Research Paper

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Evaluation of MATCONT of MATLAB for
Constructing Bifurcation Diagrams of Chemical
Process Systems
Article · September 2011
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Anand V. P. Gurumoorthy
VIT University
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Jason R. Picardo
Indian Institute of Technology Madras
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2. The IUP Journal of Chemical Engineering, Vol. III, No. 3, 2011.
Evaluation of MATCONT of MATLAB for Constructing Bifurcation Diagrams
of Chemical Process Systems
Jason R Picardo and Anand V P Gurumoorthy
Department of Chemical Engineering, VIT University, Vellore, India- 632014
Abstract
As the study of nonlinear system dynamics and bifurcation theory percolate into applied
engineering and biological disciplines, the need for a user friendly, flexible and robust
bifurcation analysis software is evident. MATCONT is a MATLAB toolbox with a graphical
user interface for the continuation of static and dynamic equilibria of nonlinear systems. This
software is relatively recent and has certain advantages over existing continuation software
which include its MATLAB foundation, in addition to new features. In this paper, this
toolbox is evaluated via a well studied illustrative example of an ethanol fermentor which
shows rich nonlinear behavior. Some computed results are displayed and a discussion of the
basic workflow in MATCONT is provided. A few difficulties encountered while studying
nonlinear systems with MATCONT are highlighted. Nevertheless, due to its merits and future
scope, as is discussed, this software is recommended for bifurcation analysis of nonlinear
systems in research as well as in the classroom.
Keywords
MATCONT, bifurcation diagrams, nonlinear systems.
1. Introduction
The advent of chaos and the science of nonlinear dynamic systems has caused the scientific
community to rethink the way it understands nature and its patterns as well as the manner in
which engineering systems are modelled, designed, operated and controlled. In chemical and
process systems engineering, much progress has been made in understanding the complex
nonlinear behavior (multiplicity, oscillations and chaos) of a variety of systems ranging from
the ideal lumped model CSTR to the distributed model packed bed reactor; from the type IV
FCCU units to more complex configurations involving autothermal and reverse flow reactors.
Distillation columns and absorbers have also been studied. Extensive reviews on this subject
are available (Elnashaie and Grace, 2007, Elnashaie et al., 2006, Elnashaie and Elshishini,
1996, Marquardt and Mönnigmann, 2005 and Lee and Chang, 1996).
Industrial interest in these phenomena remains at a low. This is partially due to the
philosophy of avoiding the troublesome operating regions altogether (Elnashaie and Grace,
2007). With the great advancements made in dynamic modelling, computational power and
digital control, the time is ripe to explore the possible advantages of higher yield and
efficiency which may be achieved in these non-conventional dynamic operating regimes. The
implications of these studies on the design, operation and control of chemical/biological
reactors cannot be ignored in a world where the cost of energy and time is on the rise. Among
the life sciences, it is becoming increasingly obvious that dynamic systems theory has a lot to
contribute to our understanding of life and nature. The inherent non-monotonic dependencies
of biological factors on environmental conditions along with the synergetic coupling between
reaction and diffusion phenomena lead to complex dynamics in many biological systems
(Murray, 2002 and Murray, 2003).

