Heteroclinic Cycles in a Class of 3-Dimensional Piecewise Affine Systems ()
1. Introduction
Since the introduction of the Lorenz system as a highly simplified model for atmospheric convection in [1] , extensive research has been conducted on chaos phenomena. The development of chaos generators has significant potential for various engineering applications. Hybrid systems have recently garnered considerable attention due to their critical role in circuit design, control theory, computer science, and biological molecular networks [2] [3] [4] [5] [6] . However, establishing the existence of singular cycles in general dynamical systems is challenging, as analytical calculations of invariant manifolds and solutions are not feasible [7] [8] [9] [10] .
Thankfully, it is possible to analytically determine the invariant manifolds and solutions of linear systems. This allows for the mathematical construction of piecewise affine systems with singular cycles, which can be utilized in chaotic generator design [11] [12] . Nevertheless, investigating the presence of singular cycles in general piecewise linear systems is not straightforward, as it involves detecting return times and potential intersections of singular cycles with switching planes, which is a complex task [6] [13] [14] . For instance, in [15] , the primary focus was on creating double-scroll chaotic generators by exploring the existence of heteroclinic cycles connecting two saddle-focus points and the associated chaotic dynamics in a specific class of 3-dimensional piecewise affine systems with a switching plane. Additionally, in [6] , the authors examined the existence of homoclinic orbits to saddle-focus and the resulting chaotic dynamics in 3-dimensional continuous piecewise linear systems in normal forms, with three parameters and a switching plane. Similarly, references [16] [17] investigated the existence of homoclinic orbits to saddle-focus and the corresponding chaotic dynamics in a specific class of 3-dimensional piecewise affine systems with a switching plane. Reference [18] delved into the existence of heteroclinic cycles that intersect two or three regions in a particular class of 3-dimensional three-zone piecewise affine systems with two switching planes. In reference [19] , the authors investigated multiple categories of planar piecewise Hamiltonian systems that feature three zones separated by two parallel straight lines. Reference [20] focuses on the study of external bifurcations of heterodimensional cycles in a 3-dimensional vector field. These cycles connect three saddle points and exhibit an orbit flip, forming a shape resembling the symbol “
”. In references [21] [22] [23] , the authors offered sufficient conditions for the coexistence of two singular cycles and the related chaotic dynamics in 3-dimensional two-zone piecewise linear systems with two parallel switching planes. Furthermore, they discovered that the coexistence of singular cycles could lead to a wider range of chaotic dynamics.
This paper is organized as follows. Section 2 gives some preliminaries of the 3-dimensional piecewise affine systems. Section 3 states the main results of this paper. Section 4 presents the proof of Theorem 1. Section 5 presents the proof of Theorem 2.
2. Statement of the Problem
Consider the 3-dimensional piecewise affine systems
(1)
where
,
are constant vectors in
,
are constant matrices in
. The eigenvalues of
are
,
with
,
,
, and the eigenvalues of
are
,
with
,
,
. Moreover, there exist invertible matrices P and Q such that
(2)
where matrices P and Q are given by
with
and
(
) being the generalized eigenvector of matrices
and
. In addition, we have
(3)
Let
Denote the switching manifold of the system (1) by
with
, where
Notice that
is an equilibrium point of the subsystem
(4)
and
is an equilibrium point of the subsystem
(5)
Moreover, assume that
and
with
,
.
From the representations of
and
in (3), the stable manifolds
,
and unstable manifolds
,
are expressed as
(6)
(7)
(8)
(9)
Suppose that
,
, and
Denote the solution of the system (4) with the initial condition
by
, and denote the solution of the system (5) with the initial condition
by
. Then we have
(10)
3. Main Results
In view of the method in [15] [17] , this section provides some theorems on the existence of heteroclinic cycles and homoclinic orbits of systems (1). For convenience, divide the region
into three parts
as shown in Figure 1.
In this article, we will only consider the case where the equilibrium point
is located in the
region. Similar methods can be used to discuss other situations.
Let
(11)
Theorem 1. If and only if System (1) exists constant real numbers
and
,
such that the following conditions hold, then there exists a heteroclinic cycle connecting
and
that intersects
transversally at
and intersects
transversally at
, as shown in Figure 2.
1)
(12)
(13)
2)
(14)
Figure 1. Graph of the switching manifold,
.
Figure 2. Graph of the heteroclinic cycle.
3)
(15)
where
(16)
and if
,
(17)
(18)
(19)
Theorem 2. If system (1) satisfies the conditions of Theorem 1 and the eigenvalues of the matrices
satisfy
then system (1) has infinite numbers of chaotic invariant sets.
4. The Proof of Theorem 1
If system (1) has a heteroclinic cycle connecting equilibrium points
and
that cross
transversely at two points, then one point is
and the other one is
.
Consider the definition of heteroclinic cycles, system (1) has a heteroclinic cycle connecting
and
which crosses
transversally at
and crosses
transversally
if and only if the following conditions hold:
1) The positive orbit of
satisfies
.
2) The negative orbit of
satisfies
.
