=Paper=
{{Paper
|id=Vol-3018/Paper_10.pdf
|storemode=property
|title=Design of Stable Periodic Regimes for one Class of Hybrid Planar Systems
|pdfUrl=https://ceur-ws.org/Vol-3018/Paper_10.pdf
|volume=Vol-3018
|authors=Valentyn Sobchuk,Oleksiy Kapustyan,Volodymyr Pichkur,Olena Kapustian
|dblpUrl=https://dblp.org/rec/conf/intsol/SobchukKPK21
}}
==Design of Stable Periodic Regimes for one Class of Hybrid Planar Systems ==
https://ceur-ws.org/Vol-3018/Paper_10.pdf
Design of Stable Periodic Regimes for One Class of Hybrid
Planar Systems
Valentyn Sobchuk, Oleksiy Kapustyan, Volodymyr Pichkur and Olena Kapustian
Taras Shevchenko National University of Kyiv, Volodymyrska St, 60, Kyiv, 01033, Ukraine
Abstract
The conditions for the existence of limit regimes for the LiΓ©nard equation, the solutions of
which are subjected to instantaneous forces of impulse nature at unfixed moments in time, are
investigated. For this system, the constructive conditions for the existence of periodic solutions
(limit cycles) are obtained such that the phase point of the system when moving along the
corresponding trajectory is subjected to π β β impulse effects for the period. It is shown that
the points that define cycles corresponding to periodic solutions satisfy the Sharkovsky order.
The conditions for the existence of at least one impulsive periodic solution and a single
discontinuous limit cycle are found. The existence of a single stable limit cycle is proved, the
phase point of which will be affected by pulsed forces π β β times. It is shown that a stable
limit cycle under given conditions will exist despite the presence of the influence of
destabilizing forces of impulsed nature.
Keywords 1
differential equation, impulse action, LiΓ©nard equation, limit cycle
1. Introduction
The rapid development of modern science and technology requires constant attention to the study of
nonlinear evolutionary dynamical systems in which there are short-term processes or which are under
the action of external forces, the duration of which can be neglected in the preparation of appropriate
mathematical models.
Such evolutionary models can be found, for example, in mechanics, chemical technology, medicine
and mathematical biology, aircraft dynamics, economics, adaptive control theory and other fields of
science and technology, where we have to study systems under the influence of short-term (pulsed)
external forces called systems with pulsed action.
In fact, it has been found that the presence of impulse action can significantly complicate the
behavior of the trajectories of such systems, even for cases of rather simple differential equations. In
the general case, in the presence of impulse action, the qualitative behavior of solutions of differential
equations (including linear problems with constant coefficients) can be significantly nonlinear and
significantly different from the behavior of such systems in the absence of impulse action.
Studying the nonlinear damping of oscillations in electric circuits, LiΓ©nard obtained a natural
generalization of the famous van der Paul equation. At the same time, the problem of the existence of
periodic regimes is important for oscillating systems in the region. Note that the limit cycle is an isolated
closed trajectory of the vector field (in other words, it is a periodic solution in some neighborhood
which has no other periodic solutions, respectively, all other trajectories from this region tend to the
limit cycle in positive or negative time). Therefore, when modeling many systems of oscillating systems
that are affected by destabilizing factors of instantaneous (impulse) nature, it is important to understand
the conditions for the existence of stable periodic regimes in them. Obtaining appropriate design
conditions allows to develop methods for supporting decision-making on the management of such
systems. Special attention should be paid to models that describe the most natural objects - dissipative
dynamical systems. A dissipative system (or dissipative structure, from the Latin dissipatio - "disperse,
II International Scientific Symposium Β«Intelligent SolutionsΒ» IntSol-2021, September 28β30, 2021, Kyiv-Uzhhorod, Ukraine
EMAIL: v.v.sobchuk@gmail.com (A. 1); kapustyanav@mail.com (A. 2); vpichkur@mail.com (A. 3); olena.kap@mail.com (A. 4)
ORCID: 0000-0002-4002-8206 (A. 1); 0000-0002-9373-6812 (A. 2); 0000-0002-5641-8145 (A. 3); 0000-0002-2629-0750 (A. 4)
Β©οΈ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
111
destroy") - is an open system that operates in the neighbourhood of equilibrium position. In other words,
it characterizes the state that occurs in an nonequilibrium environment under the condition of energy
dissipation. A dissipative system is sometimes called a stationary open system or a nonequilibrium
open system. The dissipative system is characterized by the spontaneous appearance of a complex, often
chaotic structures. Recent studies in the field of dissipative structures allow conclude that self-
organization occurs much faster in the presence of external and internal influences in the system. Thus,
such effects accelerate the process of self-organization.