3. The IUP Journal of Chemical Engineering, Vol. III, No. 3, 2011.
In order to facilitate the application of the much advanced mathematical theory of bifurcation
and nonlinear dynamics by practically oriented engineers and biologists, a robust, flexible
and user friendly software tool is necessary. Such software would require the integration of
advanced numerical algorithms for solution of differential and difference equations,
bifurcation routines and algorithms for computing various dynamical parameters of interest
(eigenvalues, periods of oscillations, floquet multipliers etc.) as well as an advanced,
interactive visualization package. It could have a similar impact to that of the now hugely
popular computational fluid dynamics (CFD) packages (ANSYS, COMSOL etc.) which have
allowed engineers studying fluid systems to focus on the engineering problem at hand and
experiment with various design and operational possibilities on a computer. This has made
the optimization of processes so much easier and brought the complex mathematics behind
CFD within the reach of non-mathematical workers.
2. Software for Bifurcation Analysis of Nonlinear Systems
Over the past decade and a half, various software packages have emerged with the objective
of aiding investigators in the exploration of dynamical systems via bifurcation analysis. The
initial versions were simple codes made available under the names of AUTO86, LINLBF,
BIFOR2, PATH and LOCA. The next generation saw interactive programs like AUTO97
(Doedel et al., 1997) XPPAUT and LOCBIF. AUTO97 was a very popular tool used by many
workers. Its latest version is AUTO07. XPPAUT is a dynamical systems analysis tool which
interfaces with AUTO. It is popular amongst mathematical biologists (Ermentrout, 2002). For
a windows version, XPP adopted LOCBIF, another continuation toolbox. LOCBIF eventually
evolved into CONTENT (Kuznetsov and Levitin, 1997). The focus of this paper is a
relatively new software package, which has been actively supported and continually
advanced since its introduction (2003), called MATCONT (Dhooge et al., 2003).
MATCONT is an improved, MATLAB version of CONTENT. It is a MATLAB toolbox with
a graphical user interface or GUI (without the GUI it is called MATCONT_CL). MATLAB
is a numerical computing environment which is now widely used in academia and industry
and is popular for its strong matrix based computational capabilities and visualization tools.
MATCONT with its GUI provides a user friendly experience with the added advantage of
generating visualizations directly without using any other software. Further, it can take full
advantage of the advanced capabilities of MATLAB including its robust integrators and
symbolic computation toolbox. A comparison of the features of MATCONT with AUTO and
CONTENT as of 2005 is presented in Table 1 (Kuznetsov, 2005). While most of these
features are not discussed here, the comparison is useful in gauging the merits of MATCONT
vis. a vis. other popular software. It is the goal of this paper to put this promising software to
the test and evaluate its performance as a tool for the non-mathematical worker who is
interested in exploring a dynamical system.
3. Isothermal Stirred Tank Ethanol Fermentor- an Illustrative Example
To evaluate the capabilities of MATCONT, an ethanol fermentor system which exhibits
multiplicity of steady states, oscillation, period doubling and chaos is adopted as an
illustrative example. In this isothermal fermentor system, the ethanol (P) is produced on
fermentation of a sugar based substrate (S) by the microorganism Zymomonas mobilis. The
dynamics of this system were studied in detail by Garhyan et al. (2003). Garhyan and
Elnashaie (2004) went on to demonstrate the new system dynamics when a selective
membrane to separate ethanol was included in the process. Ethanol acts as an inhibitor to the

4. The IUP Journal of Chemical Engineering, Vol. III, No. 3, 2011.
microorganism (Z. mobilis). The incorporation of the membrane was shown to stabilize the
chaotic regime of operation allowing the achievement of higher yields. They also
experimentally verified the model and the predictions of bifurcation theory (Garhyan and
Elnashaie, 2005). They patented their work in 2005 (US 2005/0170483 A1), as an invention
entitled- „Chaotic Fermentation of Ethanol‟. This system has been selected since it is well
studied and provides an example of the practical benefits which result from the application of
bifurcation theory to industrial systems.
Features A C M
Time integration ● ●
Poincaré maps ●
Continuation of equilibria ● ● ●
Detection of branch points and codimension 1 bifurcations of equilibria
(hopf and limit points)
● ● ●
Computation of normal forms for codimension 1 bifurcations of equilibria ● ●
Continuation of codimension 1 bifurcations of equilibria ● ● ●
Detection of codimension 2 equilibrium bifurcations ● ●
Continuation of limit cycles ● ● ●
Detection of branch points and codimension 1 bifurcations of cycles ● ● ●
Continuation of codimension 1 bifurcations of cycles ● ●
Branch switching at equilibrium and cycle bifurcations ● ● ●
Continuation of branching points of equilibria and cycles ●
Computation of normal forms for codimension 1 bifurcations of cycles ●
Detection of codimension 2 bifurcations of cycles ●
Continuation of orbits homoclinic to equilibria ●
Table 1: Comparison between AUTO, CONTENT and MATCONT. Symbols: A- AUTO, C-
CONTENT, M- MATCONT
The system model, without the selective membrane, consists of 4 nonlinear ordinary
differential equations. The model was developed by Jobses and coworkers who also verified
the same experimentally (Jobses et al., 1985, Jobses et al., 1986, Jobses, 1986). An
unsegregated structured two compartment representation was adopted in their work in which
the biomass is described in terms of two components- a key cellular component (e) which is
essential for growth and ethanol formation and the non-active component (X). The synthesis
of (e) has a nonlinear dependence on ethanol concentration. Thus, the inhibition of ethanol is
modelled as an indirect effect on specific growth rate via component (e). The four model
equations are given below and parameter values used in the simulation are given in Table 2
(Garhyan et al., 2003). The state variables are the concentrations of the active component (e),
the inactive biomass (X), the substrate (S) and ethanol (P).
  ee
S
es
PP
e
DCDC
CsK
CC
CkCkk
dt
dC







 0
2
321
(1)
XX
SS
eSX
DCDC
CK
CC
p
dt
dC







 0 (2)
SSXS
SS
eS
SX
S
DCDCCm
CK
CC
Y
p
dt
dC











 
 0
1
(3)

