The conditions for the existence of limit regimes for the Liénard 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 ∈ ℕ 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 ∈ ℕ 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. differential equation, impulse action, Liénard equation, limit cycle
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, Liénard 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,
2021 Copyright for this paper by its authors. 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 selforganization 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 t– "". 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 Lénard type under the influence of destabilizing external impulse perturbations.
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 [1115]. 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 nonautonomous 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 multivalued right-hand parts and with impulse influences at fixed moments of time are considered. The 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 Liénard equation with impulse action
Let us investigate the problem of the existence of harmonic cycles for the Liénard 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 Liénard differential equation of the form:
¨ + ( , ̇ ) ˙ + ( ) = 0, ( ∈ ⊂ ℝ3,
− phase space of the system (
= ˙, as a function of its argument is continuous. occurs at non-fixed moments of time and increases the amount of motion in the system by a certain amount ( ˙), which depends on the speed of the moving point at the time of its passage = ∗. Next impulsive action occurs, the impulsive perturbations of the moving point can be written as [27]: If ∗ is a certain moment of time at which the moving point reaches position = ∗, when an
| = ∗ = | −
| = ∗+0 = ∗−0
= − = ( ˙).
The description of the physical interpretation of the generalized Liénard equation and the
In this case, the origin of the phase plane is the only stationary point. It is covered by a family of
so(
Equation (
In the sequel, we will use the notation 0 0 ( ) = ∫ ( )
, ( ) = ∫ ( ) .
We will consider further that functions ( )and ( , )provide the condition for the existence and
uniqueness of the solution of the system (
при ≠ 0 (±∞) = ∫
( ) . ±∞ 0
( , ) ≡ ( )+ 1 2 = .
2 (− ) = − ( ) these curves are symmetrical about both coordinate axes.
Let's denote ∗ = √2 ( ∗). 3.1.
There are periodic solutions that satisfy Sharkovsky's order
(
10. The generalized Liénard differential equation (
(
( , )
)
(
для всіх , , then the energy curves can be intersected by phase trajectories only from the outside to the middle. So | ( )| ≤ ,
| ( )| ≤ для ≥ 0,
and with the help of a phase picture it is possible to be convinced that functions ( ) and ( ) or both
oscillating (have the property: for either 1 > 0 there will be a point 2 > 1, when going through
which function ( )or ( )change the sign), or both tend to zero at → ∞, with ( )it must eventually
become monotonous. In the first case, due to condition (
If we additionall assume that the equality ( , ) = 0 does not hold on any curve ( , ) = (that is, such curves should not be on the phase plane), it can be stated that lim ( ) = lim ( ) = 0.
→∞ →∞
Note that condition (
It is important that the origin of the phase plane is the only stationary point.
The periodic solutions of system (
A continuum of closed trajectories appears, apparently, when a function ( , )identically equal to zero in the annular region 1 ≤ ( , ) ≤ 2, 0 ≤ 1 < 2. Here the cycles coincide with the energy curves.
Under such conditions, describing the motion of the phase point of the system (
Consider the problem of the existence of periodic solutions of problem (
= ˙, — some parameter and 0 < ≤ |min ( )|.
The mapping ( ) = { ( − 1) − ∗, ≥ 0, −( + 1) − ∗, < 0, ( ) = − + (− ) = { ( ∗ − ), ≥ 0, ( ∗ + ), < 0, is continuous for all ∈ ℝ and has the following properties: when 0 < < 1 there is only one fixed point that is stable; if 1 < ≤ |min ( )| then we have two fixed points and the periodic point of the period 2: {
1 − ∗} and { − 2 1 − 2
∗}, { − 2 1 + 2 ∗; + 2 1 + 2 ∗}.
The points of period 3 for mapping (
1 + 3 − 2 + 3 1 − 3 ∗; ∗; + 2 − 3
1 + 3 + 2 − 3 1 − 3 ∗; ∗; + 2 + 3
1 + 3 − 2 − 3 1 − 3 ∗} , ∗} .