In the phase plane, the trajectory corresponding to such a solution is represented by the so-called
limit cycle. The limit cycle is an isolated closed curve on the phase plane, to which all integral
curves are approached in the limit case at tβ "". The limit cycle is a stationary mode with a certain
amplitude, which does not depend on the initial conditions, but is determined only by the structure of
the system.
In general, if there is some closed domain on the phase plane such that all phase trajectories crossing
the boundary of this region enter it and there is an unstable singular point within this domain, then the
latter necessarily has at least one limit cycle.
At the same time, when there is a domain on the phase plane from which the phase trajectories do
not come out and in which there are no equilibrium positions (special points), then the
limit cycle exists in this domain, and the rest of all trajectories are wound on it.
Thus, if we find on the phase plane such a two-connected domain that the directions of the phase
trajectories are inverted inside this region on the whole boundary, then we can say that the limit
cycle exists inside this domain.
In fact, this work is devoted to the study of the limit cycles existence conditions for a generalized
second-order differential equation of the LΓ©nard type under the influence of destabilizing
external impulse perturbations.
2. Analysis of literature sources
Classical statements of impulsive-perturbed problems, as well as basic notions concerning
qualitative behavior of solutions for such systems in the case of impulsive effect at fixed moments of
time, were developed in [1-3] as an adequate mathematical tool for describing physical and mechanical
phenomena where instantaneous changes of the phase state are present. Global attracting sets for
impulsive evolutionary systems, including random noise, with impulses at fixed moments of time, were
studied in [4,5]. Robust stability properties for such systems in terms of Input-to-State Stability theory
were considered in [6-8]. Limit cycles for finite-dimensional impulsive dynamical systems, that is,
systems described by ordinary differential equations whose trajectories undergo instantaneous changes
after reaching a certain surface of the phase space, were investigated in [10]. Systematic studying of the
qualitative behavior of impulsive dynamical systems infinite-dimensional spaces was carried out in [11-
15]. Global attracting sets for abstract infinite-dimensional impulsive dynamical systems were
investigated in [16-18]. A modern point of view on systems with mixed types of dynamics, i.e., systems
where there exist both continuous dynamics described by systems of differential equations and discrete,
described by difference equations, was reflected in [19], where such systems were called hybrid
systems.
In [20] the review of the most modern research methods for impulse differential equations solutions
stability and their application to problems of impulse adaptive control is carried out. In [21] the problem
of design the approximate adaptive control, including the case of impulse control functions, is
considered for some classes of infinite-dimensional problems. The well-known method of averaging
for obtaining approximate adaptive control is substantiated. The concept of an impulsive non-
autonomous evolutionary system is introduced. Questions concerning existence and properties of
impulsive attracting sets are investigated. The obtained results are applied to the study of the qualitative
behavior of the two-dimensional impulsive-perturbed Navier-Stokes system.
In [22] the recursive properties of almost periodic motions of impulsive-perturbed evolutionary
systems are studied. The obtained results are effectively applied to the study of discrete systems
qualitative behavior. In [23] the qualitative properties of stability with respect to the external (control)
perturbations for differential equations systems with impulse effects at fixed moments of time are
studied. The transparent criteria of stability conditions for classes of impulsive systems having a
Lyapunov type function are obtained. In [24], non-autonomous evolutionary problems with multi-
valued right-hand parts and with impulse influences at fixed moments of time are considered. The
112
corresponding non-autonomous multivalued evolutionary systems are designed, for which the existence
of a compact global attractor in phase space is proved.
