storace(2004)_piecewise-linear approximation of nonlinear dynamical systems

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830 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 51, NO. 4, APRIL 2004

Piecewise-Linear Approximation of Nonlinear Dynamical Systems

Marco Storace , Member, IEEE, and Oscar De Feo

Abstract—The piecewise-linear (PWL) approximation tech-nique developed by Julián et al. in the past few years is appliedto find approximate models of dynamical systems dependenton given numbers of state variables and parameters. Referringto some significant examples, i.e., topological normal forms, itis shown that a PWL dynamical system approximating a givensmooth system can preserve its main features. In particular, if the approximation accuracy increases, the equivalence betweenapproximating and approximated systems shifts from qualita-tive to quantitative. The validity of the proposed approach iseventually tested by use of a severe nonlinear example, i.e., theRosenzweig–MacArthur system, which describes the populationdynamics in a tritrophic food chain model.

Index Terms—Approximation theory, bifurcations, circuit mod-eling, nonlinear dynamics, piecewise-linear (PWL) approximation,

structural stability.

I. INTRODUCTION

T HE approximation of nonlinear dynamical systems is a keytopic in the literature on circuits and systems. The motiva-

tions for the researchers’ interest in this problem can concernmerely system theory (mathematical modeling) or may involvecircuit implementation tasks (circuit modeling).

From a mathematical modeling point of view, one isinterested in finding the simplest system of equations thatsatisfies five main requirements: well posedness,1 simulation

capability,2 qualitative similarity,3 predictive ability,4 andstructural stability5 (see, e.g., [4]). This goal needs a verysharp “Ockham’s razor” in the definition of the variables,parameters, and nonlinearities of the model in order to simplify

Manuscript received May 16, 2003; revised September 20, 2003. This workwas supported in part by the the Italian Ministry of University and Research(MIUR) under PRIN and FIRB frameworks Project 2001095333, in part by theUniversity of Genoa, Italy,and in part by the European project APEREST underGrantST-2001-34893 and Grant OFES-01.0456.This paperwas recommendedby Associate Editor C. P. Silva.

M. Storace is with the Biophysical and Electronics Engineering Department,University of Genoa, I-16145 Genova, Italy (e-mail: [email protected]).

O. De Feo is with the Laboratory of Nonlinear Systems, Swiss Federal Insti-

tute of Technology Lausanne (EPFL), CH-1015 Lausanne, Switzerland (e-mail:[email protected]).Digital Object Identifier 10.1109/TCSI.2004.823664

1A model is well posed if, when put together with other well-posed models,it does not originate absurd situations.

2When a model is defined on the basis of a given data set (typically, measuresof significant variables), it should provide good approximations also for otherdata (validation set).

3The model behavior, at least under some conditions of interest, should bequalitatively similar to that of the original system.

4The model should be able to predict previously unknown behaviors.5The qualitative properties of the model should not change due to “small”

perturbations of its parameters.

its analysis and numerical simulation, without giving up any of the aforesaid five requirements. The mathematical model of asystem can be derived by facing either a direct or an inverseproblem. In the former case, one aims to simplify as muchas possible, the description of a nonlinear dynamical systemwhose equations, state variables, and significant parameters arecompletely known (for instance, they concern a known physicalsystem). In the latter case, the models are derived from theblack-box identification of the systems, only some measures(i.e., trajectories) of which are known. In this case, the objectiveis to find the simplest approximate models of such systemsthat capture the most significant features of their flows. For

instance, nonlinear black-box modeling/identification is one of the two main steps of a very recently developed chaos-basedtemporal pattern recognition/classification technique [5]–[8].

From a circuit modeling point of view, one is interested infinding the simplest circuit structure (possibly based on a pre-scribed set of simple building blocks), whose describing equa-tions coincide with or area good approximation of a given math-ematical model. For instance, the analog emulation of neuronmodels (i.e., the first step toward the definition of biologicallyconsistent neural circuits) requires a circuit synthesis techniqueto implement complex dynamical systems, as recently proposedin [3].

We are interested in both kinds of models (either approxima-tions of known models or identified black-box models) for thepurpose of their circuit implementations. Keeping this in mind,we need:

• an approximation technique, allowing one to identify ei-ther an approximate model of a given dynamical systemor a black-box model corresponding to given dynamicaltrajectories;6

• a synthesis technique to find direct circuit implementa-tions of such models.

There are many possible methods concerning the first tech-nique. Among the works dealing with mathematical modelingof nonlinear systems, we can cite: [9], [10], for the general

system identification theory; [11], [12] for the splines theory;[13] for modeling and identification of parallel nonlinear sys-tems through Volterra and Wiener kernels; [14] for the artifi-cial neural networks theory and applications; [15] and, [16] fornonlinear time series analysis; [17] for fuzzy identification of nonlinear models of control systems; [18] for piecewise-smooth

6It is important to point out that models depend on both state variables andsystem parameters. In the case of known models of given dynamical systems,the roles played by the state variables and the system parameters are distinct,whereas, in the case of black-box identification, the same role is played. Thisfact implies that we might have to identify systems defined over domains of more than a few dimensions.

