecewise smooth, finite-dimensional, and changes the dimension of its image at the collision. In the second part of the paper we apply this general result to the class of unstable linear single-degree-of-freedom oscillators where we detect and continue numerically collisions of invariant tori. Moreover, we observe that attracting closed invariant polygons emerge at the torus collision.

Bifurcational precedences in the braids of periodic orbits of spiral 3-shoes in driven oscillators

1992 ◽  
Vol 438 (1904) ◽  
pp. 545-569 ◽  
Keyword(s):  
Differential Equation ◽  
Periodic Orbits ◽  
Poincaré Map ◽  
Topological Conjugacy ◽  
Potential Well ◽  
Poincare Map ◽  
Periodic Forcing ◽  
Smooth Potential ◽  
Subharmonic Bifurcations ◽  
The Creation

We conjecture the existence of a 3-striped spiral horseshoe under one iterate of the Poincaré map that arises in the analysis of a class of driven oscillators. These systems represent escape from a smooth potential well under periodic forcing. We assume topological conjugacy between the flow of the differential equation and an idealized suspension of a spiral 3-shoe, and deduce bifurcational precedences by consideration of intertwining in the braid of periodic orbits. We argue that many significant subharmonic bifurcations observed in such systems can then be understood in terms of the creation of this 3-shoe.

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.

BIFURCATION OF QUASI-PERIODIC OSCILLATIONS IN MUTUALLY COUPLED HARD-TYPE OSCILLATORS: DEMONSTRATION OF UNSTABLE QUASI-PERIODIC ORBITS

2012 ◽  
Vol 22 (06) ◽  
pp. 1230022 ◽  
Author(s):  
KYOHEI KAMIYAMA ◽  
MOTOMASA KOMURO ◽  
TETSURO ENDO
Keyword(s):  
Periodic Orbits ◽  
Poincaré Map ◽  
Averaging Method ◽  
Computer Algorithm ◽  
Bisection Method ◽  
Poincare Map ◽  
Periodic Oscillations ◽  
Closed Curves ◽  
Hard Type

In this paper, we obtain bifurcations of quasi-periodic orbits occuring in mutually coupled hard-type oscillators by using our recently developed computer algorithm to directly determine the unstable quasi-periodic orbits. So far, there is no computer algorithm to draw unstable invariant closed curves on a Poincare map representing quasi-periodic orbits. Recently, we developed a new algorithm to draw unstable invariant closed curves by using the bisection method. The results of this new algorithm are compared with the previously obtained averaging method results. Several new results are found, which could not be clarified by the averaging method.

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