The problems of control and decision-making in the presence of impulse perturbations were
investigated in [25], for periodic solutions were studied in [27]. In [28], the existence of global attracting
sets in multi-valued discontinuous infinite-dimensional evolutionary systems, which can have
trajectories with an infinite number of impulsive perturbations, was proved. The obtained abstract
schemes are applied to the asymptotic behavior study of the weakly nonlinear impulsive-perturbed
parabolic equations and inclusions.
In all the above works the basement of the qualitative theory of differential systems with impulsive
perturbations (jumps) are designed. In essence, the main issues of the qualitative theory of impulsive
systems were investigated with the help of the classical qualitative theory of ordinary differential
equations, methods of asymptotic integration for such equations, the theory of difference equations and
generalized functions. However, the question of the solutions existence for weakly nonlinear impulse
systems has not yet been investigated appropriately.
At the same time, the works in which important results in the field of information technologies and
social communications were obtained deserve attention. In particular, in works [29] studied applied
control algorithm functionally sustainable production processes industry.
3. Conditions for the existence of a limit cycle for the general LiΓ©nard equation
with impulse action
Let us investigate the problem of the existence of harmonic cycles for the LiΓ©nard equation, the
solutions of which are subjected to instantaneous forces of impulse nature at unfixed moments in time.
Consider a dynamical system in which motion is described by a generalized LiΓ©nard differential
equation of the form:
π₯Β¨ + π(π₯, π₯Μ )π₯Λ + π(π₯) = 0, (1)
3
(π₯ β π β β , π β phase space of the system (1), π‘ β β β time) and which is affected by instantaneous
perturbations determined by some operator ππ‘ , that at the moment of reaching a moving point of some
fixed positionπ₯ = π₯β acts according to the rule (π₯, π‘) β (π‘, ππ‘ π₯). Impulse action in such a system
occurs at non-fixed moments of time and increases the amount of motion in the system by a certain
amount πΌ(π₯Λ), which depends on the speed of the moving point at the time of its passage π₯ = π₯β . Next
we will consider that πΌ(π¦), Π΄Π΅ π¦ = π₯Λ, as a function of its argument is continuous.
If π‘β is a certain moment of time at which the moving point reaches position π₯ = π₯β , when an
impulsive action occurs, the impulsive perturbations of the moving point can be written as [27]:
ππ₯ ππ₯ ππ₯
π₯ π π‘| = π π‘| β π π‘| = ππ‘ π₯ β π₯ = πΌ(π₯Λ). (2)
π₯=π₯β π‘=π‘β+0 π‘=π‘ββ0
The description of the physical interpretation of the generalized LiΓ©nard equation and the
characteristics of its phase limit behavior are studied in detail in [30].
Equation (1) is written in an equivalent way as a system
π₯Λ = π¦,
{ (3)
π¦Λ = βπ(π₯) β π(π₯, π¦)π¦.
In the sequel, we will use the notation
π₯ π₯
πΉ (π₯) = β« π(π ) ππ , πΊ (π₯) = β« π(π ) ππ .
0 0
We will consider further that functions π(π₯) and π(π₯, π¦) provide the condition for the existence and
uniqueness of the solution of the system (3). In addition, we will assume that
π₯ π(π₯) > 0 ΠΏΡΠΈ π₯ β 0 (4)
and
Β±β
πΊ (Β±β) = β«0 π(π₯) ππ₯. (5)
In this case, the origin of the phase plane is the only stationary point. It is covered by a family of so-
called energy curves
113
1
π (π₯, π¦) β‘ πΊ (π₯) + 2 π¦ 2 = πΆ. (6)
They are all closed. Under assumption
π(βπ₯) = βπ(π₯) (7)
these curves are symmetrical about both coordinate axes.
Let's denote π¦β = β2πΊ(π₯β ).
3.1. There are periodic solutions that satisfy Sharkovsky's order
Suppose that
10. The generalized LiΓ©nard differential equation (1) is assumed to be satisfied the conditions for
the existence and uniqueness of the solution.
20. The line π₯ = π₯β is transversal to flow (1) everywhere except the trajectory for which the line
π₯ = π₯β is tangent. In this case, we assume that πΌ(0) = 0.
