On the stability of the mixing flow dynamical system perturbed with a
logistic term
ADELA IONESCU
Department of Applied Mathematics
University of Craiova
13 A.I.Cuza Str. Craiova
ROMANIA
Abstract: - This paper continues some recent work in a powerful mathematical domain, with applications in
connected scientific fields, namely the stability of dynamical systems. In fluid mechanics, stabilizing a
dynamical system is a challenging task and it can be done by various ways.
Stabilizing a dynamical system could be often easier if we approach controllable systems, because in this
form, there can be imposed some bounds on its behavior, by studying the improvement of the operators that
describe the system.
In this paper, the mixing flows dynamical systems are taken into account, more exactly the kinematics of
mixing flows. The stability analysis of the mixing flow is taken into account, in the case of perturbation with
a logistic term. The results can be extended to some other versions of the model.
Key-Words: - Kinematics of mixing; Dynamical systems in control; Lyapunov stability; Lyapunov function;
Computational methods; Algebraic methods
Received: March 11, 2023. Revised: November 22, 2023. Accepted: December 23, 2023. Published: January 24, 2024.
1 Introduction
1.1. General lines
In fluid dynamics, the turbulence is a widespread
phenomena; also, it is a basic feature for most
systems with few or infinity freedom degrees. It can
be defined as chaotic behavior of the systems with
few freedom degrees, which are far from the
thermodynamic equilibrium. A turbulent flow can
generally exhibit all of the following features: [1]
1. random behavior;
2. sensitivity to initial conditions;
3. extremely large range of length and time scales;
4. enhanced diffusion and dissipation;
In this area two important zones are distinguished:
[1]
- The theory of transition from laminar smooth
motions to irregular motions;
- Characteristic studies of turbulent systems.
Osborne Reynolds’ experiments, briefly described
in [1] and Reynolds’ seminal paper [2] of 1894 are
among the most important results produced on the
subject of turbulence.
The Reynolds’ number was identified as the only
physical parameter involved in transition to
turbulence in a simple incompressible flow over a
smooth surface. Moreover, since turbulence is too
complicated to allow a detailed understanding,
Reynolds introduced the decomposition of flow
variables into mean and fluctuating parts. After that,
a lot of studies were produced to obtain some
predictable techniques based on his viewpoint.
In hydrodynamics, the transition problem starts
at the end of last century, with the pioneering works
of Reynolds and Lord Rayleigh. The method of
considering the linear stability of basic laminar flow
until infinitesimal turbulences was highlighted as a
good investigation. Nonlinearity can act in the sense
of stabilizing the flow and therefore the primary
state is replaced with another stable motion which is
considered as secondary flow. This one can be
further replaced with a tertiary stable flow, and the
process goes on. It is thus obtained a bifurcations
sequence, and Couette-Taylor flow is a widespread
example in this sense [3].
1.2.The kinematics of mixing framework
A flow has the general mathematical formula:
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.2
Adela Ionescu
E-ISSN: 2945-0489
11
Volume 3, 2024
X
t
XX
t
x0
,
(1)
In the continuum mechanics the relation (1) is
named flow, and it is a diffeomorphism of class Ck.
It must satisfy the following equation [4]:
j
i
tX
x
XDJ detdet
(2)
Here D denotes the derivation with respect to the
reference configuration, in this case X. The equation
(2) implies two particles, X1 and X2, which occupy
the same position x at a moment. In the kinematics
of mixing, it is taken into account a basic flow
(which can be water) containing a biologic material
mixed in it. Therefore the basic measure is the
deformation gradient F, with the relation:
j
i
ij
T
tX X
x
F,XF
(3)
where
X
denotes differentiation with respect to
X. According to equation (2), F is non- singular.
After defining the basic deformation of a material
filament and the corresponding relation for the area
of an infinitesimal material surface, the deformation
measures are defined: the length deformation λ and
surface deformation η, with the following relations
[4]:
2/1
1
2/1 :det,: NNCMMC
F
(4)
where C (=FT·F) is the Cauchy-Green deformation
tensor, and the vectors M, N are the orientation
versors in length and surface respectively.
