On Period Annuli and Induced Chaos
SVETLANA ATSLEGA1,2, OLGA KOZLOVSKA3, FELIX SADYRBAEV2,4
1Department of Mathematics,
Latvia University of Life Sciences and Technologies,
Liela Street 2, Jelgava,
LATVIA
2Institute of Mathematics and Computer Science,
Rainis boulevard 29, Riga,
LATVIA
3Department of Engineering Mathematics,
Riga Technical University,
Kipsalas 6a, Riga,
LATVIA
4Institute of Life Sciences and Technology,
Daugavpils University,
Parades street 1, Daugavpils,
LATVIA
Abstract: - Nontrivial period annuli in the second order ordinary differential equation are continua of periodic
trajectories that contain inside more than one critical point. They can appear in conservative equations, which
are known to have no attractors. Nevertheless, according to some authors, their behavior may be done chaotic
by adding a periodic external force. Is the period of the external force correlated with periods of solutions in
period annuli? Is the chaotic behavior of a solution dependent on the initial value and, in turn, on a certain
periodic annulus? These, and related questions are studied in the article.
Key-Words: - Differential equations, oscillation, period annuli, sensitive dependence, chaotic behavior,
Lyapunov exponents.
Received: July 9, 2023. Revised: December 22, 2023. Accepted: January 24, 2024. Published: April 9, 2024.
1 Introduction
The second order ordinary differential equations of
the Newtonian form can have
multiple period annuli. Period annuli are
understood as continua of periodic solutions. The
trivial example is a central region in a phase plane,
associated with a critical point of the center type.
The harmonic equation is an
evident example. But there may be regions filled
with closed trajectories and surrounding more than
one critical point. An example of this can be
provided by the equation (1) with the phase plane
as in Figure 1. Imagine that g(x) is an odd degree
polynomial and the primitive G(x) has graph
resembling the mountain range like in Figure 2. If
there is a pair of mountains containing other (lower)
mountains between them, then the period annuli
appear. The periodic trajectories in these annuli
have an interesting property. The periods of
trajectories, locating close to the borders of a
period annulus, are very long (tending to infinity)
and the graph representing periods, has U-shape. A
solution with a minimal period exists in any such
period annulus. It is known that some of Duffing-
type equations when excited by adding a periodic
term on the right side of the equation can exhibit
chaotic behavior. An autonomous Duffing type
equation is the second order one, and it cannot be
chaotic due to the Poincaré-Bendixson theory.
However, there is no contradiction. The Duffing
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equation with an external force on the right side
can be written in an equivalent form as a system of
three ordinary differential equations. Such systems
can be chaotic. The classical examples are the
Lorenz and Rössler attractors. Examples and
general information can be found in textbooks on
differential equations, [1], [2], [3], [4], [5]. Do
equations of the form with multiple
period annuli possess the same property? We wish
to gather information on this subject in this article.
For this, we recall the article by the authors
with some examples. Consider first the equation:
(1)
This equation has period annuli described in Figure
1.
Fig. 1: Period annuli in equation (1). The borders
are in green and red
This equation was investigated in the work, [6].
Introduce the notation:
(2)
and add a function f(t) on the right, so the equation
becomes:
(3)
The right side can be interpreted as an external
force, which can be periodic, .
Write this equation as the system:
(4)
It is known that equations of the form (3) can
exhibit chaotic behavior (Duffing equation, for
instance). Our aim in this paper is to show, that the
second order equations of the form (3) can be
chaotic, provided that the shortened equation has
period annuli.
(5)
Remark, [7], [8]. A central region is the largest
connected region covered with cycles surrounding
the center. A period annulus is a connected region
covered with concentric cycles. A period annulus
associated with a central region is called a trivial
period annulus. It contains a single critical point of
the type center. Period annuli containing more than
one critical point will be called nontrivial period
annuli. The phase portrait depicted in Figure 1
contains exactly three trivial period annuli and two
nontrivial period annuli.
We mention the following result, which allows
easily constructing examples of equations with
multiple period annuli, [6].
Assume that g(x) is a polynomial with simple
zeros. The primitive G(x)=
may have
multiple maxima. It is easy to observe that the
equivalent system x'=y, y'=
g(x) has centers at the
point (mi,0) and saddle points at (Mj,0), where mi
and Mj are points of local minima and maxima
respectively. In that case centers and saddles
alternate. The following assertion is true.
Theorem. Let M1 and M2 (M1<M2) be non-
neighboring points of maxima for the primitive
G(x). Suppose that G(x)<min{G(M1); G(M2)} for
any x
(M1, M2).
To illustrate this, the primitive G(x) of the
function g(x) in (2) is depicted in Figure 2. There
are four maxima. The first and the third maxima
generate a nontrivial period annulus, which, in turn,
is included in a greater one, generated by two side
maxima.
