Numerical Continuation Method for Nonlinear System of Scalar and Functional Equations

2020 ◽  
Vol 60 (3) ◽  
pp. 404-410
Author(s):  
G. V. Paradezhenko ◽  
N. B. Melnikov ◽  
B. I. Reser
Keyword(s):  
Nonlinear System ◽  
Functional Equations ◽  
Continuation Method ◽  
Numerical Continuation

Numerical Solutions of Polynomial Equations for the Eight-Position Synthesis of the Cylindrical-Spherical Dyad

2012 ◽  
Author(s):  
Chintien Huang ◽  
Chenning Hung ◽  
Kuenming Tien
Keyword(s):  
Rigid Body ◽  
Numerical Solutions ◽  
Continuation Method ◽  
Quadratic Equation ◽  
Geometric Constraints ◽  
Numerical Continuation ◽  
Polynomial Equations ◽  
Quartic Polynomial ◽  
Solutions Of Equations ◽  
Position Synthesis

This paper investigates the numerical solutions of equations for the eight-position rigid-body guidance of the cylindrical-spherical (C-S) dyad. We seek to determine the number of finite solutions by using the numerical continuation method. We derive the design equations using the geometric constraints of the C-S dyad and obtain seven quartic polynomial equations and one quadratic equation. We then solve the system of equations by using the software package Bertini. After examining various specifications, including those with random complex numbers, we conclude that there are 804 finite solutions of the C-S dyad for guiding a body through eight prescribed positions. When designing spatial dyads for rigid-body guidance, the C-S dyad is one of the four dyads that result in systems of equal numbers of equations and unknowns if the maximum number of allowable positions is specified. The numbers of finite solutions in the syntheses of the other three dyads have been obtained previously, and this paper provides the computational kinematic result of the last unsolved problem, the eight-position synthesis of the C-S dyad.

Constrained Optimization Shooting Method for Predicting the Periodic Solutions of Nonlinear System

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
Haitao Liao
Keyword(s):  
Nonlinear System ◽  
Shooting Method ◽  
Continuation Method ◽  
Inequality Constraints ◽  
Original Method ◽  
Maximization Problem ◽  
Numerical Examples ◽  
Equality And Inequality Constraints ◽  
Parameter Continuation ◽  
Nonlinear Equality

An original method for calculating the maximum vibration amplitude of the periodic solution of a nonlinear system is presented. The problem of determining the worst maximum vibration is transformed into a nonlinear optimization problem. The shooting method and the Floquet theory are selected to construct the general nonlinear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the effectiveness and ability of the proposed approach are illustrated through two numerical examples. Numerical examples show that the proposed method can give results with higher accuracy as compared with numerical results obtained by a parameter continuation method and the ability of the present method is also demonstrated.

scholarly journals A sticky situation

2017 ◽  
Vol 820 ◽  
pp. 1-4
Author(s):  
P.-T. Brun
Keyword(s):  
Phase Diagram ◽  
Continuation Method ◽  
Helical Structure ◽  
Numerical Continuation ◽  
Complete Phase Diagram ◽  
Observed Behaviour ◽  
Scaling Argument

The whirling helical structure obtained when pouring honey onto toast may seem like an easy enough problem to solve at breakfast. Specifically, one would hope that a quick back-of-the-envelope scaling argument would help rationalize the observed behaviour and predict the coiling frequency. Not quite: multiple forces come into play, both in the part of the flow stretched by gravity and in the coil itself, which buckles and bends like a rope. In fact, the resulting abundance of regimes requires the careful numerical continuation method reported by Ribe (J. Fluid Mech., vol. 812, 2017, R2) to build a complete phase diagram of the problem and untangle this sticky situation.

Numerical Continuation in a Physical Experiment: Investigation of a Nonlinear Energy Harvester

2009 ◽  
Author(s):  
David A. W. Barton ◽  
Stephen G. Burrow
Keyword(s):  
Nonlinear System ◽  
Periodic Orbits ◽  
Energy Harvester ◽  
Vibrational Energy ◽  
Electrical Energy ◽  
Physical Experiment ◽  
Numerical Continuation ◽  
Unstable Periodic Orbits ◽  
Non Invasive ◽  
Nonlinear Energy

In this paper we demonstrate the use of numerical continuation within a physical experiment: a nonlinear energy harvester, which is used to convert vibrational energy into usable electrical energy. To continue a branch of periodic orbits through a saddle-node bifurcation and along the associated branch of unstable periodic orbits, a modified time-delay controller is used. At each step in the continuation the pseudo-arclength equation is appended to a set of equations that ensure that the controller is non-invasive. The resulting nonlinear system is solved using a quasi-Newton iteration, where each evaluation of the nonlinear system requires changing the excitation parameters of the experiment and measuring the response. We present the continuation results for the energy harvester in a number of different configurations.

scholarly journals Numerical bifurcation and stability analysis of an predator-prey system with generalized Holling type III functional response

2018 ◽  
Vol 36 (3) ◽  
pp. 89-102
Author(s):  
Z. Lajmiri ◽  
Reza Khoshsiar Ghaziani ◽  
M. Guasemi
Keyword(s):  
Functional Response ◽  
Bifurcation Analysis ◽  
Continuation Method ◽  
Numerical Continuation ◽  
Hopf Bifurcation Point ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Bifurcation Phenomena ◽  
Stable Periodic ◽  
Numerical Bifurcation

We perform a bifurcation analysis of a predator-prey model with Holling functional response. The analysis is carried out both analytically and numerically. We use dynamical toolbox MATCONT to perform numerical bifurcation analysis. Our bifurcation analysis of the model indicates that it exhibits numerous types of bifurcation phenomena, including fold, subcritical Hopf, cusp, Bogdanov-Takens. By starting from a Hopf bifurcation point, we approximate limit cycles which are obtained, step by step, using numerical continuation method and compute orbitally asymptotically stable periodic orbits.

Numerical Continuation in a Physical Experiment: Investigation of a Nonlinear Energy Harvester

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
David A. W. Barton ◽  
Stephen G. Burrow
Keyword(s):  
Nonlinear System ◽  
Periodic Orbits ◽  
Energy Harvester ◽  
Vibrational Energy ◽  
Electrical Energy ◽  
Physical Experiment ◽  
Numerical Continuation ◽  
Unstable Periodic Orbits ◽  
Quasi Newton ◽  
Nonlinear Energy

In this paper, we demonstrate the use of control-based continuation within a physical experiment: a nonlinear energy harvester, which is used to convert vibrational energy into usable electrical energy. By employing the methodology of Sieber et al. (2008, “Experimental Continuation of Periodic Orbits Through a Fold,” Phys. Rev. Lett., 100(24), p. 244101), a branch of periodic orbits is continued through a saddle-node bifurcation and along the associated branch of unstable periodic orbits using a modified time-delay controller. At each step in the continuation, the pseudo-arclength equation is appended to a set of equations that ensure that the controller is noninvasive. The resulting nonlinear system is solved using a quasi-Newton iteration, where each evaluation of the nonlinear system requires changing the excitation parameters of the experiment and measuring the response. We present the continuation results for the energy harvester in a number of different configurations.

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