Very often, in practice is used the scalar form of (4),
details can be found in [4].
A central study point in the kinematics of mixing is
the deformation efficiency, which can be naturally
quantified. In this context, the basic qualitative
quantities in the kinematic of mixing are defined;
the first one is the deformation efficiency in length
[4]:
1
2/1
:
/ln DD
DtD
e
(5)
Similarly, there is defined the deformation efficiency
in surface, eη : in the case of an isochoric flow (the
jacobian equal 1), the following equation holds:
1
2/1
:
/ln DD
DtD
e
(6)
where D is the deformation tensor, obtained by
decomposing the velocity gradient in its symmetric
and non-symmetric part.
The flows with a special form of the
deformation gradient F have a great interest,
because in this class there are contained the
Constant Stretch History Motion –CSHM flows.
[4,5]
2 Computational stability analysis.
Recent results
2.1. Computational Lyapunov analysis
When considering a differential equation
modelling the evolution of a phenomenon, we must
always take into account the fact that the
reproduction of the initial conditions is never
entirely identical. Therefore is very important to
study how small variations in the initial conditions
will introduce small variations in the phenomenon
evolution.
Stability is a well-known property of the
solutions of differential equations in Rn of the form
by which, given a “reference" solution
, any other solution starting
close to
remains close to
for long times.
Although the converse theorems provided a great
help for this problems, starting with 1950’s, they are
not very constructive in practice, since they use the
solution trajectory of the system to construct the
Lyapunov function, but the solution trajectories are
generally not known [6].
The Lyapunov theorem is of great importance in
system theory, giving the possibility of establishing
stability or asymptotic stability of equilibrium points
without explicitly computing trajectories. [6,7].
Theorem 1 (Lyapunov). Let be an
equilibrium point for the system (1). Let
be a positive definite continuously differentiable
function.
1. If is negative semi-definite, then
xe is stable;
2. If is negative definite, then xe is
asymptotically stable.
The theorem gives the existence of a Lyapunov
function but does not provide a method to compute
one. If for linear systems, this issue arises naturally,
in general computing a Lyapunov function is an
open problem giving rise to different methods to
construct it.
The two basic Lyapunov criteria are very useful
for finding a Lyapunov and control Lyapunov
function. The first one is based on eigenvalues
analysis, and the second one on the monotonicity of
the function V [7].
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.2
Adela Ionescu
E-ISSN: 2945-0489
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Volume 3, 2024
The first Lyapunov criterion is based on the
eigenvalues analysis.
Let us consider the following continuous-time
nonlinear system:
(7)
In the vicinity of the equilibrium point , let
us consider the corresponding linearized system:
(8)
This criterion has three distinct cases for the
eigenvalues λi of the matrix A [7]:
(i) If for all i, then is
asymptotically stable;
(ii) If there exits at least one i such as
then is unstable;
(iii) If there exits at least one i such as
and for all other λj, j≠i,
, then we cannot conclude
anything about the stability of .
In this case we say that the criterion is
not effective
We concern if Lyapunov functions always exist.
How could we find such a function? For the first
question the answer is generally positive but,
finding a Lyapunov function is not immediate, since
the converse theorems assume the knowledge of the
solutions of the system (7) [6,7]. Therefore, refining
the definition of Lyapunov function and establishing
a more specific context for it is very necessary.
Between the specific directions of Lyapunov
function research, two are very wide used in
applications: the control Lyapunov functions and
sum of squares (SOS) Lyapunov functions [7,8,9].
We recall briefly in what follows the definition of
control Lyapunov functions.