Fig. 2: Primitive G(x) of the function g(x), defined
in (2). Each maximum point corresponds to a
critical point of the type center
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2 Chaotic Behavior
Chaotic behavior in dynamical systems is studied
extensively. There are multiple definitions and
criteria for chaos. One of the essential signs of
chaotic behavior is the sensitive dependence of
solutions on the initial data. To detect sensitive
dependence, computational tools are widely used.
Let us mention the mathematical apparatus, based
on the Lyapunov exponents. The Lyapunov
exponents reflect the evolution of the difference
between two initially close solutions of a
dynamical system. These differences for the three-
dimensional systems, projected on the coordinate
axis, result in three Lyapunov curves. If at least one
of these curves is positive, this is a sign of sensitive
dependence.
More on chaotic behavior in systems of
ordinary differential equations and Lyapunov
exponents can be learned from the book, [9] and
the article [10], as well as from many other relevant
sources. The chaotic behavior of trajectories may
be observed near attractors. An extract from Table
1.1 in the book, [9], (Table S in a current text)
shows the relation between types of attractors and
the Lyapunov exponents.
Table S.
λ1
λ 2
λ 3
Attractor
Dimension
Dynamic
Equilibrium
0
Static
0
Limit cycle
1
Periodic
+
0
Strange
2 or 3
Chaotic
Numbers , , indicate locations of the
Lyapunov curves as positive (+), lying on the
horizontal zero-axis, and negative (). In the same
book, [9], the equation:
(6)
was considered, which was called conservative
meaning that it does not contain a damping term.
This equation was shown to have chaotic behavior
(chaotic sea with islands of periodicity). We will do
the same for equations with period annuli. An extra
question we wish to address is how the chaotic
behavior depends on the initial conditions and the
structure and number of period annuli.
3 Results
Example 1.
Consider equation:
(7)
where is given:
(8)
is a parameter, is a coefficient, The
primitive G(x) is depicted in Figure 3 and the phase
plane is shown in Figure 4. This equation (7) is
equivalent to the system:
(9)
We will examine system (9) and show that it can be
chaotic.
Consider system (9), provided , ,
. Recall that in a
scalar form, the equation (7) with zero right side
has three trivial period annuli, and two nontrivial
ones.
Fig. 3: The primitive G(x) of g(x) above
Fig. 4: Three nontrivial period annuli in the
equation where g(x) is as in (8)
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Figure 5, Figure 6 and Figure 7 respectively
reflect the phase portrait, graphs of solutions and
Lyapunov curves for the perturbed system (9).
Fig. 5: Phase portrait for the system (9), h=1,
=3.2, (x(0),y(0),z(0)) = (1, 0, 0.4)
Fig. 6: Solutions of the system (9), h=1, =3.2,
(x(0),y(0),z(0)) = (1, 0, 0.4)
Fig. 7: Lyapunov curves for the system (9), h=1,
=3.2, (x(0),y(0),z(0)) = (1, 0, 0.4). Lyapunov
exponents (0.3552, 0, 0.3552)
These data are characteristic of chaotic
behavior. The following table shows the dynamics
of Lyapunov numbers under the change of the
amplitude h.
Table 1. The Lyapunov numbers for the equation
(7) with
=3.2, (x(0),y(0),z(0)) = (1, 0, 0.4) and
varying h
λ1
λ 2
λ 3
0.2
0.281788
0
-0.281789
0.4
0.239252
0
-0.239252
0.6
0.376682
0
-0.376683
0.8
no data
1.0
0.355192
0
-0.355192
For all h the sum of three Lyapunov exponents
is zero. One of the exponents is positive, and that
refers to chaotic behavior (Table 1).
Example 2.
One of the simple equations that can have a
nontrivial period annulus, is equation (5), where
g(x) is:
(10)
The vector field associated with (5) is depicted
in Figure 8.
Fig. 8: The period annuli (three trivial ones, and
one nontrivial spreading to infinity) in the equation
where g(x) is as in (8)
Consider equation:
(10)
and the equivalent system :
(11)
A series of computational experiments were
performed. They yielded that for h=3.9, =0.6 the
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trajectory that starts at x0=0.62, y0=0.9, z0=0.4,
exhibits irregular behavior. The Lyapunov
exponents indicate chaos. This is shown in Figure
9. The Lyapunov curves and graphs of solutions to
the system (9) are shown in Figure 10 and Figure
10a respectively.
Fig. 9: Phase portrait for the system (11)
The Lyapunov exponents also indicate chaotic
behavior.
Fig. 10: Lyapunov exponents for the system (11)
Fig. 10a: Solutions of the system (11)
The two, Table 2 and Table 3 show the
dynamics of the Lyapunov exponents when the
amplitude h and the coefficient change.
