Modeling and Analysis of the Monotonic Stability
of the Solutions of a Dynamical System
Abstract: - This study aims to develop an approach for the qualitative analysis of the monotonic stability of
specific solutions in a dynamical system. This system models the motion of a point along a conical surface,
specifically a straight and truncated circular cone. It consists of two nonlinear ordinary differential equations of
the first order, each in a unique form and dependent on a particular parameter. Our proposed method utilizes
traditional mathematical analysis of a function with a single independent variable, integrated with
combinatorial elements. This methodology enables the precise determination of various qualitative cases where
the chosen function's value monotonically decreases as a point moves along the conical surface from a specified
starting point to a designated point within a final circular region. We assume that the system's partial solutions
include a finite number of inflection points and multiple linear intervals.
Key-Words: - stability, mathematical analysis, monotonic functions, conical surface, nonlinear differential
equations, partial solution
1 Introduction
Analyzing the stability of solutions in nonlinear
dynamic systems is a crucial challenge in
contemporary science and technology. The
conventional method for examining the stability of
dynamic systems, represented by ordinary
differential equations, involves the second
Lyapunov method, which is widely used to assess
solution stability [1-4]. However, this traditional
approach has a notable limitation: it presumes the
knowledge of the Lyapunov function.
Therefore, developing alternative methods to
evaluate the stability of partial solutions in systems
of ordinary differential equations is both
scientifically and practically significant. One such
approach is a method based on the concept of
monotonic stability in stable partial solutions of
nonlinear differential equations. This methodology
is explored in several articles [5-7], with the distinct
aspect of these studies being their focus on
mathematical models framed as systems of first-
order ordinary differential equations.
The objective of this work is to perform a
qualitative analysis of the monotonic stability of
solutions within a dynamic system that describes the
motion of a point on a conical surface. We assume
that these partial solutions encompass a finite
number of inflection points and multiple linear
intervals. Our aim is to establish conditions for the
monotonic stability of these partial solutions. In
achieving the primary results, we propose
employing classical mathematical analysis
techniques for functions with a single independent
variable, combined with combinatorial elements.
This method allows for the precise identification of
various qualitative scenarios where the value of a
selected function decreases monotonically as a point
traverses a conical surface from a specified initial
point to a designated point within a finite circular
area.
2 Preliminaries
Let us assume that all spatial curves depicting the
behavior of specific solutions to the dynamical
system under consideration are confined to the
surface of a right, truncated circular cone. The upper
boundary of this cone's surface is a circle with the
largest radius
0
, while the lower The cone's vertex
is situated at the origin O of the rectangular
Cartesian coordinate system OXYZ. We will refer to
this surface as the stability cone, which is illustrated
in Fig. 1.
In the mathematical model, we employ a
spherical coordinate system. Assume that the initial
position of a point moving along the trajectory of
the solution to the dynamical system is situated on
the upper circle, with its coordinates being
0 0 0
( , , )
.
VLADISLV V. LUYBIMOV,
Department of Further Mathematics,
Samara National Research University,
34, Moskovskoe shosse, Samara, 443086
RUSSIA
5HFHLYHG0DUFK5HYLVHG2FWREHU$FFHSWHG1RYHPEHU3XEOLVKHG'HFHPEHU
PROOF
DOI: 10.37394/232020.2023.3.13
Vladislv V. Luybimov
E-ISSN: 2732-9941
84
Volume 3, 2023
Fig. 1. Stability cone
Additionally, the final position of the point on the
trajectory is located on the lower circle, with
coordinates
1 1 1
( , , )
. It is important to note that
the condition
01
const
is satisfied. Figure 2
depicts the trajectory of a point demonstrating
monotone stability of the solution on the stability
cone.
3 Equations of motion of a point along
a conical surface
Let us consider a system of continuous differential
equations modeling the motion of a point
( , , )R
along a conical surface within a spherical coordinate
system
10
( , )
df
dt
,
0
const
, (1)
20
( , )
df
dt
.