5. The IUP Journal of Chemical Engineering, Vol. III, No. 3, 2011.
PPXP
SS
eS
PX
P
DCDCCm
CK
CC
Y
p
dt
dC












 0
1
(4)
The dilution rate (D) and substrate concentration in the feed (Cso) were taken as bifurcation
parameters since they are most easily manipulated during design and manufacturing. A
detailed study involving several bifurcation diagrams was presented by Garhyan et al. (2003).
These results are reproduced using MATCONT. Three of these diagrams are presented below
to aid in the demonstration and discussion of MATCONT‟s capabilities.
Table 2: Parameters used in simulation
parameter description and unit valuea
k1 regression coefficient (h-1
) 16.0
k2 regression coefficient (m3
/kg∙h) 4.97×10-1
k3 regression coefficient (m6
/kg2
∙h) 3.83×10-3
mS maintenance coefficient (substrate utilization) (kg/kg∙h) 2.16
mP maintenance coefficient (product formation) (kg/kg∙h) 1.1
YSX yield coefficient of biomass from substrate (kg/kg) 2.44498×10-2
YPX yield coefficient of product from biomass (kg/kg) 5.26315×10-2
KS saturation constant (kg/m3
) 0.5
p maximum growth rate (h-1
) 1
D dilution rate (h-1
) 4.0
V fermentor volume (m3
) 0.003
CX0,P0,e0 concentrations in feed (kg/m3
) 0,0,0
ρ medium density (kg/m3
) 789
a
Parameter values of the system possessing oscillatory behavior as demonstrated
experimentally by Jobses and coworkers (Jobses et al., 1985, Jobses et al., 1986 and Jobses,
1986). The same parameter values were adopted by Garhyan and coworkers (Garhyan et al.,
2003).
4. Plotting Bifurcation Plots in MATCONT
4.1. One parameter bifurcation plots
One parameter bifurcation plots describe the change in asymptotic behavior of a system as
one parameter is varied while all others are held constant. Fig (1) and (2) show the
bifurcation diagrams when D and CSO are held constant respectively. The presence of an
incomplete hysteresis type static bifurcation in both figures leads to multiple steady states for
certain parameters. One limit point (saddle node point) at which a saddle state and a node
merge and cancel out is seen in each figure. The supercritical hopf bifurcation (H) is the
parameter value at which the steady state becomes dynamically unstable leading to the
gradual onset of stable oscillations (limit cycles) of period one. As the limit cycle approaches
the saddle states, a period doubling bifurcation occurs and the period of oscillations double
from 1 to 2 to 4, finally terminating in a homoclinic bifurcation on colliding with the saddle
states. Note that for these parameter values chaos is not observed as the cycles terminate
homoclinically before the onset of chaos via period doubling.

6. The IUP Journal of Chemical Engineering, Vol. III, No. 3, 2011.
Fig. 1. Bifurcation diagram with D as the bifurcation parameter and CS0 at 150.3 kg/m3
. H-
hopf point, LP- limit point and PD- period doubling. The dotted line denotes unstable
states while the solid line denotes stable states
Fig. 2. Bifurcation diagram with CS0 as the bifurcation parameter and D at 0.045 h-1
.
Nomenclature same as Fig. 1
The major steps involved in plotting these diagrams in MATCONT are described below:
1. Firstly, the system of equations is defined in an interactive window, in which the state
variables and parameters are to be specified and the equations typed out in simple
MATLAB syntax. The MATLAB symbolic toolbox can be used to compute analytic
expressions of derivatives necessary for computation. This significantly improves
computation time especially when high accuracy is desired.
2. In order to begin the continuation of static equilibria, an initial equilibrium point
(steady state) is required. This can be obtained by time integration at the desired
parameter values. This task is made easy by MATCONT which allows the user to
integrate the equations in time using any of MATLAB‟s ordinary differential equation
integrators. This operation can be carried out in the same GUI framework which
eliminates the necessity of writing a separate m-file (MATLAB code file) for the task.
Moreover, once a steady state is reached, the value of the last point can be
interactively selected and the continuation of static equilibria initiated. Garhyan et al.
(2003) have employed a separate set of codes using IMSL FORTRAN routines for
time integration.