30. Impulsive operator ππ‘ is assumed to be continuous with respect to its variables.
Calculate the complete derivative of the derivative with respect to the system (3)
ππ(π₯, π¦)
( ) = π(π₯)π₯Μ + π¦π¦Μ = βπ (π₯, π¦)π¦ 2 . (8)
ππ‘ (3)
It follows that
π (π₯, π¦) β₯ 0 Π΄Π»Ρ Π²ΡΡΡ
π₯, π¦, (9)
then the energy curves can be intersected by phase trajectories only from the outside to the middle. So
|π₯(π‘)| β€ π, |π¦(π‘)| β€ π Π΄Π»Ρ π‘ β₯ 0, (10)
and with the help of a phase picture it is possible to be convinced that functions π₯ (π‘) and π¦(π‘) or both
oscillating (have the property: for either π‘1 > π‘0 there will be a point π‘2 > π‘1 , when going through
which function π₯(π‘) or π¦(π‘) change the sign), or both tend to zero at π‘ β β, with π₯ (π‘) it must eventually
become monotonous. In the first case, due to condition (7), the amplitudes of the function π₯(π‘) decrease
monotonically.
If we additionall assume that the equality π(π₯, π¦) = 0 does not hold on any curve π (π₯, π¦) = πΆ (that
is, such curves should not be on the phase plane), it can be stated that
lim π₯ (π‘) = lim π¦(π‘) = 0. (11)
π‘ββ π‘ββ
Note that condition (4) is sufficient for all bounded solutions of system (3) to have either oscillating
coordinates π₯(π‘) and π¦(π‘), or coordinates that satisfy (11), and the function π₯(π‘) monotonic (for large
values π‘). This conclusion can be extended to systems
π₯Λ = π¦, π¦Λ = βπ(π₯) β π(π₯)π¦,
π₯Λ = π¦, π¦Λ = βπ(π₯) β πΉ(π¦) (πΉ (0) = 0).
It is important that the origin of the phase plane is the only stationary point.
The periodic solutions of system (3) correspond to the cycles surrounding the origin.
A continuum of closed trajectories appears, apparently, when a function π (π₯, π¦) identically equal to
zero in the annular region πΆ1 β€ π (π₯, π¦) β€ πΆ2 , 0 β€ πΆ1 < πΆ2 . Here the cycles coincide with the energy
curves.
Under such conditions, describing the motion of the phase point of the system (1), (2), we construct
a Poincare map for the line π₯ = π₯β , which is used to study the question of the existence of periodic
regimes of problem (1), (2). It is obvious that in this case the problem of the existence of periodic
solutions of system (1), (2) is reduced to the problem of the existence of periodic and fixed points of
some mapping of a segment into the same segment, which is determined by the formula
π(π¦) = βπ¦ + πΌ(βπ¦), (12)
where πΌ(βπ¦) < π¦, π¦ β 0, π¦ = π₯Λ.
114
Consider the problem of the existence of periodic solutions of problem (1), (2), when the impulsive
function has the form:
(π β 1)π¦ β π π¦β , π¦ β₯ 0,
πΌ(π¦) = { (13)
β(π + 1)π¦ β π π¦β , π¦ < 0,
where π¦ = π₯Λ, π β some parameter and 0 < π β€ |minπΊ(π₯)|.
The mapping
π (π¦β β π¦), π¦ β₯ 0,
π(π¦) = βπ¦ + πΌ(βπ¦) = { (14)
π (π¦β + π¦), π¦ < 0,
is continuous for all π¦ β β and has the following properties: when 0 < π < 1 there is only one fixed
point that is stable; if 1 < π β€ |minπΊ(π₯)| then we have two fixed points
π πβπ2
{ π¦ } and { π¦ },
1βπ β 1βπ2 β
and the periodic point of the period 2:
πβπ2 π+π2
{ π¦; π¦ }.
1+π2 β 1+π2 β
The points of period 3 for mapping (8) form two cycles
πβπ2βπ3 π+π2βπ3 π+π2 +π3
{ π¦β ; π¦β ; π¦β } ,
1+π3 1+π3 1+π3
πβπ2+π3 π+π2βπ3 πβπ2 βπ3
{ π¦β ; π¦β ; π¦β } .