Theorem 1. Suppose that for differential equation (
( , ) ≤ 2, 0 ≤ 1 < 2. Then equation (
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 ( , ) ≢ 0. The problem of the existence and
uniqueness of limit cycles for system (
Theorem 2. Assume (0,0) < 0 and moreover Assume that there exist 1 > 0 such that ( , ) ≥ 0 for
| | ≥ 0 > 0,
( , ) ≥ −ℱ for | | ≤ 0.
1
0
∫ ( , )
system (
We construct a ring domain that satisfies the requirements of Bendixon's theorem [30]. To do this, we use inequality
We assume that the function ( ) satisfies the conditions ( ) > 0 for ≠ 0 and therefore the phase picture on the plane has the only stationary point = = 0.
At 1 ≥ 0⁄2 we will receive
0 This inequality remains true even when 1 < 0⁄2. Indeed, we have 1 − 0 = ∫ ( , ) .
1 1 − 0 ≤ − 2 0 ∫ ( , ) ≤ −5ℱ 0 0. 1 = 21 12 + ( 1)< 81 02 + 41 02 + ( 0)= = 0 − 81 02 = 0 − 4 0√ ( 1)− ( 0).
1
Since (0,0)< 0 and function ( , ) continuous, then from (
Because ≤ 0 для | | ≥ 0, then in this area to form the outer boundary the curves ( , )= const can be used. Let's put 0 = √ ( 1)− ( 0)and construct a closed curve 0: ( , )= 0, where (see Fig.1 )
0 = ( 0)+ 12 02.
Consider also the trajectory coming from the point 0( 0, 0) ( 0 > 0) of this curve. For ≥ 0 the trajectory passes inside 0, approaching the axis and crosses the vertical for the first time = 1 at some point 1( 1, 1).
Let 1 = ( 1, 1), then
1
0
1
0
2′
Hence, given inequality (
If the trajectory is at a point 2( 2, 2)the lower half-plane returns to the vertical 0, then 2 ≤ 1 and, accordingly, 2 < 0 − 5ℱ 0 0.
Let the trajectory intersect the line = − 0 on the arc ̂23, then
− 0 0 3 − 2 = ∫ ( , ) = − ∫ ( , )| | ≤ 0 0
− 0 ≤ ℱ ∫ | | ≤ 2ℱ 0 0.
− 0
If on the set ̂23 we have | | ≥ 0, and at the first time 2′ = 0 at point ′2( 2′, 2′), − 0 ≤ 2′ < 0, then and On the other hand
2′ − 2 ≤ ℱ ∫ | | ≤ 2ℱ 0 0
1( 02 − 22)< 2ℱ 0 0 + [ ( 0)− ( 2′)] ≤ 2ℱ 0 0 + ( 0).
2
1( 02 − 22)> 5ℱ 0 0
2
and, accordingly, using inequality (
( 0)> 3ℱ 0 0 = 6ℱ 0√ ( 1)− ( 0)≥ 6 ( 0), which is impossible.
So the point ′2 does not exist everywhere on the arc ̂23 we have | | ≤ 0. From here we find similar to how it was for the arc ̂23. below the point 0.
3 − 0 = ( 3 − 2)+ ( 2 − 0) < 2ℱ 0 0 − 5ℱ 0 0 = −3ℱ 0 0.
( , )falls along the arc ̂34, where 4 — next for 3 the point of intersection of the trajectory of the line = − 0, then fair inequality 4 − 0 < −3ℱ 0 0.
Let the trajectory point 5 lies vertically = 0. Then for the arc ̂45 we get the inequality ≤ 0, From here 5 − 4 ≤ 2ℱ 0 0 and, accordingly, 5 − 0 < −ℱ 0 0. So the point 5 should be If the phase trajectory does not reach the vertical = − 0, then it crosses the axis at some point , where − 0 < < 0. Then, accordingly, we get ( ) < 2 + 2ℱ 0 0 < 0 − 3ℱ 0 0 5 < ( )+ 2ℱ 0 0 < 0 − ℱ 0 0. and of the annular region.