1057-7122/04$20.00 © 2004 IEEE



STORACE AND DE FEO: PWL APPROXIMATION OF NONLINEAR DYNAMICAL SYSTEMS 831

(PWS) modeling; [19] and [20] for implicit piecewise-linear

(PWL) modeling and analysis; [1]–[3] for explicit PWL mod-

eling and analysis; and [21] for PWL modeling and analysis of

control systems. On the contrary, as far as the synthesis tech-

nique is concerned, apart from some general attempts based on

mixed analog/digital structures [22], [23], circuit implementa-

tions are available mainly for specific classes of models.

A possible method applying both needed techniques is thePWL approximation technique developed in the past few years

by Julián et al. [1]–[3]; it constitutes one of the corner stones of

this paper. In the general context of the approximation/identifi-

cation and circuit synthesis of nonlinear systems, this paper will

be focused on the approximation aspects. The black-box identi-

fication of PWL models and the circuit synthesis will be treated

elsewhere. We only anticipate that possible circuit implementa-

tions of PWL functions can be obtained by resorting either to

mixed analog/digital structures (e.g., by properly generalizing

the solution proposed in [24]) or to completely analog circuits,

by exploiting CMOS architectures used in the field of fuzzy sys-

tems (for instance, see [25]).

The PWL approximation technique mentioned above was re-cently used [3] and developed [26], [27] to define a method

for the synthesis of nonlinear multiport resistors. However, in

these works, the influence of the approximations of the multiter-

minal resistors on the behavior of the overall (dynamical) circuit

containing such resistors was only touched upon. In this paper,

we shall face the problem not addressed in [3], [26], [27] by

analyzing the qualitative behaviors of dynamical systems char-

acterized by PWL flows that approximate the flows of some

significant examples. Such an analysis will be carried out by

using bifurcation theory, which is the second corner-stone of

this paper. The dynamical behaviors of the original systems as

well as those of the approximate ones will be analyzed by re-

sorting to some packages. The first package is a toolbox devel-

oped by the authors in the Matlab environment;7 it allows the

application of the PWL approximation method to quite a large

class of dynamical systems. Such a toolbox permits one to easily

obtain the phase portraits of both the original and PWL-approx-

imate dynamical systems for given parameter configurations.

This is a preliminary step useful to choose a PWL approxima-

tion candidate for the final circuit synthesis. To guarantee that

the dynamical behavior of the PWL-approximate flow will be

faithful to that of the original system for any values of some

significant parameters (i.e., to verify the structural stability—in

a given limited domain—of the original system to the pertur-

bation induced by the approximation), we shall obtain a com-plete bifurcation scenario of the approximate system. Toward

this end, we shall resort to other packages, such as CONTENT

[28] and AUTO2000 [29], which allow one to numerically an-

alyze the bifurcations of dynamical systems by means of con-

tinuation methods [30]. As an essential prerequisite for using

such methods is flow smoothness, we shall replace the PWL

flow with a PWS version of it. Such a replacement is not com-

pletely “painless,” but does not substantially compromise either

7The toolbox containing routines for PWL approximation and sim-ulation of dynamical systems can be found at the WWW addresshttp://www.dibe.unige.it/department/ncas/STORACE/storace.html.

the approximation accuracy or the long-term target of the re-

search field concerning the circuit realization of dynamical sys-

tems, as the possible circuit realizations of the PWL basis func-

tions will be necessarily smooth. Despite the existence of some

packages specific for the analysis of dynamical PWL systems

(for instance, see [31]), none of them provides a complete char-

acterization of the dynamics in terms of significant bifurcation

parameters for the class of systems considered in this paper.The rest of the paper is organized as follows. In Section II,

we shall briefly recall some basic definitions concerning the

PWL approximation of continuous-time dynamical systems. In

Section III, some topological normal forms will be considered

and some PWL approximations of these systems will be dis-

cussed by making reference to either equilibrium manifolds or

phase portraits. Section IV concerns the bifurcation analysis of

the smoothed versions of the most significant PWL approxi-

mations of the considered normal forms. A more complex ex-

ample (the Rosenzweig–MacArthur system) will be given and

analyzed in Section V. In Section VI, some concluding remarks

will be made.

II. PWL APPROXIMATION: BASIC DEFINITIONS

In this paper, we shall focus on continuous-time dynamical

systems of the kind

(1)

where the flow , the state vector ,

and the parameter vector are column vectors ( and

are positive integer numbers). We aim to approximate the flow

through a proper PWL function obtained as the weighted sum of

a set of basis functions according to the technique proposed

by Julián et al. We shall denote by a continuous PWL ap-

proximation of over the -dimensional compact domain

, i.e., , where is a hyperrectangle

(rectangle, if ) of the kind

(2)

Each dimensional component of the domain (generically

denoting a component of or ) can be subdivided into

subintervals of amplitude ; thus, a boundary con-

figuration is obtained [1]. Each hyperrectangle contains

nonoverlapping hypertriangular (triangular, if )

simplices. As a result, turns out to be partitioned (simpli-

cial partition) into hyperrectangles and to contain

vertices. The domain associated with asimplicial boundary configuration (i.e., the numbers ’s)

can be completely described by the triplets

.