1βπ3 1βπ3 1βπ3
Thus, the theorem is valid.
Theorem 1. Suppose that for differential equation (1) the function π(π₯, π¦) identically equal to zero
in the region πΆ1 β€ π (π₯, π¦) β€ πΆ2 , 0 β€ πΆ1 < πΆ2 . Then equation (1) with impulse action (2), (14), where
π¦ = π₯Λ, at 1 < π β€ |minπΊ(π₯)|, has π(π)β periodic regimes such that the phase point of this system
when moving along the corresponding trajectory undergoes exactly π impulse actions for the period
where π is an arbitrary natural number. The points defining the cycles corresponding to the periodic
regimes of the problem (1), (14) satisfy the Sharkovskyβs order.
3.2. The case of isolated periodic solutions
Further, let us investigate the case when in some domain there are limit cycles, i.e., there are isolated
periodic solutions of the system, so in this domain π(π₯, π¦) β’ 0. The problem of the existence and
uniqueness of limit cycles for system (2) is considered in the following theorems.
Theorem 2. Assume π(0,0) < 0 and
π (π₯, π¦) β₯ 0 for |π₯| β₯ π₯0 > 0,
moreover
π (π₯, π¦) β₯ ββ± for |π₯| β€ π₯0 . (15)
Assume that there exist π₯1 > π₯0 such that
π₯1
β« π (π₯, π¦) ππ₯ β₯ 10β±π₯0 , (16)
π₯0
where π¦ = π¦(π₯) > 0 is an arbitrary continuously decreasing function. Under such conditions the
system (3) has at least one periodic regime.
Proof. We construct a ring domain that satisfies the requirements of Bendixon's theorem [30]. To
do this, we use inequality
πΊ (π₯0 )
βπΊ(π₯1 ) β πΊ (π₯0 ) β₯ max {20β±π₯0 , }, (17)
β±π₯0
We assume that the function π(π₯) satisfies the conditions π₯π(π₯) > 0 for π₯ β 0 and therefore the
phase picture on the plane π₯π¦ has the only stationary point π₯ = π¦ = 0.
115
Since π (0,0) < 0 and function π(π₯, π¦) continuous, then from (8) we obtain π β₯ 0 in some
neighborhood of the origin. Therefore, as the inner boundary of the annular region, you can choose a
curve π (π₯, π¦) = πΆ > 0 with a sufficiently small value of the parameter πΆ.
Because π β€ 0 Π΄Π»Ρ |π₯| β₯ π₯0 , then in this area to form the outer boundary the curves π (π₯, π¦) =
const can be used. Let's put π¦0 = βπΊ (π₯1 ) β πΊ (π₯0 ) and construct a closed curve π²0 : π (π₯, π¦) = π0 ,
where (see Fig.1 )
1
π0 = πΊ (π₯0 ) + π¦02 .
2
Consider also the trajectory coming from the point π΄0 (π₯0 , π¦0 ) (π¦0 > 0) of this curve. For π₯ β₯ π₯0
the trajectory passes inside π²0 , approaching the axis π₯ and crosses the vertical for the first time π₯ = π₯1
at some point π΄1 (π₯1 , π¦1 ).
Let π1 = π (π₯1 , π¦1 ), then
π₯1
π1 β π0 = β« π(π₯, π¦)π¦ ππ₯.
π₯0
At π¦1 β₯ π¦0β2 we will receive
π₯1
1
π1 β π0 β€ β π¦0 β« π (π₯, π¦) ππ₯ β€ β5β±π₯0 π¦0 .
2
π₯0
This inequality remains true even when π¦1 < π¦0β2. Indeed, we have
1 1 1
π1 = π¦12 + πΊ (π₯1 ) < π¦02 + π¦02 + πΊ(π₯0 ) =
2 8 4
1 2 1
= π0 β π¦0 = π0 β π₯0 βπΊ(π₯1 ) β πΊ (π₯0 ).
8 4
Hence, given inequality (16), we obtain π1 < π0 β 5β±π₯0 π¦0.