(1)
In this context, t is an independent real variable,
and
( ), ( )R t t
are non-negative, twice continuously
differentiable functions that serve as specific
solutions to the dynamical system (1), defined over
the interval
01
[ , ]t t t
. Additionally,
10
( , )f
is a
known nonpositive, continuously differentiable
function, defined over the same interval, while
20
( , )f
is a nonnegative, continuously
differentiable function,also defined within
01
[ , ]t t t
.
Furthermore, the function
0
R(t ) (t ) / cos
represents the distance from the origin (0,0,0) to the
point
( , , )R
. Note that the constant value
0
acts
as a parameter in the system of equations (1).
4 Analysis of the Monotonic
Stability of the Solutions of a
Dynamical System on a Conical
Surface
Let us define the concept of monotone stability for a
specific solution
R R(t )
within the dynamical
system (1)–(2). Consider a non-negative solution
R R(t )
of system (1) that meets the following
conditions on interval
01
[ , ]t t t
on the conical
surface at
0
const
:
(i) the function
R R(t )
is defined and twice
continuously differentiable;
(ii) the derivative
2
2
dR
dt
consistently maintains its
sign between inflection points
0 1 2 , , , , ,k m m
or (and) within intervals where the derivative
2
20
dR
dt
(
01
[ , ]t t t
).
Definition 1. A non-negative solution
R R(t )
of
system (1) is considered monotone stable on interval
01
[ , ]t t t
if it satisfies conditions (i)–(ii), and the
solution decreases monotonically on the stability
cone at
0
const
on this interval.
Theorem 1. (Sufficient condition for monotonic
stability). If a non-negative solution
R R(t )
of
system (1) fulfills conditions (i)–(ii), and the
derivative
()dR t
dt
is negative on interval
01
t [t ,t ]
,
then this solution is monotone stable in the interval.
Note. The proof of Theorem 1 is straightforward.
It relies on the fundamental sufficient condition for
a strictly decreasing function of one variable, while
also considering the fulfillment of the inequality
0
( ) 1 0
cos
dR t d
dt dt
and conditions (i)–(ii).
Note. The term “qualitatively different cases of
monotonic stability of solutions” refers to monotone
stable solutions
R R(t )
of system (1) that
demonstrate a unique type of convexity compared to
other solutions exhibiting monotonic stability.
For further analysis of monotone stability, we
will consider the first and second derivatives of the
solution
R R(t )
. Regarding the enumeration of
qualitatively different cases of monotonic stability
of solution
R R(t )
, the following theorem applies:
Theorem 2. If a solution
R R(t )
of the system
equation (1)–(2) satisfies Def. 1, and the number of
inflection points in this solution is 0, 1, 2, ..., m, or
the number of linear intervals in this solution is
1,2,3, ..., n, then the number of qualitatively
PROOF
DOI: 10.37394/232020.2023.3.13
Vladislv V. Luybimov
E-ISSN: 2732-9941
85
Volume 3, 2023
different cases of monotonic stability of this solution
is
2m 3 1 8n 1
CC
.
Proof. To prove Theorem 2, let us divide the
argument into two parts. First, we determine the
number of qualitatively different cases of monotonic
stability for particular solutions
R R(t )
that
conform to the conditions of Def. 1 and have n and
m inflection points in the interval
01
[ , ]t t t
. Initially,
we identify the count of distinct monotonic stability
cases in the absence of inflection points for the
function
R R(t )
over the interval
01
[t ,t ]
. There are
three such cases. In the first linear case
01
t [t ,t ]
,
the derivatives of the particular solution have the
signs:
dR(t) 0
dt
and
2
2
d R(t) 0
dt
. In the second case,
the derivatives of the solution exhibit the signs:
dR(t) 0
dt
and
2
2
d R(t) 0
dt
. In the third case, the
derivatives of the solution have the signs:
dR(t) 0
dt
and
2
2
d R(t) 0
dt
. Next, consider the scenario where
the function
R R(t )
has one inflection point within
this interval
01
[t ,t ]
. The function
R R(t )
is
continuously differentiable in the interval
01
[t ,t ]
.