As shown in [1] and [2], the class of continuous PWL func-

tions that are linear over each hypertriangular simplex

constitutes an -dimensional linear space , which is

defined by the domain , its simplicial partition , and a proper

inner product (see [26] for details). Each function belonging to

can be represented as a sum of basis functions

(arbitrarily organized into a vector), weighted by an -length

coefficient vector . The coefficients determine the shape of

uniquely.




There are many possible choices for the PWL basis func-

tions, each of which is made up of (linearly independent)

functions belonging to . For instance, there are bases

more convenient for performing function interpolation or func-

tion approximation, from a computational or a circuit-synthesis

point of view. However, any basis can be expressed as a linear

combination of the elements of the so-called basis, which can

be defined by recursively applying (up to times) the fol-lowing function [1]–[3]:

(3)

As shown in [3], the weighting coefficients can be easily

found by applying optimization techniques (e.g., a least-squares

criterion) to a set of properly distributed samples of over do-

main .

In the Section III, we shall approximate three specific

dynamical systems by applying the PWL technique recalled

above. Different approximations will be characterized by

different values of the triplets .

In particular, for a domain fixed for each system, we shallconsider different PWL approximations by varying the numbers

(subdivisions) and (samples of ) along any dimensional

component of .

III. SOME TOPOLOGICAL NORMAL FORMS AND THEIR

PWL APPROXIMATIONS

The analysis of a dynamical system can be carried out by

finding out its bifurcation diagram, which represents very

compactly all possible behaviors of the system and transitions

(bifurcations) between them under parameter variations. The

bifurcation diagram of a given dynamical system can be

very complicated, but, at least locally, bifurcation diagrams

of systems in many different applications can look similar

(topological equivalence). The concept of topological equiva-

lence leads to the definition of topological normal forms, i.e.,

polynomial forms that provide universal bifurcation diagrams

and constitute one of the basic notions in bifurcation theory.

Another central concept is the structural stability of a dynam-

ical system, that is, the topological equivalence between such

a system and any other system obtained by slightly changing

some parameters [30], [32].

In this section, we shall find PWL approximations of

some codimension-2 topological normal forms (cusp, Bautin,

and Bogdanov–Takens), and we shall get an idea about the

structural stability of such approximations to the number of exploited basis functions. For details concerning such normal

forms, the reader is referred to [30].

A. Cusp Bifurcation

The topological normal form for the cusp bifurcation is

(4)

In this case, and .

Fig. 1 shows the bifurcation diagram of the normal form (4).

Equation (4) can have from one to three equilibria. A fold bifur-

cation with respect to the parameter occurs at a bifurcation

Fig. 1. Bifurcation diagram of the normal form (4).

curve , which di-

vides the plane into two regions, labeled by A and B in

Fig. 1. The two branches and meettangentially atthe cusp

point (0, 0), marking a cusp bifurcation of equilibria. In region

A, above , there are three equilibria of (4), two stable and one

unstable. The right (left) stable equilibrium collides with the un-

stable one and both disappear if we cross from region A

to region B at any point other than the origin. If one approaches

the cusp point from inside region A, all three equilibria merge

into a triple root of the right-hand side of (4). Fig. 2(a) shows the

equilibrium manifold , i.e., the locus of the points

fulfilling the equation , near the cusp bifurcation.

The PWL approximations of the flow are obtained

over the domain

with

The domain is partitioned by performing subdivisions

along the state component , and and subdivisions along

the parameter components and , respectively. The coeffi-

cients of the basis were derived from a set of samples of

corresponding to a regular grid of points over the

domain . We point out that the flow is linear with respect to

the dimensional component , then we can fix . With




Fig. 2. Equilibrium manifolds of (a) the cusp normal form and (b), (c), (d) its PWL approximations: coarse (A1), medium (A2), and fine (A3), respectively. Stableregions are shown in dark gray, unstable regions in light gray, and the fold bifurcation border is marked in black.

this caveat in mind, we now focus our attention on the following

three PWL approximations of

(A1)

(A2)

(A3)

Fig. 2(b)–(d) shows the equilibrium manifolds for

the PWL approximations (A1), (A2), and (A3), respectively.

With reference to Fig. 2(a), it is easy to deduce that:

• approximation (A1) is very rough: its equilibrium mani-

fold does not exhibit any qualitative similarity to the orig-

inal one;

• approximation (A2) is qualitatively good: the fold bifur-

cation border is similar to that of the original one, eventhough its shape is quite saw toothed;

• approximation (A3) is good from both the qualitative and

quantitative points of view.