If the trajectory is at a point π΄2 (π₯2 , π¦2 ) the lower half-plane returns to the vertical π₯0 , then π2 β€
π1 and, accordingly, π2 < π0 β 5β±π₯0 π¦0.
Let the trajectory intersect the line π₯ = βπ₯0 on the arc π΄Μ 2 π΄3 , then
βπ₯0 π₯0
π3 β π2 = β« π (π₯, π¦)π¦ ππ₯ = β β« π(π₯, π¦)|π¦| ππ₯ β€
π₯0 βπ₯0
π₯0
β€ β± β« |π¦| ππ₯ β€ 2β±π₯0 π¦0 .
βπ₯0
If on the set π΄Μ β² β² β² β² β²
2 π΄3 we have |π¦ | β₯ π¦0, and at the first time π¦2 = π¦0 at point π΄2 (π₯2 , π¦2 ), βπ₯0 β€ π₯2 <
π₯0 , then
π₯0
π2β² β π2 β€ β± β« |π¦| ππ₯ β€ 2β±π₯0 π¦0
π₯2β²
and
1 2
(π¦ β π¦22 ) < 2β±π₯0 π¦0 + [πΊ(π₯0 ) β πΊ (π₯2β² )] β€ 2β±π₯0 π¦0 + πΊ (π₯0 ).
2 0
On the other hand
1 2
(π¦ β π¦22 ) > 5β±π₯0 π¦0
2 0
and, accordingly, using inequality (17), we obtain
πΊ(π₯0 ) > 3β±π₯0 π¦0 = 6β±π₯0 βπΊ(π₯1 ) β πΊ (π₯0 ) β₯ 6πΊ (π₯0 ),
which is impossible.
So the point π΄β²2 does not exist everywhere on the arc π΄Μ
2 π΄3 we have |π¦ | β€ π¦0. From here we find
116
π3 β π0 = (π3 β π2 ) + (π2 β π0 ) < 2β±π₯0 π¦0 β 5β±π₯0 π¦0 = β3β±π₯0 π¦0 .
Since the function π (π₯, π¦) falls along the arc π΄Μ3 π΄4 , where π΄4 β next for π΄3 the point of intersection
of the trajectory of the line π₯ = βπ₯0 , then fair inequality π4 β π0 < β3β±π₯0 π¦0.
Let the trajectory point π΄5 lies vertically π₯ = π₯0 . Then for the arc π΄Μ
4 π΄5 we get the inequality π¦ β€ π¦0 ,
Μ
similar to how it was for the arc π΄2 π΄3 .
From here π5 β π4 β€ 2β±π₯0 π¦0 and, accordingly, π5 β π0 < ββ±π₯0 π¦0. So the point π΄5 should be
below the point π΄0 .
If the phase trajectory does not reach the vertical π₯ = βπ₯0 , then it crosses the axis π₯ at some point
π, where βπ₯0 < π < 0. Then, accordingly, we get
πΊ(π ) < π2 + 2β±π₯0 π¦0 < π0 β 3β±π₯0 π¦0
and
π5 < πΊ (π ) + 2β±π₯0 π¦0 < π0 β β±π₯0 π¦0 .
Complementing the arc π΄Μ Μ
Μ
Μ
Μ
Μ
Μ
Μ
0 π΄5 segment π΄5 π΄0 to a closed loop, construct the desired upper boundary
of the annular region.
If the phase trajectory we are studying ends at some point in the segment 0 < π₯ < π₯0 axis π₯, then
you just need to increase π¦0, that such an end point was a point π₯ = π₯0 . Placing π΄5 = (π₯0 , 0), we obtain,
as before, the outer boundary of the annular region. The theorem is proved.
Now let us consider the uniqueness result. We denote by β+ and ββ domains in the phase plane
π₯π¦, in which the function π (π₯, π¦) is positive or negative. The part of the curve π (π₯, π¦) = π, that
belongs to βΒ± we will denote by βΒ± (π ).