7. The IUP Journal of Chemical Engineering, Vol. III, No. 3, 2011.
3. As the continuation proceeds, MATCONT plots the results on a graphic window in
real time. When a bifurcation point is reached MATCONT pauses and returns the
precise location of the point along with dynamic parameters of interest. For e.g., at a
hopf point the first lyapunov coefficient is returned which is characteristic of the
stability of the emergent limit cycle (the emergent cycle is stable for a negative
coefficient as is the case in Fig 1 and 2). This computation results in a static
bifurcation diagram which depicts the steady states (equilibria), their stability and the
hopf and limit points.
4. In order to analyze the behavior of the oscillations as a parameter is changed, a
dynamic bifurcation diagram involving limit cycle continuation is required. The limit
cycle continuation can be started from the hopf point detected in the static
continuation. Else it can also be started from a limit cycle in the same manner as the
static continuation was started from an equilibrium point. In this regard, MATCONT
provides another useful feature in which a limit cycle can be detected from the output
of a time integration, provided a limit cycle does exist for the chosen parameter values
(which can be determined from the static bifurcation diagram) and is asymptotically
reached from the selected initial conditions.
5. The limit cycles are plotted as straight lines extending over the range of state variable
values covered by a cycle. The plotted cycles are spaced much closer in Fig 1 than in
Fig 2. This is due to the different step sizes used for advancing the bifurcation
parameter in each figure which was done for illustratory purposes. The period
doubling bifurcation is detected just before the cycles meet the saddle states after
which a homoclinic termination occurs. Further study of the doubling cycles is best
accomplished by the use of Poincare bifurcation diagrams and since MATCONT does
not have this feature, they are not discussed in the present paper.
6. All the above operations were carried out by a few clicks of the mouse. Numeric
windows can be opened to display important quantities in real time as the
continuation proceeds, such as eigenvalues and floquet multipliers which allow the
determination of the stability of equilibria and limit cycles respectively. The graphical
window can be opened as 2D or 3D plots and the axes variables and their range easily
adjusted. Visualizations can be saved in a variety of high quality formats. Saving
them as a MATLAB figure file (.fig) allows the user to reopen them in MATLAB and
make cosmetic changes as required. The saddle and unstable equilibria branches have
been represented with dotted lines (MATCONT represents them as solid lines) by
making suitable modifications to the figure file.
4.2. Two parameter bifurcation plot
A two parameter bifurcation diagram is useful as a parameter map from which regions of
multiplicity and oscillatory behavior operation can be identified. The bifurcation points such
as hopf and limit points are plotted as two parameters are varied. This gives a broader, though
less detailed picture of multiplicity, oscillations and their interaction via homoclinic
terminations. This diagram for varying values of D and Cso is given below (Fig. 3). The
curves were manually color coded after the image was generated. It is plotted in MATCONT
by selecting a bifurcation point from a one parameter bifurcation diagram and then starting a
two parameter continuation.

8. The IUP Journal of Chemical Engineering, Vol. III, No. 3, 2011.
Fig. 3. Two parameter bifurcation diagram with CS0 and D as bifurcation parameters.
5. Critical Comments
In this section the difficulties encountered while studying various nonlinear systems, using
MATCONT, are discussed.
1. In the analysis of an electrochemical oscillator (Koper and Sluyters, 1991) the model
definition required a conditional statement (if-else). This statement cannot be included
in the GUI system definition window. Hence, it was necessary to make suitable
modifications to the MATLAB m-file which defines the system equations after it was
generated by MATCONT.
2. A novel continuously stirred decanting reactor (CSDR) studied by Khinast et al
(1998) shows rich dynamic behavior. The model consists of 13 differential algebraic
equations (DAEs) with a state dependent mass matrix. Unfortunately, MATCONT
does not have the capability to work with DAE models and could not be used to study
this system.
3. MATCONT does not have option to modify plotting features such as color, number of
points plotted, markers etc. in the GUI. Thus, as mentioned previously, changes must
be made after the figure is plotted which can be tedious. Further, the limit cycles are
represented by a number of points along the cycle which is in contrast to the standard
practice of using markers for the extreme values of oscillation of the dependent
variable. This makes visualization difficult; moreover the number of markers on a
limit cycle cannot be modified even after generating the plot.
6. Conclusion
In this work, the utility of MATCONT as a tool for studying multiplicity and dynamic
oscillations of nonlinear systems has been demonstrated. It has proved to be user friendly
especially for users of MATLAB. Its strengths include convenient access to powerful
visualization tools and integrators and a large variety of features which encompass all the
capabilities of present bifurcation software, with some additions (Table 1). MATCONT has

9. The IUP Journal of Chemical Engineering, Vol. III, No. 3, 2011.
several advanced features, not discussed in this paper, which are helpful in studying complex
dynamical behavior including quasi-periodicity and chaos. Apart from systems of differential
equations it can also be used to study discrete dynamical maps (systems of iterative
difference equations). The possibility of interfacing MATCONT with other MATLAB
toolboxes holds promise for the future. For e.g., a module which interfaces between
MATCONT and the MATLAB optimization toolbox would make an excellent tool for
optimizing systems while considering changes in output due to bifurcations.
The MATCONT team continues to work on adding new features and fixing bugs. This
software is freely available for download at Source Forge
(http://sourceforge.net/projects/matcont) and can be installed and run easily by a MATLAB
user. It is possible to use MATCONT to investigate nonlinear systems with a minimum of
mathematical knowledge, thus allowing practically oriented engineers and biologists to
analyze nonlinear systems with just an intuitive feel for the related mathematics and
knowledge of essential rules and theorems. Such a tool would be particularly useful in
applied engineering studies as well as in classroom courses on nonlinear dynamics and
bifurcation theory.
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