B. Bautin (Generalized Hopf) Bifurcation

The Bautin (generalized Hopf) bifurcation can be described

by the following polynomial normal form:

(5)

In this case, and .




Fig. 3. Bifurcation diagram of the normal form (5). The coordinates of thethree black points a; b, and c are: a = ( 0 0 : 5 ; 0 : 9 ) ; b = ( 0 0 : 1 5 ; 0 : 9 ) , andc = ( 0 : 3 ; 0 : 9 ) .

Fig. 3 shows the bifurcation diagram of the normal form (5). The

plane ispartitionedinto three regions bytwocurves:the

vertical line , made up of the two half-lines (for

) and (for ), corresponds to a Hopf bifurca-

tion, and the half-parabola

marks a nondegenerate tangent bifurcation of cycles [30].

The three regions are labeled by A B, and C, as shown in Fig. 3.

The system (5) has only one equilibrium point, i.e., the origin of

the plane . The origin has purely imaginary eigenvalues

along and is stable for and unstable for .

A stable limit cycle bifurcates from the origin if we cross thehalf-line from region A to region C. On the contrary, an

unstable limit cycle appears if the half-line is crossed from

region C to region B. Two limit cycles coexist in region B, and

collide and disappear on the curve . The codimension-2 point

marks a degenerate Hopf bifurcation of the equi-

librium (i.e., the first Lyapunov coefficient of the Hopf bifurca-

tion changes its sign).


over the domain

with

The domain is partitioned by performing and subdi-

visions along the state components and , respectively, and

and subdivisions along the parameter components

and , respectively. The coefficients of the -basis were de-

rived from a set of samples of corresponding to a regular grid

of points over the domain . We obtained

the following three PWL approximations of :

(B1)

(B2)

(B3)

Fig. 4 shows some phase portraits obtained by numerically

integrating the dynamical system having on the right-hand side

either or one of the three . The phase portraits were

achieved for the parameters set to the values of points a (first

row of figures), b (second row), and c (third row) in Fig. 3. The

first column of figures shows the phase portraits of the original

system(5), andthe second, third,and fourthcolumns show those

of the (B1), (B2), and (B3) approximations, respectively. For all

the trajectories, we chose as starting points the following pairs

: , and . Referring

to the first column, it is easy to deduce that:

• approximation (B1) is unacceptable: at the point a, there is

a stable limit cycle instead of the stable equilibrium point,

whereas, at the other two points b and c, the only attractor

is an equilibrium point (then, this PWL approximation is

not qualitatively similar to the original system);

• approximation (B2) is qualitatively good: the attractors/re-

pellors of the original system hold on for all points a b,

and c in the parameter space, even if they change their

shapes/positions in the state space; in particular, the pres-

ence of the unstable cycle at the point b is evident (it sep-

arates the basin of attraction of the stable focus from thatof the stable limit cycle);

• approximation (B3) is very good from both the qualitative

and quantitative points of view.

C. Bogdanov–Takens (Double-Zero) Bifurcation

The Bogdanov–Takens (double-zero) bifurcation can be de-

scribed by the following polynomial normal form:

(6)

In this case also, and .Fig. 5 shows the bifurcation diagram of the normal form (6).

System (6) can have from zero to two equilibria. A fold bifur-

cation occurs at the bifurcation parabola

, which has two branches, (for ) and

(for ). To the right of (region A in Fig. 5), system

(6) does not have any equilibria, whereas two equilibria, a node

and a saddle , ap-

pear if one crosses from right to left. The equilibrium

undergoes a node-to-focus transition (which is not a bifurca-

tion) through a curve (not shown in Fig. 5) located in region

B between and the vertical axis . The lower half of

the axis , marks a




Fig. 4. Phase portraits corresponding to the parameter values at points a (first row), b (second row), and c (third row). The first column shows the phase portraitsfor the original system (5), and the second, third, and fourth columns show those for the (B1), (B2), and (B3) PWL approximations, respectively.

Fig. 5. Bifurcationdiagram ofthe normal form (6). Thecoordinates ofthe fourblack points a; b; c, and d are: a = ( 0 : 2 ; 0 0 : 7 ) ; b = ( 0 : 0 5 ; 0 0 : 7 ) ; c =

( 0 0 : 0 5 ; 0 0 : 7 ) , and d = ( 0 0 : 2 ; 0 0 : 7 ) .