Similar with the system (7), we can define a
control system as follows:
(9)
where is the control. The control is an
open-loop control if u is function of time, u = u(t)
and closed-loop if u = k(x). The closed-loop
control is in fact the feedback control. If the
feedback has been fixed, u=k(x) and the equilibrium
in the origin has a desired stability property, then we
have a feedback stabilized system [7,10].
The system (9) is called locally, asymptotically null-
controllable, [8] if for every in a neighborhood of
the origin there is an open-loop control u such that
the solution of the system with initial value tends
asymptotically towards the origin. In this context,
Sontag [10] introduced the control Lyapunov
function (CLF) as follows:
(10)
Where V is a positive definite function and γ is a
comparison function [8]. Asymptotic null-
controllability cannot be characterized by smooth
control Lyapunov functions. Therefore, more
general definitions of differentiability like the Dini-
or the proximal sub-differential must be taken into
account. Details about the refinements of the
relation (10) can be found in [10].
2.2 Recent results
Lyapunov functions are not always “energy-
like” functions, but they have some similarities with
energy-like functions in certain contexts. In the
stability theory, Lyapunov functions are used to
analyse the stability of an equilibrium point of a
dynamical system. Energy functions often have a
similar role in physical systems; still Lyapunov
functions can be more general.
For a linear system case , finding a
Lyapunov function implies finding a matrix P such
that is negative definite [11]. Then the
associated Lyapunov function is given by
. But although this seems not very
complicated, it was seen only recently that, in this
context, both are sum of squares
functions!
In the case of non-linear systems, there are
involved supplementary requirements for the
quantities implied in calculus. Therefore related
mathematical methods provided recently a useful
help. Between them, the convex analysis through
the semidefinite-programming provided very useful
algorithms.
The sum of squares (SOS) technique has
significant impact not only in optimization, but also
in convex analysis and especially in control theory.
The SOS technique generalizes a computational
appliance in control theory, “Linear Matrix
Inequalities” – LMI. Parrilo and Ahmadi had
important contributions in this field. [9]. With this
technique, the models can be easier handled and
most solutions can be found numerically.
In recent papers, [12,13,14] it was taken into
account the stability of the basic form for the mixing
flow dynamical system. When studying the mixing
flow phenomena, one starts from the widespread
kinematic 2d model [4]:
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.2
Adela Ionescu
E-ISSN: 2945-0489
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Volume 3, 2024
(11)
Although this is a linear model, when associating
the corresponding initial condition
(12)
it is obtained a complex solution for the Cauchy
problem (11)-(12) [3,4]. The geometric standpoint is
very interesting, the streamlines of the above model
satisfy the relation
and this is
corresponding to some ellipses with the axes rate
if
K is negative, and to some hyperbolas
with the angle
between the
extension axes and x2, if K is positive [4].
To this broad isochoric flow, one can easily
associate the corresponding 3d dynamical system
[3]:
(13)
with the third component standing for the movement
velocity of the system.
In the 3d case the nonlinear system has a complicate
behaviour, the influence of the parameters leading to
a far from equilibrium model. The perturbed model
was taken into account and it was found again a
strong sensitivity with respect to parameters [3].
In 2d case, some perturbations of the mixing
flow dynamical system were taken into account. In a
first stage, the feedback linearization [7] of the
model in a slightly perturbed form was performed,
namely for the system:
(14)
and an interesting conclusion was found, namely it
was found a different parameters’ distribution for
the feedback linearized form of the model. Also, a
SOS Lyapunov function was found both for the
initial and for the feedback linearized model [12].
In a next stage, a control Lyapunov function
was found for the model (14) in a controlled form
[14]. It is important to notice that this can be
realized in feasible conditions for the parameters.
Further, in [13] it was realized a good comparison
analysis concerning the existence of a Lyapunov
function, for the initial form versus the feedback
linearized form, for the mixing flow model
perturbed with a logistic type term.