Complementing the arc ̂05 segment ̅̅̅̅ ̅̅0̅ to a closed loop, construct the desired upper boundary 5 as before, the outer boundary of the annular region. The theorem is proved.
If the phase trajectory we are studying ends at some point in the segment 0 < < 0 axis , then you just need to increase 0, that such an end point was a point = 0. Placing 5 = ( 0, 0), we obtain,
Now let us consider the uniqueness result. We denote by ℛ+ and ℛ− domains in the phase plane , in which the function ( , ) is positive or negative. The part of the curve ( , ) = , that belongs to ℛ± we will denote by ℛ±(
).
Теорема 3. Assume conditions of Theorem 2, and assume that the function ( , ) has continuous
derivatives of the first order. Additionally, we assume that for every value of parameter , for which
the set ℛ±(
)exists, the infimum of the function
( , ) =
1
2 + ( , )
1
( , )
(
Let us consider the behavior of system (
1
10. 20. The line
Generalized Liénard differential equation (
= ∗ is transversal to the flow (
Let us consider coordinates of the phase point ( ( ), ̇ ( )), where ( ) = ( , 0, ̇ 0, 0) for = 1 + 0, i.e., afterimpulsive perturbation. Then
Then, if ( 1, ̇1), when 1 = ∗, and ̇1 = ̇ ( 1 + 0)is defined by (
∗> 1
Note, that if the property does not hold, i.e., ( ∗, ̇1) ∉ ∗, then the phase point of the system
(
Assume that we have constructed members of the sequence { , ( , ̇ )}, = 1, , where 1 =
min { ∗: ( ∗, −1, ̇ −1, −1) = ∗}, ∗> −1 = ( , −1, ̇ −1, −1) = ∗, = 1, , ̇ = ̇ ( , −1, ̇ −1, −1)+ ( ̇ ( , −1, ̇ −1, −1)).
It is clear that ( , ̇ ) = ( ∗, ̇ ) ∈ ∗, where = 1, − 1. Under condition ( , ̇ ) = ( ∗, ̇ ) ∈ ∗, it is possible to construct ( + 1)-th member of this sequence. Otherwise, it consists of only members. P- In general case, for arbitrary values of initial data ( 0, ̇0) the sequence 1, 2, … 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 ∉ ∗, then the system (
If the sequence has a finite (not empty) number of points containing, for example, exactly k≥1 elements, then the condition | ̇ + ( ̇ )| ∉ ∗, ( ∗) = 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 and for all ∈ [− ∗, ∗] ∪ (−∞, − ∗)∪ ( ∗, ∞)we have | ( )| = ∗, where ( ) is the -th iteration of the function ( ) = − − (− ), = ̇ , then the sequence { , ( , ̇ )}, ∈ ℕ has an infinite number of points. Moreover, ( , ̇ ) = ( ∗, ̇ ) = ∗ for all .
It follows from the analysis that problem (
Theorem 4. Assume that
1) Function ( , ) has continuous derivatives of the first order, and, moreover, there
exist positive 1, 2 such that
(
and 3) (− 1) = ( 2).
( , ) < 0, inf ( , ) = −ℱ on ( 1, 2) ( , ) ≥ 0 otherwise; ∫ 02 ( , ) where 0 = min( 1, 2)is sufficiently large and = ( )is an arbitrary nonincreasing function;
Then the impulsive system (
The detailed analysis of the qualitative behavior of the system (
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 ( )-periodic regimes are proved such that the phase point of the system when moving along the corresponding trajectory undergoes exactly impulsive disturbances for the period where is an arbitrary positive integer. The points that define cycles that correspond to periodic regimes satisfy the Sharkovsky’s order.
The conditions for the existence of at least one periodic regime and a single limit cycle are found — the only one with precision to shift in time of the periodic regime.
For system (
This work was partially supported by the National Research Foundation of Ukraine (project F81/41743).