Hopf bifurcation of , which originates (if one crosses from

right to left) a stable limit cycle in region C, whereas re-

mains a saddle. If one further decreases , the cycle grows and

approaches the saddle; it turns into a homoclinic orbit when it

touches the saddle. The locus of the points where the cycle turns

into a homoclinic orbit is the curve

. Then, in region D there are no

cycles but only an unstable focus (node) and a saddle, which

collide and disappear at the branch of the fold curve.


over the domain

with

The domain is partitioned by performing and subdi-

visions along the state components and , respectively, and

and subdivisions along the parameter components

and , respectively. The coefficients of the basis were de-

rived from a set of samples of corresponding to a regular grid

of points over the domain . In particular,

as the flow is linear in the dimensional component , we can




Fig. 6. Phase portraits corresponding to the parameter values at points a (first row), b (second row), c (third row), and d (fourth row). The first column showsthe phase portraits for the original system (6), and the second, third, and fourth columns show those for the (C1), (C2), and (C3) PWL approximations, respectively.

fix and . By doing so, we obtained the following

three PWL approximations of

(C1)

(C2)

(C3)

The phase portraits shown in Fig. 6 were obtained by fixing

the parameters at the values of points a (first row of figures), b

(second row), c (third row), and d (fourth row). The last three

columns of figures show thephaseportraits forthe PWLapprox-

imate flows (C1), (C2), and (C3), respectively. The comparisons

of the last three columns with the first column (phase portraits

of the original system (6)) show that only approximation (C3) is

both qualitatively and quantitatively very good. Approximation

(C1) is already unsatisfactory from a qualitative point of view:

for instance, it has a stable equilibrium point and a saddle for

Fig. 7. Induction of a transition region from stationary to cyclic regime due tothe PWS approximation. (a) In the original smooth case the transition is markedby a Hopf bifurcation. (b) In the PWL approximate system, the transition ismarked by a degenerate fold bifurcation of cycles (involving one or severalother unstable cycles not shown in figure) and the cycle appears already “big. ”(c) In the PWS approximate system, the two aforesaid cases intermingle and thetransition takes place in a (very small) region 1 where regular folds of cyclesand Hopf bifurcations alternate.




Fig. 8. Cusp bifurcationsinducedby thesmoothingfunction(7) inthe referencesystem _x = p ( x ; x ) + p + p x , where p = 0 : 0 5 . Curves _x ( x ) correspondingto the PWL case (black curves) and to the PWS case with a = 4 0 (dark-gray curves), a = 6 0 (gray curves), and a = 1 2 0 (light-gray curves). (a) p = 0 : 2 andp = 0 0 : 0 7 5 e 0 3 . (b) p = 0 : 9 and p = 0 0 : 0 7 5 e 0 3 . (c) Equilibrium manifolds of the reference system, for both the PWL (wireframe) and PWS cases witha = 4 0 .

Fig. 9. Bifurcation diagrams versus ( p ; p ) for the smoothed PWL approximations (B2) (a) and (B3) (b). The bifurcation diagram of the original system (5)is shown (thick dashed curves) for comparison. The black lines mark Hopf bifurcations, the light-gray and gray lines mark tangent bifurcations of equilibriaand cycles, respectively. The black dots indicate degenerate Hopf bifurcations. (c) Sketch of the bifurcations occurring at the boundaries of the spurious closedregions (here magnified) characterizing the bifurcation diagrams for a sufficiently good PWL approximation and for a = 4 0 . The thick gray curves are qualitativetrajectories in the phase plane corresponding to the regions A; B, and C in the parameter plane.

and fixed at a, anda stable limit cyclefor and fixed at d.

On the other hand, approximation (C2) is qualitatively good, but

its phase portraits exhibit some quantitative differences, as com-

pared with the corresponding phase portraits in the first column

(e.g., see the case corresponding to point c).

IV. CONTINUATION ANALYSIS

From the results presented in the Section III, it can be

concluded that the overall qualitative behavior of a dynamical

system, i.e., its structural stability to the parameters, is already

mimicked by a PWL approximation characterized by relatively

small ’s, i.e., a relatively small number of basis functions.

Of course, too rough a partition of the domain causes the qual-

itative behavior of the approximate system to differ from the

original one. If one increases the subdivisions ’s along some

dimensional component of the domain (and then the number

of basis functions), the equivalence shifts from qualitative

to quantitative. This statement is further corroborated by the

results of the bifurcation analysis carried out for the best PWL

approximations of the Bautin and Bogdanov–Takens normal

forms and presented in this section. Such results were obtained

by resorting to the continuation packages CONTENT [28] and

AUTO2000 [29]. In order to meet the smoothness requirements

imposed by the continuation methods, we used a smoothed

version of the function by simply replacing the absolute

value function in (3) with the following function:

(7)

where the parameter controls the degree of smoothness. Of

course, the smoothed (PWS) versions of the functions still

form a basis, provided that the parameter is not too small (in

our continuations, we fixed ).

The transition from PWL to PWS functions results in two

main consequences. If an equilibrium point of the PWL flow is

located within a simplex (i.e., not on a simplex border), such an




equilibrium point cannot undergo a Hopf bifurcation, as

is linear. The Hopf bifurcation of an equilibrium point lying

within a simplex is replaced by a degenerate fold bifurcation of

cycles (the more accurate the PWL approximation, the smaller

the cycles), as pointed out in Fig. 7. Thus, even if the bifurca-

tion process leading to the cycle generation is locally altered,

a good PWL approximation is structurally stable. On the other

hand, the PWS version of admitsHopf bifurcationsof anyequilibrium point. Then, the first consequence of the smoothing

process is the possibility of finding Hopf bifurcation curves, as

in the original case. This phenomenon, which will affect the bi-

furcationscenarios of approximate systems (as we shall see later

on), is sketchedin Fig. 7 where thedeformationof thecycle birth

due to PWL or PWS approximations is illustrated in the control

space.

