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0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 3, pp. 383-388. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230906
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 70K40, 34C15
A Note on Forced Oscillations in Systems on a Plane
I. Yu. Polekhin
A sufficient condition for the existence of forced oscillations in nonautonomous systems on a plane is presented under the assumption that the magnitude of the nonautonomous perturbation is small. An advantage of the results presented over analytic methods is that they can be applied in degenerate systems as well.
Keywords: periodic systems, forced oscillations, systems on a plane, degenerate systems, center
1. Introduction
This note considers the topological method of proving the existence of forced oscillations on a plane for systems of the following form:
x = Uo(x,y) + ul(x,y,t,e),
y = Vo(x, y) + vi(x, y, t, e)
under the assumption that the system
x = u0(x, y),
0( , (1.2) y = vo(x, y)
has a periodic solution and the functions ul and vl are periodic in t.
Received March 17, 2023 Accepted June 28, 2023
This work was supported by the Russian Science Foundation (Agreement No. 19-71-30012).
Ivan Yu. Polekhin ivanpolekhin@mi-ras.ru
Steklov Mathematical Institute of the Russian Academy of Sciences,
ul. Gubkina 8, Moscow, 119991 Russia
Moscow Institute of Physics and Technology,
Institutskiy per. 9, Dolgoprudny, 141701 Russia
Lomonosov Moscow State University,
Leninskie Gory, Moscow, 119234 Russia
Demidov Yaroslavl State University,
ul. Sovetskaya 14, Yaroslavl, 150000 Russia
As an example, we will prove the following result. It is well known that in a harmonic oscillator with external forcing x = —x + e sin t all solutions go to infinity if e = 0. Conversely, if one considers the system
x = A(x, y) ■ y, y = —\(x, y) ■ x + e sin t
in which A: R2 R is a positive smooth function which takes values smaller than 1 in any neighborhood (including an arbitrarily small one) of some point (x0, y0) and is equal to 1 outside this neighborhood, then for small e the system has a solution that does not go to infinity. Moreover, it has a 2^-periodic solution.
2. The main results
Theorem 1. Suppose that in the system (1.1) the functions u0, v0: £ C2 (R2, R), and the functions u1, v1 C2 R3 x I, R), where I C R is an open interval containing zero. Also, let the functions u1 and v1 be T-periodic in t. Suppose that the system (1.2) has a T1-periodic solution j, u1(x, y, t, 0) = v1(x, y, t, 0) = 0 and T = kT-^ k £ N. Then for e > 0 sufficiently small the system (1.1) has a T-periodic solution. It is assumed that j is not an equilibrium point.
Proof. The main idea of the proof consists of two steps. In the first step, instead of the system (1.1) we consider a modified system of the following form:
x = u0(x, y) + Uy,(x, y)%(x, y, t, e),
(2.1)
y = v0(x, y) + Uy(x, y)v1(x, y, t, e),
where the smooth function uy £ C™ (R2, R) is equal to zero on the curve j (we will use the symbol j both for the solution and for the trajectory of this solution) and is equal to 1 everywhere except in a small neighborhood of the curve j. Using the Lefschetz theorem on a fixed point, one can show that the modified system (2.1) has a T-periodic solution which is contained in the region spanned by the solution j.
Using the fact that the solutions to the systems (1.1) and (2.1) which start sufficiently close to the curve j are close on finite time intervals at small e > 0, we show that the found T-periodic solution cannot pass through points at which uy = 1.
We will use the following notation. We will denote the solution to the system (1.1) with initial conditions (x0, y0) at time t0 by (x(t; x0, y0, e), y(t; x0, y0, e)), and the solution to the modified system (2.1), by (xm(t; x0, y0, e), ym(t; x0, V0, e)).
Thus, we choose a ¿-neighborhood Os (j) of the curve j such that it is tubular, i.e., consists of normals to the curve j. Such a neighborhood exists [1].
Choose a smooth function uy(x,y) in such a way that at distance r from the curve j its value is defined as follows:
1 if \r\ > 5.
Consider the map for period T of the modified system (2.1). This map has a fixed point
that corresponds to a T-periodic solution which does not leave the region U bounded by the curve j. Indeed, since the solutions starting in U cannot leave U (cannot cross the trajectory
of the solution 7), we have a map of U U dU = U U 7 into itself. This set is compact and U is homeomorphic to a two-dimensional disk by the Jordan-Schoenflies theorem [2], then by the Lefschetz theorem on a fixed point [3] we have a fixed point of the map for a period. Moreover, the fixed point cannot lie on the boundary since period 7 is not equal to T.
For the modified system, a T-periodic solution exists for all 5 sufficiently small and for all e. We show that, if 5 and e are small, then the T-periodic solution to the modified system cannot pass through points Os (7).
Consider some neighborhood V of the curve 7. For any point (x, y) G 7, any initial time t0 and for any p > 0 there exists a A > 0 such that, if in V
and
then
(x, y)u1 (x, y, t, e)\ < A, (x, y)vi(x, y, t, e)\ < A
\x0 - x\ < A, \y0 - y\< A,
\xm(to + T; xQ, yQ, e) - x(tQ + T; x, y, 0)| < p, \ym(t0 + T; xQ, yQ, e) - y(t0 + T; x, y, 0)| < p.
In particular, since 7 is a T1 -periodic solution and T = kTx, k G N, for p sufficiently small we find that
(xm (t0 + T; x0, y0, e), ym(t0 + T; x0, y0, e)) G Op(x, li)
and the solution (xm(t; x0, y0, e), ym(t; x0, y0, e)) is not T-periodic. Choose e > 0 to be sufficiently small in such a way that the following inequalities are satisfied in V:
\ui(x, y, t, e)\ < A, \vi(x, y, t, e)\ < A.