3 Results for the mixing flow
dynamical system perturbed with a
logistic type term
The mixing flow dynamical system in 3d case
is a nonlinear model. If we start to perturb it, then
we get a strongly nonlinear model. In [3] it was
analysed the solution of the model in the non-
perturbed case and with a perturbation with a
logistic type term. So the following model was
taken into account:
)(
.
3
121
.
2
2
.
1
constcx
xxGxGKx
xGx
(15)
For this model a complex solution was found, with
the expression:
33
2
121
2
1
121
12
2
121
1
121
1
exp
342
2
2
exp
342
2
2
exp
342
2
2
exp
342
2
2
Xtcx
tKG
K
XXX
K
tKG
K
XXX
Kx
tKG
K
XXX
tKG
K
XXX
x
(16)
In (16) the following notations were made in order
to simplify the expressions:
2
341
,
2
341
21
K
K
K
K
Taking into account the stability analysis for a
model like (15) is easy to observe that this is a
challenging task. Therefore, starting with the 2d
case would bring useful information.
2
0
2
0
2
;
1
0
1
0
1XtxxXtxx
.,11,
.
3
1
.
2
2
.
1
constcK
cx
xGKx
xGx
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.2
Adela Ionescu
E-ISSN: 2945-0489
14
Volume 3, 2024
For the aim of the present paper, the 2d case is taken
into account for the mixing flow perturbed with a
logistic type term:
(17)
As it is about a nonlinear model, finding a
Lyapunov function (in the form CLF or SOS) is
quite difficult, since it is difficult to fulfill the
criteria. Therefore, the stability analysis is
approached according to the eigenvalues criterion.
In order to have the system in the form (7), we
consider it formally in a “controlled” form, adding a
control on the first component:
(18)
It is obvious that the origin is solution for (18).
The matrix associated to linearized system around
the origin is:
(19)
Consequently, the characteristic equation is
(20)
According to the eigenvalues criterion for stability,
we impose for the discriminant of (20) the condition
which implies, after the calculus, the
condition
(21)
This is equivalent to
(22)
From the above inequality, taking into account that
we have two possibilities:
(23)
It is easy to evaluate each of the situations in (23),
starting from each hypothesis for G and then
evaluate G itself from the second inequality.
After all calculus, taking into account the basic
conditions for G and K, we obtain the situations:
(24)
So from the inequalities (24), we deduce that the
control p depends on the parameter K.
The above relations are feasible from the parameters
standpoint, therefore we can assess that in the form
(18) of the mixing flow model, it is feasible to apply
the eigenvalues criterion.
Going further and calculating the eigenvalues, we
find:
(25)
Thus, the eigenvalues criterion is realized, namely
in the conditions (24) for the parameters, we have
the situations:
a) If G<0 then the origin is asymptotically
stable;
b) If G>0 then the origin is asymptotically
unstable
Conclusions
The mixing flow dynamical system
perturbed with a logistic type term provides a
strongly nonlinear model. In [13] it was realized a
Lienard type construction of the model and based on
it, a Lyapunov function was found both for the
initial and the feedback linearized model.
In the present paper the stability analysis is
approached for the mixing flow model perturbed
with a logistic term. Namely, it is found that in a
controlled form and some feasible conditions for the
parameters, the first stability criterion is feasible and
some simple feasible conditions for the stability of
the origin are found.
Thus, although a nonlinear model, the
dynamical system modeling the kinematics of
mixing can reach the stability. This enables us to
take into account the construction of a Lyapunov
function, in a suitable form for the controlled model.
Also, further versions of the model and some 3d
versions of vortex models will be taken into account
in order to complete the panel of events for the
mixing flow theory.
International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.2
Adela Ionescu
E-ISSN: 2945-0489
15
Volume 3, 2024
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Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article
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(Attribution 4.0 International, CC BY 4.0)
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International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2024.3.2
Adela Ionescu
E-ISSN: 2945-0489
16
Volume 3, 2024
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