The second consequence is that the smoothed version of a

PWL flow can inducedifferent bifurcation diagrams foran equi-

librium point, depending on the value of the smoothing param-

eter . For instance, Fig. 8 illustrates the cusp bifurcations in-

duced by the smoothing function (7) in the reference system

, where . In the PWLcase (black curves and wireframe in Fig. 8), there is only one

equilibrium point for any pair . After the smoothing,

the bifurcation diagram changes qualitatively, as the equilib-

rium undergoes a cusp bifurcation [see the equilibrium man-

ifold in Fig. 8(c), plotted versus the parameters and ].

The smaller the value of , the more evident the cusp bifur-

cation. The smooth curves in the first two plots correspond to

(dark-gray curves), (gray curves), and

(light-gray curves). In particular, [cf. Fig. 8(a)], for

and , there is only one equilibrium for both

the system with the PWL flow(black line) and its smoothed ver-

sions, for any considered value of the smoothing parameter. On

the contrary [cf. Fig. 8(b)], for and ,

the smoothed versions for small values of the smoothing param-

eter (e.g., and ) have three equilibria, whereas

for the PWL case as well as for large values of the smoothing

parameter (e.g., ) the system has only one equilibrium.

We point out that the clusters of spurious solutions generated by

small values of are made up of equilibria that turn out to be

numerically very close to one another. Then, the smoothing of

the flow can give rise to numerical problems in the continuation

of bifurcation curves, but does not substantially affect the dy-

namics of the approximate system.

This kind of phenomenon can concern the bifurcations not

only of equilibria but also of any invariant of the PWL flow. Inparticular, the transition from stationary to cyclic regime will

be marked not by a sharp bifurcation manifold in the parameter

space (Hopf bifurcation) but by a transition region where Hopf

bifurcations and fold bifurcations of cycles alternate, as pointed

out in Fig. 7.

A. Bautin (Generalized Hopf) Bifurcation

The bifurcation diagrams for approximations (B2) and (B3)

are shown in Fig. 9. They are superimposed, for comparison,

upon the bifurcation diagram (thick dashed curves) of the orig-

inal system (5). The role played by the spurious closed regions

that characterize such diagrams can easily be understood by

Fig. 10. Bifurcation diagram versus ( p ; p ) for the smoothed PWLapproximation (C3). The bifurcation diagram of the original system (6)is shown (thick dashed curves) for comparison. The black line is the foldbifurcation curve, the light-gray line is the homoclinic curve, and the gray lineis the Hopf bifurcation curve. The black dot denotes the Bogdanov-Takensbifurcation.

making reference both to the bifurcation diagram sketched in

Fig. 9(c), where they aremagnified, andto thetwo consequences

of the smoothing of the PWL flow. As pointed out in Fig. 9(c),

the smoothing induces regions of coexistence of equilibria (cf.

the shaded area in region A), and a transition region D (shaded

region around the original Hopf) from a stable equilibrium to

a stable cycle. As discussed above, several periodic solutions

close to one another coexist in D. Apart from these local differ-

ences, thegeneral layoutof thebifurcationdiagram is preserved,

at least for sufficiently large values of the smoothing parameter

and for sufficiently accurate PWL approximations. The bifur-

cation diagram for approximation (B1) is not shown in Fig. 9

as the spurious closed regions become dominant in the diagram,

thus resulting in a system with much richer dynamics than in the

original one (the parameter space contains more regions charac-

terized by different qualitative behaviors, such as the new equi-

libria and limit cycles shown in the phase portraits in the second

column of Fig. 4). On the other hand, a proper approximation

by relatively small numbers ’s of subdivisions along each

dimensional component of the domain , like (B2), not only

preserves the qualitative behaviors of the original system for

givenparameter values (cf. third column in Fig. 4) but also keeps

the qualitative partitioning of the parameter space, while main-

taining the quantitative differences (cf. first diagram in Fig. 9).

Finally, if we further increase the number of hyperrectangles

that partition (i.e., the number of basis functions employed

for the PWL approximation), as in the case (B3), we obtain an

almost perfect matching between the original and approximate

systems in terms of simulation capability, qualitative similarity,

and structural stability.

B. Bogdanov – Takens (Double-Zero) Bifurcation

Fig. 10 shows the bifurcation diagram for the PWS version of

the finest PWL approximation of system (6), i.e., (C3).




Fig. 11. Equilibrium manifolds of: (a) the Bogdanov–Takens normal form and (b) its fine PWL approximation (C3). Stable regions are shown in dark gray,unstable regions in light gray, and the fold bifurcation border is marked in black.

