A piecewise linear approximation method for

the evaluation of Lyapunov Exponents of

polynomial nonlinear systems

B. Cannas, F. Pisano

Electric and Electronic Engineering Department, University of Cagliari, Cagliari, Italy

Abstract

Lyapunov exponents of a dynamical system give information about its long-term evolution.

Exponents estimation is not an easy task; it is computationally costly and, in presence of

chaotic dynamics, it exhibits numerical difficulties. In a previous paper, the authors pro-

posed an algorithm for Lyapunov exponents estimation in piecewise linear systems that

strongly reduces the computational time. In this paper, the algorithm is applied also to cha-

otic systems with polynomial nonlinearity. Firstly, a suitable piecewise linear approxima-

tion for the polynomial nonlinear function is evaluated by means of a Multi-Layer Percep-

tron neural network with linear and saturating linear transfer functions. Then, the linearity

of the new state equation and of the variational equation, obtained resorting to the piece-

wise linear approximation of the nonlinear function, are exploited to accurately evaluate

Lyapunov exponents of the approximated system with a reduced execution time.

1 Introduction

Lyapunov exponents play a crucial role on description of dynamical systems, because of

their connection with the presence of chaotic dynamics. Lyapunov exponents are numerical

parameters which quantify sensibility of nonlinear dynamical systems to initial conditions.

In particular, they estimate the rate of long term divergence of trajectories starting from

nearby initial conditions. A positive Lyapunov exponent may be taken as one of the signa-

tures of chaos.

Generally, it is not possible to analytically calculate Lyapunov exponents, thus mostly nu-

merical techniques based on variational equation integration are used. Each step requires

the integration of the original differential equation but, in presence of chaotic dynamics, the

estimation of Lyapunov exponents has many numerical difficulties related to the divergence

of trajectories starting from nearby initial conditions.

In [1], an algorithm which optimizes Lyapunov exponents estimation in piecewise linear

systems has been developed. The algorithm exploits the linearity of the state and variational

equation to solve them in closed-form strongly reducing the computational time without

compromising the precision of the calculation.

2

In this paper, the piecewise linear approximation method for the evaluation of Lyapunov

exponents presented in [1] is applied also to chaotic systems with polynomial nonlinearity.

2 The PWL algorithm

The methodology for the estimation of the Lyapunov exponent (LE) spectrum for piecewise

linear systems proposed in [1] is briefly described. It exploits the linearity of the state and

variational equation to solve them in closed-form strongly reducing the computation time.

The algorithm is handy from a computational point of view and rather easy to implement.

Let us consider a n-th order continuous-time dynamical system defined by the state equa-

tion

tfdtd xx (2.1)

The solution is denoted by φt(x0,t0)n, where x0 is the state at initial time t0. Let us con-

sider the variation of the solution when the initial conditions are perturbed,

Y(t)=φt(x0,t0)/x0, and the Jacobian matrix A(t)=f/x. The equation

IYYΑY )(),()( 0tttdtd (2.2)

is called variational equation. The Lyapunov exponents i, i=1,...n, for regular systems, can

be evaluated as

tdtt

it

i ln1

lim0

(2.3)

where di(t) are the square root of the eigenvalues of YT(t)Y(t). In presence of at least one

positive LE, the elements of the matrix YTY diverge [1]. Moreover, YTY is ill-conditioned,

because its column vectors tend to line up along the local direction of most rapid growth.

These problems can be overcome considering the growth rate of n-dimensional volumes [2-

4]. These volumes can be computed by QR method applied on integration of system (2.2)

[5].

In practical applications, it is important to know how sensitive the result is, before its com-

putation is attempted. Thus, for a small perturbation E in matrix A, Relative Condition

Number (RCN) has to be computed

AAEA

AE

AA eeeRCN

suplim)(

0. (2.4)

For PWL systems, the variational equation in each region j is Ẏ(t)=AjY(t), and its solution is

given by Y(tk)=CjY(tk-1). Thus, if at time tk-1 the trajectory lies in region j, the variation of

the solution in region j caused by the perturbation xk-1 is

3

xexp(Aj(t–tk-1))xk-1. (2.5)

Therefore,

001021211 teeet

tttttt

kjkkkjkkkj YY

AAA (2.6)

where j(k–1) identifies the region at the time instant tk-1.