Consequently, as a point moves through an
inflection point, there is a noticeable change in the
convexity of the function’s graph. The introduction
of one inflection point creates a new interval with a
consistent convexity of two possible types of the
function
R R(t )
, yielding two additional
qualitatively distinct cases of monotonic stability,
distinguished by the sign of the second derivative
over this new interval. Each subsequent inflection
point similarly contributes the potential for two
more qualitatively distinct cases. Thus, for m
inflection points, the number of qualitatively distinct
cases of monotonic stability is given by the
equation:
2m 3 1
C 2m 3
. Secondly, consider the
number of qualitatively different cases of monotonic
stability for particular solutions
R R(t )
that, in
addition to non-linear sections with constant
convexity, also have n linear sections over the
interval
01
[ , ]t t t
. It is demonstrated that the number
of such cases equals
8n 1
C
, as proven using
mathematical induction. For n=1, t there are
81
C8
qualitatively different cases of monotonic stability.
This result can be explained as follows: The
formation of a single linear section at the beginning
of the interval
01
[ , ]t t t
results in two distinct cases
of monotonic stability. These cases are
characterized by distinctly different types of
protrusions in the final nonlinear section of the
interval
01
[ , ]t t t
. Similarly, it is demonstrated that
the emergence of a single linear section in the
interval’s final part
01
[ , ]t t t
results in two distinct
cases of monotonic stability. Specifically, these two
cases are differentiated by the nature of the
protrusion in the initial nonlinear section of the
interval
01
[ , ]t t t
. When a linear section is formed in
the inner part of the interval
01
[ , ]t t t
, it can result
in four qualitatively different cases of monotonic
stability. These cases are distinguished by their
unique constant convexity patterns, which differ in
the two nonlinear sections immediately adjacent to
the linear section on both the right and left sides. As
a consequence, the emergence of a single linear
section in the solution
R R(t )
can give rise to
eight distinct cases of monotonic stability. The
foundation of the method of mathematical induction
has thus been established. The proof for the second
part of this method relies on the fact that each
distinct case with k linear sections in a specific
solution
R R(t )
corresponds to two distinct cases
in a solution with k+1 linear sections. This
relationship is straightforward and does not
necessitate further elaboration. Therefore, if a
formula
8k 1
C
defines the number of distinct cases of
monotonic stability for k linear sections in a
particular solution
R R(t )
, then for k+1 linear
sections in the solution, the number of different
cases of monotonic stability equals
8k 1
C2
. In this
scenario, the equality
8k 1 8(k 1) 1
C 2 C
holds true.
Consequently, the second part of the method of
mathematical induction, the induction step, is also
verified. Therefore, when n linear sections are
formed in a solution
R R(t )
, the number of
qualitatively different cases of monotonic stability is
equal to
8n 1
C
. Summarizing, the total number of
distinct cases of monotonic stability arising from the
formation of 0, 1, 2, ...,m inflection points or 1,2,3,
...,n linear sections in specific solutions
R R(t )
is
equal to
2m 3 1 8n 1
CC
. Thus, the theorem is proven.
Note. Theorem 2 does not undertake a qualitative
analysis of the cases of monotonic stability in
solutions
R R(t )
that simultaneously contain
inflection points and linear sections.
Let us illustrate the application of the established
equality
2m 3 1 8n 1
CC
with an example.
Example. Calculate the number of qualitatively
different cases of monotonic stability when there are
4 inflection points or 3 linear sections on the
solution curve within a given interval.
Solution. Referring to Theorem 2, the number of
distinct cases of monotonic stability can be
PROOF
DOI: 10.37394/232020.2023.3.13
Vladislv V. Luybimov
E-ISSN: 2732-9941
86
Volume 3, 2023
determined as
2m 3 1 8n 1
C C 11 24 35
cases,
considering m = 4 and n=3.