Π’Π΅ΠΎΡΠ΅ΠΌΠ° 3. Assume conditions of Theorem 2, and assume that the function π (π₯, π¦) has continuous
derivatives of the first order. Additionally, we assume that for every value of parameter π, for which
the set βΒ± (π )exists, the infimum of the function
1 1 π
πΉ (π₯, π¦) = 2
+ π(π₯, π¦) (18)
π¦ π¦π(π₯, π¦) ππ¦
on β+ (π ) is positive and no less than its supremum on ββ (π ). Then system (3) has a unique limit
regime.
3.3. Existence of limit cycle under impulsive perturbations
Let us consider the behavior of system (3) under assumptions of theorem 2,3 in domain
1
π0 = πΊ (π₯0 ) + π¦02 .
2
Assume that
10. Generalized LiΓ©nard differential equation (1) is assumed to be satisfied the conditions of
existence and uniqueness of solution.
20. The line π₯ = π₯β is transversal to the flow (1) everywhere everywhere except the trajectory for
which the line π₯ = π₯β is tangent. And in this case we assume that πΌ(0) = 0.
30. The impulsive operator ππ‘ is assumed to be continuous with respect to veriables (π₯, π₯Μ ).
Let for the initial data (π₯0 , π₯Μ 0 ) of the problem (3), (12) the following property holds:
π: π₯0 < π₯β and π₯Μ 0 > π₯β , or π₯0 > π₯β and π₯Μ 0 < π₯β , or (π₯0 , π₯Μ 0 ) β πβ Ρ |π₯0 | > π₯β
and conditions of theorems 2,3 hold for all (π₯, π¦). Then the phase point (π₯0 , π₯Μ 0 ) moves along trajectory
π₯ (π‘β , π₯0 , π₯Μ 0 , π‘0 ) = π₯β , when it is affected by impulsive perturbation (12). Let
π‘1 = min {π‘β : π₯(π‘β , π₯0 , π₯Μ 0 , π‘0 ) = π₯β }.
π‘β>π‘0
Let us consider coordinates of the phase point (π₯(π‘), π₯Μ (π‘)) , where π₯ (π‘) = π₯ (π‘, π₯0 , π₯Μ 0 , π‘0 ) for π‘ =
π‘1 + 0, i.e., afterimpulsive perturbation. Then
117
π₯(π‘1 + 0) = π₯β ,
(19)
π₯Μ (π‘1 + 0) = π₯Μ (π‘1 , π₯0 , π₯Μ 0 , π‘0 ) + πΌ(π₯Μ (π‘1 , π₯0 , π₯Μ 0 , π‘0 )).
Then, if (π₯1 , π₯Μ 1 ), when π₯1 = π₯β , and π₯Μ 1 = π₯Μ (π‘1 + 0) is defined by (19), as a new initial data for the
problem (3), (12), the property π holds, i.e., (π₯β , π₯Μ 1 ) β πβ, then there exists a moment of time π‘β such
that π₯ (π‘β , π₯β , π₯Μ 0 , π‘0 ) = π₯β , when the phase point of (3), (12) is again affected by impulsive perturbations
(12). We denote
π‘2 = min {π‘β : π₯ (π‘β , π₯β , π₯Μ 1 , π‘1 ) = π₯β }.
π‘β>π‘1
Note, that if the property π does not hold, i.e., (π₯β , π₯Μ 1 ) β πβ, then the phase point of the system
(3), (12) for π‘ > π‘1 will not have impulsive perturbations. This situation is possible only if π₯β = π₯ββ² ,
when the hyperplane does not intersect the limit cycle π²0 at any point. In such a case the system (3),
(12) has only one impulsive influences. After that we get a new initial data for (3) which has a limit
cycle due to theorem 2,3, and the phase point in the sequel will move without any impulses.