The evaluation of the matrix exponential and the choice of the crossing time tc are crucial

for the accuracy of the algorithm. The matrix exponential is evaluated using Padé approxi-

mation implemented in Matlab and RCN is computed using the Matrix Function Toolbox

[6]. The sampling time is fixed and it is chosen in order to guarantee a good RCN in the

evaluation of the matrix exponential.

Since the sampling time is fixed, the transition between two regions will not exactly occur

at the time tk, because in tk the trajectory has already passed over the hyperplane separating

the two regions. If the distance of x(tk) from the separating hyperplane is less than the dis-

tance of x(tk-1), tc is set to tk otherwise to tk-1. If, at the time instant tk-1, the trajectory lies in

the region j, the solution of variational equation at time tk is

1111

kjkk

tt

k ttetet jkkj YCYYYAA

(2.7)

It is worth noting that this procedure do not require additional exponential matrix calcula-

tions. Moreover, since the integration step is fixed, the matrix exponential Cj is evaluated,

for the system state equation, only s times, where s is the number of possible regions, re-

quiring a limited additional computational cost and strongly reducing the computational

cost of the algorithm. Moreover, the use of (2.7) reduces round-off errors.

3 Neural network PWL approximation

Piecewise-linear or, more correctly, piecewise-affine models can approximate any nonline-

ar function with arbitrary accuracy, provided that the function domain is partitioned in a

large enough number of subdomains where the function is approximated by an affine sys-

tem.

A suitable piecewise linear approximation f(x) for the nonlinear function is evaluated by

means of a neural network with saturating linear transfer functions in the hidden layer and a

linear output layer.

For example, in fig. 1 the structure of a neural network approximating a nonlinear odd func-

tion passing through the origin is shown. The number of hidden neurons of the neural net-

work represents the number of knee points in the positive subdomain of the piecewise line-

ar approximation. After the network training, it is possible to extrapolate the parameters of

the PWL approximation, the knee points γi and the slopes mi.

4

Generally speaking, neural networks with several hidden networks better approximate the

desired behavior. Nevertheless, the transitions between two regions during the LE evalua-

tion penalizes the precision of the algorithm. Thus, a compromise between the two re-

quirements is made in order to choose the number of hidden neurons.

Fig. 1. Neural network structure for approximating nonlinear odd functions

4 Results

The linearity of the new state equation and of the variational equation, obtained resorting to

the piecewise linear approximation of the nonlinear function, are exploited to accurately

evaluate Lyapunov exponents of the approximated system with a reduced execution time.

The technique has been applied to the Chua’s circuit with cubic nonlinearity [7], with state

equations:

ydt

dz

zyxdt

dy

xcxydt

dx

3

(4.1)

When α = 10, β = 16 and c = –0.143 the system (4.1) has a chaotic behaviour, showing a

double scroll attractor (fig. 2).

The cubic non linearity f(x) = x3 has been approximated with a PWL function fa(x), using

the neural network shown in fig.1. In table 1 the mean squared error between f(x) and the

PWL approximation, varying the number of regions M in the positive subdomain, is shown.

5

Fig. 2. Double scroll attractor of the Chua’s circuit with cubic nonlinearity

Table 1. Mean squared error (MSE) on approximation of the cubic nonlinearity varying the

number of regions

n° of regions 2 3 4 5 6 7 8 9

MSE 1.442∙10-5

2.378∙10-6

7.592∙10-7

2.822∙10-7

1.324∙10-7

6.881∙10-8

4.061∙10-8

2.547∙10-8

n° of regions 10 11 12 13 14 15 16 -

MSE 1.631∙10-8

1.113∙10-8

7.627∙10-9

5.786∙10-9

4.210∙10-9

3.357∙10-9

2.675∙10-9

-

For each value of M, the approximated PWL system

ydt

dz

zyxdt

dy

xfcxydt

dxa

(4.2)

has been integrated. Fig. 3 shows the attractors corresponding to the values M = 2, 4, 8 and