Definition 2. The qualitative analysis of the
monotonic stability of the partial solution
R R(t )
of dynamic system (1) in the interval
01
t [t ,t ]
refers
to examining the convexity of a given strictly
monotonically decreasing solution in the interval.
The theorem is established [7].
Theorem 3. To conduct a qualitative analysis of the
monotonic stability of an unknown non-negative
solution
R R(t )
of system (1), the following
conditions must be met:
(i) the particular solution
R R(t )
adheres to
Def.1 in the interval
01
t [t ,t ]
and this solution is
not linear;
(ii) the first derivative
d (t)
dt
of the known
continuously differentiable function
(t)
is defined
in the interval
01
t [t ,t ]
and retains a consistent
sign throughout this interval;
(iii) the initial conditions
R(0) 0
,
(0)
, and
the final value
1
(t )
are known.
Note. The proof of Theorem 3 closely aligns with
the proof presented in Theorem 3 of the article [7].
Essentially, this proof outlines a method for the
qualitative analysis of the monotonic stability of an
unknown particular solution
R R(t )
. Let us
examine this method.
Proof (Method for qualitative analysis of monotonic
stability of an unknown nonlinear particular
solution
R R(t )
).
Assume that the conditions of Def. 1 are satisfied
and that the solution
R R(t )
is not linear. It's
evident that the function
R R(t )
decreases
monotonically within the interval
01
t [t ,t ]
. The
second derivative of this function,
2
2
dR
dt
, either
maintains its positive or negative sign, or it changes
sign at a finite number of the inflection points of the
function
R R(t )
in the interval
01
t [t ,t ]
. It is
important to note that the function
R R(t )
may be
unknown. Let us outline a method for analyzing the
sign of the second derivative
2
2
dR
dt
of this function
R R(t )
in the interval
01
t [t ,t ]
for the
aforementioned case. To do this, we need to express
the second derivative
2
2
dR
dt
of this function
R R(t )
.
Given that function
R R(t )
is a twice-
differentiable function, we proceed as follows:
21
20
df
d R 1 d .
cos d dt
dt
(2)
(2)
According to system (1), the sign of the first
derivative
d
dt
is constant and known
01
t [t ,t ]
.
Therefore, to ascertain the sign of the second
derivative
2
2
dR
dt
of the function
R R(t )
for all t in
the interval
01
[t ,t ]
, it is essential to determine the
sign of the derivative
1
df
d
for all t over the interval.
Notably, the first derivative
1
df
d
is a known smooth
function
1
df F( )
d
. Given that the argument φ of
the function
F( )
changes strictly monotonically
01
t [t ,t ]
, if the initial value
(0)
is known, then
the sign of the derivative
1
df F( )
d
can be deduced
by directly calculating the values of
F( )
for all φ
within the specified interval
1
[ (0), ]
. This outlines
a method for analyzing the sign of the second
derivative (2) of this function
R R(t )
in the
interval
01
t [t ,t ]
. Once the sign of this second
derivative of the function
R R(t )
is determined for
the entire interval
01
t [t ,t ]
, it enables us to
ascertain the convexity type of the strictly
decreasing solution
R(t )
at each point
01
t [t ,t ]
.
Consequently, in line with Def. 2, we have
effectively conducted a qualitative analysis of the
monotonic stability of the solution
R R(t )
of
system (1), which is strictly decreasing within
interval
01
t [t ,t ]
. With this, the proof of Theorem 3
is concluded.
5 Examples of mathematical
models with monotonic stability
Firstly, let us consider an example where the partial
solution to the system of ordinary differential
equations (1) is represented by a function whose
curve forms a conical helix [8]:
0.2 cos ,x t t
0.2 sin ,y t t
(3)
(3)
0.25 .zt
PROOF
DOI: 10.37394/232020.2023.3.13
Vladislv V. Luybimov
E-ISSN: 2732-9941
87
Volume 3, 2023
This curve is expressed in a spherical coordinate
system as follows:
0.2 ,t
0(1.25) ,acrtg const
(4)
.t
In this scenario, the function
R(t )
is linear.