Then we find that for a given point (x,y) G 7 there exists a A-neighborhood, e0 > 0, ¿0 > 0 such that all solutions to the modified system which start in this A-neighborhood are not T-periodic for all e G [0, e0] and for all 5 G [0, ¿0].
(xm(t0 +T; x, y, 0), ym(i0 +T; x, y, 0))
Fig. 1. The solid lines denote three periodic solutions to the system at e = 0. The dash-and-dot line denotes three solutions in each of the regions from Theorems 1 and 2
Consider a finite covering of the curve 7 with such A-neighborhoods. To each such neighborhood there correspond an e0 and a 50. Choose 5 such that 0~(y) is contained in the finite
covering. Then for all e which are smaller than the minimum from the set of quantities e0 we find that the solution (2.1), which starts in 0~(y), is not T-periodic.
Consequently, the resulting solution to the modified system cannot pass through the points at which aYf = 1. Hence, this solution exists in the initial system (1.1) as well. □
Theorem 2. Suppose that in the system (1.1) the functions u0, v0: e C2 (R2, R) and the functions u1, e C2 (R3 x I, R), where I C R is an open interval containing zero. Also, let the functions u1 and v1 be T-periodic in t. Let the system (1.2) have a set of periodic solutions Yi, i = 0, ..., l, l > 1, and let the periods of these solutions be Ti and T = kTi, k e N. Assume that the curves Yi, i > 0, are contained in the region bounded by y0, but are not contained in each other, i. e, Yi is not contained in the region bounded by Yj with j = 0. If u1(x, y, t, 0) = = v1 (x, y, t, 0) = 0, then for e > 0 sufficiently small the system (1.1) has a T-periodic solution. It is assumed that all Yi are not equilibrium points.
Proof. The proof repeats the proof of Theorem 1 except that the existence of a fixed point of the map for a period of the modified system follows from the fact that this map is in a natural way homotopic to an identity map and the Eulerian characteristic of the region is not equal to
Thus, when the conditions of Theorem 2 are satisfied, we obtain l + 1 periodic solutions: l of them are contained inside the curves y1, ..., Yi, and another solution is contained inside the curve Yo, but outside the curves y1, ..., Yl. Combining Theorems 1 and 2, one can obtain estimates on the number of periodic solutions for more complex arrangements of the curves.
As an example, we consider the system (1.3). If the function X(x,y) satisfies the conditions described above, the period of the trajectory that passes through point (x0,y0) is strictly larger than T. Hence, all conditions of Theorem 1 are satisfied.
0
(a)
V 1.4
1.2
1.0
0.8
0.6
\ f % ...
\ \
0.6 0.8
1.0
(b)
1.2 1.4
Fig. 2. A periodic solution (a) passing through the region in which the equation of the harmonic oscillator was changed (b). The dotted line denotes some solution to the modified system which goes to infinity and which passes through the region of variation. One can see a "kink" of the trajectory where it passes through the region in which the system has been modified (b)
We present some numerical results concerning this example. Choosing a specific form of the function A(x, y), one can obtain the following result (Fig. 2): for e > 0 sufficiently small there
exists a 2^-periodic trajectory passing through the region in which A = 1. In the same region there are solutions with unbounded trajectories.
As an example illustrating Theorem 2, we consider the perturbation of a Hamiltonian system with the Hamiltonian
2 2 4
y X X
The system has three types of periodic trajectories: two about the equilibrium points x = 1, y = 0 and x = -1, y = 0 and one that corresponds to sufficiently high levels of energy (Fig. 3). The periods of motion along these trajectories depend on energy. Therefore, there exist three periodic solutions satisfying the conditions of Theorem 2. At small e > 0 the system
x = y,
o
y = x — x + e sin t
will have three 2^-periodic solutions (Fig. 3).
Fig. 3. The solid lines denote three periodic solutions to the system for e = 0. The dash-and-dot line denotes three solutions in each of the regions from Theorems 1 and 2
We note that the results presented in the paper admit a generalization to systems of large dimensions and to more specific classes of differential equations (for example, to Hamiltonian systems) for which there exist special theorems on the existence of periodic solutions which can be used instead of general theorems suitable for all ordinary differential equations.
The main difference of the proposed method from the Poincare method of a small parameter [4, 5] is that, firstly, there is no need to seek a solution of the required period of the unperturbed (e = 0) system, secondly, there is no need to verify the fulfilment of the conditions of the implicit function theorem, which requires, in particular, solving a linear nonautonomous equation in variations. In the proposed approach, it suffices to find one solution whose period is, in contrast, different from the period of the external perturbation.
Conflict of interest
The author declares that he has no conflict of interest.
References
[1] Hirsch, M.W., Differential Topology, Grad. Texts in Math., vol. 33, New York: Springer, 1976.
[2] Newman, M. H. A., Elements of the Topology of Plane Sets of Points, 2nd ed., New York: Dover, 1992.
[3] Lefschetz, S., On the Fixed Point Formula, Ann. of Math. (2), 1937, vol. 38, no. 4, pp. 819-822.
[4] Malkin, I. G., The Methods of Lyapunov and Poincaré in the Theory of Nonlinear Oscillations, 2nd ed., Moscow: URSS, 2010 (Russian).
[5] Hartman, Ph., Ordinary Differential Equations, 2nd ed., Classics Appl. Math., vol. 38, Philadelphia, Penn.: SIAM, 2002.