16.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

x

y

6

Fig. 3. Attractors corresponding to the values of a) M = 2, b) M = 4, c) M = 8 and d) M = 16

0 0.1 0.2 0.3 0.4 0.5

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

xa

y a

a)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

xa

y a

b)

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

xa

y a

c)

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

xa

y a

d)

7

As it can be noticed, approximations with two branches are not able to reproduce a double

scroll attractor, whereas with more than three branches they show qualitatively equivalent

motion between the smooth system and the PWL approximation. This fact has been con-

firmed by the evaluation of the Lyapunov Exponents.

In fig. 4a the values of the Lyapunov Exponents for each value of M are plotted and com-

pared with those obtained by LET toolbox for Matlab [8]. Red lines represent the 3σ confi-

dence intervals of Lyapunov Exponents of system (4.1), evaluated using LET starting from

different initial conditions.

The calculation time is of the order of 2 minutes independent of the number of regions.

LET lasts 1 hour and 50 minutes to perform the same calculation. The graphical interface

has been disabled since it causes a growth of calculation time

In Table 2, the simulation parameters are listed.

Fig. 4. (a) LEs values; LET (red), PWL algorithm (blue) b) calculation time varying the value of

M for the PWL algorithm

Table 2. Parameter values for LE estimation

Time step N° iterations Initial conditions

5ms 2∙106 [0 0 0.1]

2 3 4 5 6 7 8 9 10 11 12 13 14 15 160.2

0.3

0.4

number of regions

1

a)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16-0.01

0

0.01

number of regions

2

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16-3

-2.8

-2.6

number of regions

3

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16116

117

118

119

120

121

122

number of regions

calc

ula

tion t

ime

(s)

b)

8

5 Conclusions

An algorithm for Lyapunov exponents estimation in piecewise linear systems has been ex-

tended to non linear systems with polynomial nonlinearity. As a first step, a suitable piece-

wise linear approximation for the polynomial nonlinear function is evaluated by means of a

Multi-Layer Perceptron with linear and saturating linear transfer functions. Then, the algo-

rithm is applied to a new system with the piecewise linear approximation of the nonlinear

function. Comparisons with results obtained by LET with disabled graphical interface are

reported in the case of Chua’s circuit LEs evaluation. Results confirm that the proposed al-

gorithm allows one a fast evaluation of LEs without compromising the accuracy of the cal-

culation.

6 References

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systems. NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: Int.

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2011, vol 1389, Issue: 1, 1844-1848.

[2] Benettin G, Galgani L, Giorgilli A, Strelcyn JM (1980), Lyapunov characteristic exponents

for smooth dynamical systems and for Hamiltonian systems; a method for computing all of

them, Meccanica 15:9-20.

[3] Benettin G, Galgani L, Giorgilli A, Strelcyn JM (1980), Lyapunov characteristic exponents

for smooth dynamical systems and for Hamiltonian systems; a method for computing all of

them, Meccanica 15:21-30.

[4] Shimada I, Nagashima T (1979) A numerical approach to ergodic problems of dissipative

dynamical systems. Prog. Theor. Phys. 61:1605-1616.

[5] Dieci L, Russell RD, Van Vleck ES (1997), On the Computation of Lyapunov Exponents for

Continuous Dynamical Systems. SIAM J. Numer. Anal. 34 :402-423.

[6] Higham NJ (2005) The Scaling and Squaring Method for the Matrix Exponential Revisited.

SIAM Journal On Matrix Analysis and Applications 26 (4): 1179-1193.

[7] Huang AS, Pivka L, Wu CW, Franz M. (1996) Chua’s Equation with Cubic Nonlinearity, In-

ternational Journal of Bifurcation and Chaos 12(A):2175-2222.

[8] http://www.mathworks.cn/matlabcentral/fileexchange/233-let