Assume that we have constructed π members of the sequence {π‘π , (π₯π , π₯Μ π )}, π = 1, π, where
π‘1 = min {π‘β : π₯(π‘β , π₯πβ1 , π₯Μ πβ1 , π‘πβ1 ) = π₯β }, (20)
π‘β>π‘πβ1
π₯π = π₯(π‘π , π₯πβ1 , π₯Μ πβ1 , π‘πβ1 ) = π₯β , π = 1, π, (21)
π₯Μ π = π₯Μ (π‘π , π₯πβ1 , π₯Μ πβ1 , π‘πβ1 ) + πΌ(π₯Μ (π‘π , π₯πβ1 , π₯Μ πβ1 , π‘πβ1 )). (22)
It is clear that (π₯π , π₯Μ π ) = (π₯β , π₯Μ π ) β πβ, where π = 1, π β 1. Under condition (π₯π , π₯Μ π ) = (π₯β , π₯Μ π ) β
πβ, it is possible to construct (π + 1)-th member of this sequence. Otherwise, it consists of only π
members. P- In general case, for arbitrary values of initial data (π₯0 , π₯Μ 0 ) the sequence π‘1 , π‘2 , β¦ can be
infinite, or it can be finite, in particular, it may consists of only one element, or it can be empty.
If the sequence of moments of time consists of one point, which is possible, for example, for the
case when for all π₯ β πβ, then the system (4), (12) undergoes an impulse action only once and for it
there is a limit cycle in the requirements of Theorems 2, 3.
If the sequence has a finite (not empty) number of points containing, for example, exactly kβ₯1
elements, then the condition |π₯πΜ + πΌ (π₯πΜ )| β πβ , πΌ(πβ ) = 0, and the system has exactly k times the
action of impulses, and there is a limit cycle in the requirements of Theorems 2,3.
It easy to see that if for some π and for all π¦ β [βπ¦β , π¦β ] βͺ (ββ, βπ¦β ) βͺ (π¦β , β) we have |π π (π¦)| =
πβ, where π π (π¦) is the π-th iteration of the function π(π¦) = βπ¦ β πΌ(βπ¦), π¦ = π₯Μ , then the sequence
{π‘π , (π₯π , π₯Μ π )}, π β β has an infinite number of points. Moreover, (π₯π , π₯Μ π ) = (π₯β , π₯Μ π ) = πβ for all π.
It follows from the analysis that problem (3), (12) will have a single limit cycle under the conditions
of Theorem 3, where the phase point will be affected by impulse perturbation when the sequence {π‘π },
π = 1,2, β¦ is infinite. The following theorem takes place
Theorem 4. Assume that
1) Function π(π₯, π¦) has continuous derivatives of the first order, and, moreover, there
exist positive π₯1 , π₯2 such that
π(π₯, π¦) < 0, inf π(π₯, π¦) = ββ± on (π₯1 , π₯2 )
and
π (π₯, π¦) β₯ 0 otherwise;
ππ¦
2) π¦ ππ₯ β₯ 0 and
π₯
β«π₯ 2 π (π₯, π¦)ππ₯ β₯ 10β±π₯0 ,
0
where π₯0 = min(π₯1 , π₯2 ) is sufficiently large and π¦ = π¦(π₯) is an arbitrary nonincreasing
function;
3) πΊ (βπ₯1 ) = πΊ (π₯2 ).
Then the impulsive system (3), (12) has a unique limit cycle, and the phase point has π β β
impulsive perturbations
118
.
Figure 1: limit cycle with the hyperplane of impulsive effect
The detailed analysis of the qualitative behavior of the system (1), (2), (12) demonstrates the
complex behavior of the generalized LΓ©nard equation (1) with impulse action (2). The effective criteria
for the existence of a single stable limit impulsive regime for such an equation are investigated.
4. Conclusions
The paper investigates the conditions for the existence of boundary cycles for the Lienard equation,
the solutions of which are affected by instantaneous forces of momentum nature at unfixed moments in
time.
For this system, the constructive conditions for the existence of π(π)-periodic regimes are proved
such that the phase point of the system when moving along the corresponding trajectory undergoes
exactly π impulsive disturbances for the period where π is an arbitrary positive integer. The points that
define cycles that correspond to periodic regimes satisfy the Sharkovskyβs order.
The conditions for the existence of at least one periodic regime and a single limit cycle are found β
the only one with precision to shift in time of the periodic regime.
For system (3) with impulsive action (12), the existence of a single stable limit regime is proved, the
phase point of which will be affected by pulsed forces π β β times. It is shown that a stable limit regime
under given conditions will exist despite the influence of impulse forces according to the law (12).
5. Acknowledgements
This work was partially supported by the National Research Foundation of Ukraine (project
F81/41743).
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