The bifurcation diagram of the original system is presented

for the sake of comparison (thick dashed curves). Both the ho-

moclinic and Hopf curves are very similar to their dashed coun-

terparts (actually, for the Hopf curve there is a transition region

as in the case of the Bautin bifurcation, but it is not reported

here for the sake of clarity). The fold curve shows some loops,

which follow directly from the PWL approximation, as shown

in Fig. 11.Assuming that the PWL approximation of the first (linear)

component of the original flow is almost perfect, we repre-

sented the equilibrium manifolds for , in

both the original and approximate cases (see Fig. 11(a) and (b),

respectively). The irregular shape of the equilibrium manifold

in the approximate flow (see in particular the thick black curves

evidenced on both the equilibrium manifolds) determines the

presence of loops in the projection of the three-dimensional fold

bifurcation border on the parameter plane (shown in both fig-

ures for the sake of comparison). The loops are originated by

the folds of the equilibrium manifold in the approximate flow,

whereas the equilibrium manifold in the original flow is much

more regular (see also Fig. 2 for another example). These loopsinduce some topological differences between the approximated

and the original system, as locally, inside the loops, the approxi-

mated systemhas an equilibrium point not present in theoriginal

one. However, from a global point of view, the parameter space

remains partitioned into two main regions, one without equilib-

rium points and one with two equilibrium points, as illustrated

in Fig. 10. The only difference is in the border between the two

regions, being a sharp line in the original system and a more ir-

regular narrow transition region in the approximated one (as in

the case of the Bautin Bifurcation). The width of such a transi-

tion region will become thinner and thinner as we increase the

accuracy of the approximation.

Fig. 12. Detail of the bifurcation diagram of (8) in the ( p ; p ) plane.

V. MORE COMPLEX EXAMPLE: THE

ROSENZWEIG–MCARTHUR SYSTEM

The so-called Rosenzweig–MacArthur system is taken from

biology and is one of the most common models describing the

population dynamics in a tritrophic food chain model composed

of a prey, a predator, and a top-predator [33]–[35]. The bifurca-

tion scenario of such a system is quite complicated and repre-

sents a suitable benchmark to test the efficiency of the proposed

approximation approach.




Fig. 13. Trajectories of the original (first row) and PWL (second row) flows corresponding to points a (first column), b (second column), c (third column),d (fourth column), and e (last column) in the parameter plane ( p ; p ) .

The Rosenzweig–MacArthur model is given by the following

system of ordinary differential equations (see [33] for moredetails):

(8)

The reference parameter values used in this paper are the same

as used in [33], namely,

. The parameter is fixed to the value , and

the remaining parameter is varied to perform the bifurcation

analysis.

It has been shown—first in [34] and in greater detailin [35]—that the parameter space (in particular, the plane

) admits regions of populations-coexistence solutions

(i.e., regions where each solution remains strictly positive

for any ) and that it is possible to partition such regions into

five subregions characterized by strongly qualitatively different

behaviors of the solutions:

• steady-state solution, equilibrium;

• bursting (slow/fast) cycle;

• bursting chaos;

• nonbursting chaos;

• nonbursting cycle.

Such asymptotical behaviors correspond to regions

A B C D, and E in Fig. 12, respectively.As we are interested in reproducing the above scenario of

coexistence solutions, we shall not approximate system (8) to

avoid problems in the numerical simulations due to the invari-

ancy of all the coordinate planes and axes ( is always a

solution). We shall approximate the logarithmically scaled ver-

sion of system (8), obtained by resorting to the variable change

(9)

The PWL approximation of such a system is obtained over the

domain

with

The domain is partitioned by performing , and

subdivisions along the state components , and , respec-

tively, and subdivisions along the parameter component .

The coefficients of the basis were derived from a set of sam-

ples of corresponding to a regular grid of

pointsoverthe domain . In particular, we shall consider the fol-

lowing PWL approximation of

.

The simulation results of the original and approximate

systems are shown in Fig. 13, where each column displays

the trajectories corresponding to points a b c d, and e in

the plane . The axes’ limits are not displayed for the

sake of clarity: they are the same for the pairs of figures in

the same column, whereas they are different and adapted

to the specific points in the plane along each row.The column-by-column comparison between the two rows of

figures points out that:

a the system state settles down at an equilibrium point in

both cases (there are some negligible differences in the

locations of such points and in the local behaviors of

the trajectories);

b c the correspondences between the two stable bursting

cycles and between the two bursting chaotic attractors

are almost perfect;

d the original and the approximate normal (nonbursting)

chaotic attractors are very similar (there are some neg-

ligible differences in the shapes of the attractors);




e the system state asymptotically reaches a nonbursting

cycle in both the original and approximate cases.

To sum up, we can conclude that the approximate flow works

very well, from both a qualitative and a quantitative point of

view. Concerning the structural stability, it has been assessed

only at five points of the parameter plane; thus, nothing could

be stated about the general bifurcation scenario in two param-

eters (the continuation of the PWS case would require con-

siderable computational resources and time). However, in [35]

the strict relationships between the bifurcation scenario and the

shape and quality of the trajectories was pointed out. Such re-

lationships, together with the results reported in Section IV for

normal forms (even for PWL approximations rougher than in

the present case), allow us to conclude that the very close simi-

larities between the trajectories of the original and approximate

systems imply at least a qualitative equivalence between the two

bifurcation scenarios.

