On the numerical approximation of trajectories in the phase plane

doi:10.1016/S0096-3003(00)00148-X

Applied Mathematics and Computation

Volume 126, Issue 1, 15 February 2002, Pages 119-132

https://doi.org/10.1016/S0096-3003(00)00148-X Get rights and content

Abstract

In this paper, our work contributes to study the problem which consists in determining discrete trajectories which have relatively simple expressions by using Implicit Runge–Kutta methods and relatively simple computations.

Introduction

We consider the differential equation: $y^{′} (t)=Vy(t)+ϵg(t,y(t)); y(t)=(y_{1} (t),y_{2} (t))∈R^{2}; ϵ∈R, where g:R×R^{2} →R^{2} is a continuous non-linear map .$ This equation can generate an attractor. A simple attractor is for example a stable periodical solution. In more complicated situations, one has for example a strange attractor or a chaotic attractor. One has defined concepts to characterize these attractors, e.g., Lyapunov numbers, Hausdorff dimension, capacity dimension,…[3]. It is important to determine an efficient set T of approximations of trajectories in the phase space to estimate these quantities, also it is important to use efficient computations to obtain the elements of T. We think that discrete trajectories having relatively simple expressions can belong to T. It is interesting to use efficient or simple computations because, for example, they can generate a weak cost in the context of numerical computation in classical computers. A discrete trajectory having a relatively simple expression can present advantages, because it can be numerically computed (or it can be numerically evaluated) with a weak error in classical computers, or one can determine easily the magnitude of this error, it is well known that generally, the elementary operations (x×y,x+y,…) can generate errors. Also a relatively simple discrete trajectory can be exploited in the context of symbolic computation, because for example it can depend only on some parameters. In iterative processes which permit to obtain approximations of periodical solutions [1], [2], it is interesting to use also an efficient set of discrete trajectories. In this paper, our work contributes to study the problem which consists in determining discrete trajectories which can belong to T, more precisely we construct discrete trajectories which have relatively simple expressions, with IRK methods and with relatively simple computations. We do not search IRK methods which generate good approximations of solutions of (1.1), [6], [7], in particular our work does not consist to find IRK methods which generate a small phase error or a small amplification error [4]. Also, we construct IRK methods which have not necessarily good stability properties for example the A-stability, [5], but these methods can generate efficient computations, or relatively simple computations. Recall that IRK methods are suitable for stiff problems, and Eq. (1.1) can generate stiff problems.

In Section 2, we recall some properties of the IRK method having a number of internal stages s∈{2,3}, we present also some notations that we will use in the other sections.

In Section 3, we consider the case $V= 01 −w_{0}^{2} 0, w_{0} ≠0, ϵ=0.$ We construct with IRK methods, discrete trajectories having relatively simple expressions and which depend principally on parameters of the considered IRK method. We present also some properties of the obtained trajectories.

In Section 4, we give some elements which can contribute to generalize the result of Section 3 to the general two-dimensional case (1.1).

Section snippets

IRK methods

The following matrices define an IRK method: $A=(a_{ij})_{i,j=1,…,s}, s∈{2,3}; c=(c_{1},…,c_{s})^{T} ∈[0,1]^{s}; b=(b_{1},…,b_{s})^{T} .$ If s=2, then $A= c_{1}^{2} −2c_{1} c_{2} 2(c_{1} −c_{2}) c_{1}^{2} 2(c_{1} −c_{2}) c_{2}^{2} 2(c_{2} −c_{1}) c_{2}^{2} −2c_{1} c_{2} 2(c_{2} −c_{1}), b= 1−2c_{2} 2(c_{1} −c_{2}), 1−2c_{1} 2(c_{2} −c_{1})^{T} .$ If s=3, then $A= (13 c_{1}^{3} −(c_{2} +c_{3})c_{1}^{2} /2+c_{1} c_{2} c_{3}) (c_{1} −c_{2})(c_{1} −c_{3}) (13 c_{1}^{3} −(c_{1} +c_{3})c_{1}^{2} /2+c_{1}^{2} c_{3}) (c_{2} −c_{1})(c_{2} −c_{3}) (13 c_{1}^{3} −(c_{1} +c_{2})c_{1}^{2} /2+c_{1}^{2} c_{2}) (c_{3} −c_{1})(c_{3} −c_{2}) (13 c_{2}^{3} −(c_{2} +c_{3})c_{2}^{2} /2+c_{2}^{2} c_{3}) (c_{1} −c_{2})(c_{1} −c_{3}) (13 c_{2}^{3} −(c_{1} +c_{3})c_{2}^{2} /2+c_{1} c_{2} c_{3}) (c_{2} −c_{1})(c_{2} −c_{3}) (13 c_{2}^{3} −(c_{1} +c_{2})c_{2}^{2} /2+c_{1} c_{2}^{2}) (c_{3} −c_{1})(c_{3} −c_{2}) (13 c_{3}^{3} −(c_{2} +c_{3})c_{3}^{2} /2+c_{2} c_{3}^{2}) (c_{1} −c_{2})(c_{1} −c_{3})$