Table 2. The Lyapunov numbers for the equation
(10) with
=0.6, (x(0),y(0),z(0)) = (0.62, 0.9, 0.4)
and varying h
λ1
λ 2
λ 3
2.8
0.0271906
-0.027196
0
3.1
0.173493
-0.1735
0
3.5
0.196616
-0.196622
0
3.7
0.114328
-0.114336
0
3.9
0.188477
-0.188485
0
Table 3. The Lyapunov numbers for the equation
(10) with h=3.9, (x(0),y(0),z(0)) = (0.62, 0.9, 0.4)
and varying
λ1
λ 2
λ 3
0.1
0.0739052
-0.0739115
0
0.2
0.109937
-0.109944
0
0.4
0.145782
-0.145789
0
0.6
0.188477
-0.188485
0
2
0.302086
-0.302113
0
Both Table 2 and Table 3 for various h and
contain a positive Lyapunov exponent.
Example 3.
Consider equation (5) and equation (12)
(12)
where
(13)
The vector field for the unperturbed equation
(12) is described by Figure 11.
Fig. 11: The two trivial period annuli and one
nontrivial in the equation where
g(x) is as in (13)
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The corresponding system :
(14)
differs from that in Example 1. Let the parameters
be h=3.9, ,
. The vector field, the Lyapunov
curves and the graphs of solutions for the system
(14) are shown in Figure 12, Figure 13 and Figure
14 respectively.
Fig. 12: Phase portrait for the system (14)
Fig. 13: Lyapunov curves for the system (14). The
Lyapunov exponents are (0.184021, 0.184494,
0.)
Fig. 14: Solutions of the system (14)
Example 4.
Finally, let us consider an example from the
beginning of the article.
Consider equation (5) and equation (15)
(15)
where
which can be written as a system of the
(16)
The amplitude of oscillations in period annuli
is larger than in the previous examples. To affect
the solutions in period annuli the amplitude h
should be large also. Let h=176 , ,
. We provide
below the phase portrait (Figure 15), the Lyapunov
exponents (Figure 16) and the graphs of solutions
(Figure 17) for the system (16). The solutions
exhibit irregular behavior.
Fig. 15: Phase portrait for the system (16)
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Fig. 16: Lyapunov curves for the system (16). The
Lyapunov exponents are (0.667735,0.678565, 0)
Fig. 17: Solutions of the system (16)
4 Conclusion
The second order ordinary differential equations
with an external force can be chaotic. For this, they
have to be nonlinear (the necessary condition). The
nonlinearity should be suitable. The conditions for
the nonlinearity can be found using computational
experiments. We tried to detect chaotic behavior in
periodically exited equations, which, without
external forcing, have period annuli. We have
examined four examples of equations of Newtonian
form that possess one or several nontrivial period
annuli. These equations were excited by a periodic
external force with amplitude h and the period
2π/. As a rule, for appropriate values of h and
the chaotic behavior was observed. In Example 4
the amplitudes of oscillations in period annuli are
significantly greater than in the previously
considered examples. Consequently, the parameter
h in the excited equation is large to induce the
chaotic behavior. For the criteria for chaotic
behavior, the graph of the Lyapunov curves and
Lyapunov exponents was chosen. The positivity of
the Lyapunov curve was an indicator of chaos in
the equation. The damping term is not needed for
obtaining the chaotic behavior.
References:
[1] L. Cveticanin. Strongly Nonlinear
Oscillators. Springer, 2014
[2] Guo-hua Chen, Zhao-Ling Tao, Jin-Zhong
Minc. Notes on a conservative nonlinear
oscillator. Computers & Mathematics with
Applications. Vol. 61, Issue 8, 2011, 2120-
2122,
https://doi.org/10.1016/j.camwa.2010.08.086.
[3] Guo-hua Chen, Zhao-Ling Tao, Jin-Zhong
Minc. Notes on a conservative nonlinear
oscillator. Computers and Mathematics with
Applications, 61 (2011) 2120-2122.
[4] R.E. Mickens. Truly nonlinear oscillations.
World Scientific, 2010, Singapore.
[5] R. Reissig, G. Sansone, R. Conti. Qualitative
theory of nonlinear differential equations.
Moscow, “Nauka”, 1974 (Russian).
[6] S. Atslega, F. Sadyrbaev. Solutions of two-
point boundary value problems via phase-
plane analysis. Electronic Journal of
Qualitative Theory of Differential Equations.
Proc. 10th Coll. Qualitative Theory of Diff.
Equ. (July 1-4, 2015, Szeged, Hungary)
2016, No. 4, 1-10; doi:
10.14232/ejqtde.2016.8.4 .
[7] S. Atslega, & F. Sadyrbaev (2013). On
periodic solutions of Liénard type equations.
Mathematical Modelling and Analysis, 18(5),
708-716.
https://doi.org/10.3846/13926292.2013.8716
51.
[8] M. Sabatini. On the period function of ′′
′ J. Diff. Equations, 196
(2004), 151-168.
[9] J.C. Sprott. Elegant Chaos: Algebraically
Simple Chaotic Flows, World Scientific,
2010.
[10] M. Sandri. Numerical calculation of
Lyapunov exponents. The Mathematica
Journal, Vol. 6 (1996), Issue 3, 79-84.
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