VI. CONCLUDING REMARKS

By combining simulations with advanced numerical contin-

uation techniques, we have shown that the PWL approximation

technique developed by Julián et al. in the past few years can

be successfully used to approximate smooth dynamical systems

dependent on given numbers of state variables and parameters.

We point out that to approximate known systems, as in the pro-

posed examples, it would be unnecessary to find out PWL ap-

proximations of functions dependent on both state variables and

parameters (the overall approximation/analysis process would

be greatly simplified). But the proposed method is also aimed

at finding PWL approximations of unknown systems, as stated

in the Introduction; then the paper should be read from a per-spective of approximation/identification of dynamical systems.

From such a standpoint, it is important to treat the parameters

as if they were unknown.

Generally speaking, the PWL approximation accuracy grows

if one increases the number of subdivisions along each dimen-

sional component of the domain. However, when the function

to be approximated shows particular symmetry properties with

respect to the th-dimensional component, an odd number

of subdivisions could yield better results than a larger even ,

and vice versa. Moreover, if is linear with respect to the th-di-

mensional component, it is useless to increase (see Sections

III-A and III-C).

The reported results have shown that, if we increase the ap-

proximation accuracy up to a sufficient degree, the approxi-

mate dynamical systems preserve both the dynamical (trajecto-

ries) and structural-stability (bifurcations) arrangements of the

original systems, also in regions of the parameter space charac-

terized by chaotic behaviors. However, the lower bound to the

approximation accuracy needed to guarantee the structural-sta-

bility equivalence to the original systemis still an open issue and

very probably an unsolvable one. In the absence of a general cri-

terion, one can find case by case the configurations thatguar-

antee both the smallest number of basis functions (i.e., the

lowest complexity of the circuit implementation of the approxi-

mate system) and the structural-stability equivalence. We point

out that atnow itis not possible to find a threshold for the quanti-

tative error in the approximation of the “static” function

necessary to achieve good qualitative results in the approxima-

tion of the corresponding dynamical system , as

static and dynamic approximations are not strictly related. As a

matter of fact, the dynamical system is particularly sensitive to

the accuracy of the approximation of in some specific

regions of the domain, e.g., those corresponding to invariant

manifolds, as shown in some of the examples considered in this

paper.

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Marco Storace (M’01) was born in Genoa, Italy,in 1969. He received the Laurea (M.Sc.) five-yeardegree (summa cum laude) in electronic engineeringand the Ph.D. degree in electrical engineering fromthe University of Genoa, Genoa, Italy, in 1994 and1998, respectively.

Since 1999, he has been an Assistant Professor inthe Department of Biophysical and Electronic Engi-neering, University of Genoa. He was a Visiting Re-

searcher to EPFL, Lausanne, Switzerland, in 1998and 2002. His main research interests are in the areaof nonlinear circuit theory and applications, with emphasis on circuit models of nonlinear systems (e.g., hysteresis, biological neurons), methods for the piece-wise-linear approximation of nonlinear systems and for the consequent circuitsynthesis, methods of analysis and synthesis of cellular circuits, and nonlineardynamics. He is the author or coauthor of about 40 scientific papers, more thanhalf of which have been published in international journals.

Oscar De Feo received the Bachelor’s degree(summa cum laude) in industrial electronics fromMaxwell High School, Milan, Italy, the M.Sc. degree(summa cum laude) in computer science engineeringfrom Politecnico di Milano, Milan, Italy, and thePh.D. degree in technical sciences from the SwissFederal Institute of Technology Lausanne (EPFL),Lausanne, Switzerland, in 1990, 1995, and 2001,respectively.

He has been involved in research assignmentsat EPFL, the International Institute for Applied

Systems Analysis (IIASA), Laxenburg, Austria, École normale supérieure,Paris, France, University College Dublin, Dublin, Ireland, Fondazione EniEnrico Mattei, Milan, Italy, and the Research Institute for Mathematics andInformatics, Amsterdam, The Netherlands. He is currently with the Laboratoryof Nonlinear Systems, EPFL, where he is in charge of the “Nonlinearphenomena” and “Dynamical System Theory for Engineers” courses. His mainresearch interests are in the fields of bifurcation and nonlinear systems theorywhich he applies to several fields: chaotic-based representation of uncertaintyin biological neural network, chaos-based modeling of signals, nonlinearcircuit design, biodiversity and life-history traits modeling, exploitation of ecosystems, ecologically sustainable development and environmental impact,and numerical methods for nonlinear systems analysis.

Dr. De Feo received the Mikhalevich Award from the IIASA, the Best PaperAward at the European Conference on Circuit Theory and Design (ECCTD’99),and the prize from the Chorafas Foundation for the work accomplished duringhis Ph.D. thesis.

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