Discrete trajectories generated by IRK methods with $s ∈ {2,3}$

We consider case (1.2) $V= 01 −w_{0}^{2} 0, w_{0} ≠0, ϵ=0.$ Assume that $w_{0} ⩾3,$ then we can consider the following particular case of (2.1): $c_{1} =c= 1 w_{0}, c_{2} =3c= 3 w_{0}$ consequently, one has $A= 5 4w_{0} −1 4w_{0} 9 4w_{0} 3 4w_{0}, b= 6−w_{0} 4, w_{0} −2 4^{T} .$ In the following proposition, we are going to show that one can obtain a discrete trajectory having a relatively simple expression, when (3.3) is applied to (1.2) with an appropriate initial condition and an appropriate step h.

Proposition 1

Assume that $y_{0}^{(2)} =2hw_{0} y_{0}^{(1)}; 0<h< 150 .$ Then, when (3.3) is applied to (1.2), we obtain

Extension to the general case

The work in the general case $y^{′} (t)=Vy(t)+ϵg(t,(y_{1} (t),y_{1} (t))),$ $V= v_{11} v_{12} v_{21} v_{22}$ consists firstly to construct IRK methods which depend only on the matrix V and the parameter ϵ, and such that this dependence can be exploited efficaciously. Consequently, for example it is interesting to study the following case: $v_{11} v_{12} v_{21} v_{22} = 1/a_{11} 1/a_{12} 1/a_{21} 1/a_{22}; ϵ= 1 a_{11},$ where $A= a_{11} a_{12} a_{21} a_{22}$ defines an IRK method, $(a_{11} +a_{12} =c_{1}, a_{21} +a_{22} =c_{2})$ .

It is clear that choice (4.1) of the parameters V and ϵ is arbitrary, nevertheless the

Conclusion

The work of this paper is simply a first exploration of the problem which consists in determining with a judicious choice of:

•
IRK methods;
•
parameters h;
•
initial conditions (y_n₀⁽¹⁾,y_n₀⁽²⁾);

discrete trajectories having relatively simple expressions, and such that the computations which permit to obtain these discrete trajectories are relatively simple. It is interesting also to study this problem with IRK methods having higher order [7], or with diagonally multistage methods [6].

References (7)

W.-H. Steeb et al.
Chaos in limit cycle systems with external periodic excitations
Int. J. Non-Linear Mech.
(1987)
Z. Jackiewicz et al.
Variable stepsize diagonally implicit multistage integration methods for ordinary differential equations
Appl. Numer. Math.
(1995)
H. Kawakami
Bifurcations of periodic responses in forced dynamic nonlinear circuits, computation of bifurcation values of the system parameters
IEEE Trans. CAS, USA
(1984)

There are more references available in the full text version of this article.

Cited by (0)

View full text

Applied Mathematics and Computation

On the numerical approximation of trajectories in the phase plane

Abstract

Introduction

Section snippets

IRK methods

Discrete trajectories generated by IRK methods with s∈{2,3}

Extension to the general case

Conclusion

Chaos in limit cycle systems with external periodic excitations

Int. J. Non-Linear Mech.

Variable stepsize diagonally implicit multistage integration methods for ordinary differential equations

Appl. Numer. Math.

Bifurcations of periodic responses in forced dynamic nonlinear circuits, computation of bifurcation values of the system parameters

IEEE Trans. CAS, USA

Discrete trajectories generated by IRK methods with $s ∈ {2,3}$