Indeed, since
0
R(t ) (t )/ cos
, where
0.2 ,t
0
cos cos( (1.25)).acrtg
Consequently, we obtain:
0 2 1 25R(t ) . t / cos(acrtg( . ))
. (5)
Letting the variable t vary from −6.5π to 0 and
differentiating function (5) with respect to t twice ,
we find the first and second derivatives:
0
0 2 0R(t ) . / cos
,
0R(t )
. According to
Theorem 1, this implies that the particular solution
(4) is monotonically stable.
Figure 3 illustrates this conical curve, depicting
the monotonically stable solution (3) in a Cartesian
coordinate system.
Fig.3. Conical curve of a monotonically stable solution (3)
Figure 4 shows a conical curve describing a
monotonically stable solution (3), located on the
stability cone.
Let us consider another example where the partial
solution to the system of ordinary differential
equations (1) is represented by a function that forms
a cylindrical conical helix, described as follows [8]:
Fig.4. Conical curve of a monotonically stable solution (3) on
the stability cone
0.6
2.5 cos ,
t
x e t
0.6
2.5 sin ,
t
y e t
(6)
0.6
3.5 .
t
ze
(6)
This curve, when expressed in a spherical
coordinate system, is represented by:
0.6
2.5 ,
t
e
0(1.4) ,acrtg const
(7)
.t
In this case, the function
R(t )
is nonlinear. This
is evident from the relationship
0
R(t ) (t )/ cos
,
where
0.6
2.5 ,
t
e
0
cos cos( (1.4)).acrtg
Consequently, we derive:
06
2 5 1 4
.t
R(t ) . e / cos(acrtg( . ))
. (8)
When we let the variable t vary from 0 to 10 and
differentiate function (8) with respect to t twice, we
obtain the first and second derivatives:
06
1 5 1 4 0
.t
R(t ) . e / cos(acrtg( . ))
,
06
0 9 1 4 0
.t
R(t ) . e / cos(acrtg( . ))
.
Based on Theorem 1, this suggests that the
particular solution (4) is monotonically stable. It is
apparent that the curve (4) is convex downward
throughout the specified interval for the variable t.
Figure 5 illustrates the cylindrical conical curve,
depicting the monotonically stable solution in a
Cartesian coordinate system.
PROOF
DOI: 10.37394/232020.2023.3.13
Vladislv V. Luybimov
E-ISSN: 2732-9941
88
Volume 3, 2023
Fig.5. Cylindrical conical curve of a monotonically stable
solution (6)
Figure 6 depicts the cylindrical conical curve,
which represents a monotonically stable solution as
per equation (6). This curve is situated on the
stability cone, visually illustrating the stability
characteristics of the solution within the context of
the system.
Fig.6. Cylindrical conical curve of a monotonically stable
solution (3) on the stability cone
6 Conclusion
The mathematical model describing the motion of a
point on a conical surface was formulated as a
system of two nonlinear ordinary differential
equations, dependent on a specific parameter. These
equations were defined using a spherical coordinate
system. Mathematical analysis was employed to
establish conditions for the monotonic stability of
the point's motion on the conical surface. Utilizing
combinatorial methods, an expression was derived
to calculate the number of monotonic stability cases,
incorporating the presence of inflection points and
linear sections on the curve being analyzed.
Furthermore, the article presents a method for the
qualitative analysis of the monotonic stability of the
solution, focusing on examining its convexity. Two
illustrative examples are provided, demonstrating
monotonically stable motion of a point on conical
surfaces.
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PROOF
DOI: 10.37394/232020.2023.3.13
Vladislv V. Luybimov
E-ISSN: 2732-9941
89
Volume 3, 2023
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