3) The positive orbit of
satisfies
.
4) The negative orbit of
satisfies
.
5) Transversal condition:
Since
and
, then the negative orbit of
is a straight line connecting
and
, the negative orbit of
is a straight line connecting
and
. Hence,
We will prove that the positive orbit of
satisfies
which is equivalent to
(20)
If
, then we have
. And if
, the inequalities in (20) are equivalent to
(21)
and
(22)
where
(23)
(24)
(25)
Consider the expressions of
and
, notice that
,
, so the functions
,
are the periodic oscillation attenuation functions and
,
as
. To prove (21) and (22), we only need to consider the first local minimal points of the corresponding functions
and
in
.
From (21)-(22), we have
(26)
(27)
(28)
(29)
(30)
In the sequel, we will prove that
holds if and only if the second inequalities in (14) and the first inequality in (15) hold.
From (26), the local minimal points satisfy
(31)
If
satisfies Equation (31), then it satisfies
(32)
or
(33)
We can verify that and
for
satisfying Equation (33), so the
satisfying Equation (33) is not local minimal points. And
for
satisfying Equation (32), so
has the unique local minimal value in
at
in (32) with
(34)
In addition, from Equations (32) and (23), we have
then
Since
we must have
to ensure
for
. Moreover, consider the transversal condition
, then we have
. Thus
and we obtain that
Recall that
, we have
Therefore,
holds for
if and only if the second inequality in (14) and the first inequality in (15) hold.
Next, we will prove that
holds
if and only if the third inequality in (14) and the second inequality in (15) hold. Consider the
the first local minimal point
of the function
in
.
(35)
Similar to the discussions of
, the local minimal point
of
in
satisfies
(36)
and
(37)
To have
for
, we must have
, which is the third inequality in (15). If
similar to the discussions of
, we must have
.
From Equation (36), we have
Consider formula (24), we have
for
, and
for
. Then the local minimum point
satisfies
for
, and
for
.
Since
, we have
for
. If
and
, then we have
Thus, we obtain that
and
If
and
, then we have
Thus, we obtain that
and
Therefore,
holds for
if and only if the third inequality in (14) and the second inequality in (15) hold. Of course, using the aforementioned method, we can obtain the conditions for
.
From conditions of theorem (1), we have
In conclusion, conditions 1) - 5) hold if and only if conditions in Theorem 1 hold. The proof of Theorem 1 is completed.
5. The Proof of Theorem 2
In this section, we will prove Theorem 2 through a two-step process based on the methodology presented in reference [15] .
Construct the Poincaré Map
At first, if system (1) fulfills the conditions stated in Theorem 1, it possesses a heteroclinic cycle Γ which connects the fixed points
and
. This heteroclinic cycle Γ transversely intersects
at the points
and transversely intersects
at the points
, as depicted in Figure 2.
For a small real constant number
, let
, and
,
, and
where
,
and
, they are small enough such that
,
. According to the aforementioned definition of the Poincaré sections
, the heteroclinic cycle Γ intersects each
at a single point for
. Suppose
(38)
(39)
then there exist
,
, such that
(40)
(41)
To create the Poincaré map of the system (1), we first require the subsequent outcomes.
Let's consider the mapping
from
to
. For a point
is defined as the first intersection of the trajectory
with
. By (10), we get
(42)
Denote as
(43)
then we have
Denote the upper boundary of
as
, the lower boundary of
as
, the left boundary of
as
, and the right boundary of
as
.
Under the coordinate system
, utilizing polar coordinates for the x and y components on
, the transformation of the four boundaries by the function
can be described as follows:
From the above results, we have
, as
.
For sufficiently large values of k, we have
. Based on the above calculations, we can roughly draw the graph of
as Figure 3. Using the same approach used to define
, we define the mapping
from
to
:
(44)
Denote as
(45)
then we have
Similar to the representation of
, when n is sufficiently large, we can roughly draw the graph of
as Figure 4. Now, we introduce the mapping
from
to
. Note that the flight time of a point
to reach
corresponds to the largest negative solution of the equation
(46)
By the first equality of (39),
can be expressed as
Figure 3. Geometric structure of
and
.
Figure 4. Geometric structure of
and
.
Given the second equivalence in (40)
, we can conclude
from (46), we have
Therefore, according to the implicit function theorem, there exists a neighborhood
of
such that
where
are constant real numbers.
Then, neglecting the
terms, the expression for
is given by
(47)
where
are constant real numbers.
Note that
is an affine map. Using the same method to define
, we can also define the following map. The transformation
from
to
. There exists a neighborhood
of
such that the affine transformation
is defined as
(48)
where
are constant real numbers.
The map
from
to
. There exists a neighborhood
of
such that
:
(49)
where
are real constant numbers.
The map
from
to
. There exists a neighborhood
of
such that
:
(50)
where
are real constant numbers.
In the end, we construct the Poincaré map P as follows
(51)
For convenience, let
(52)
(53)
then we have
.
Note that
, and
are all affine mappings. Based on the diagrams of
and
shown in Figure 3 and Figure 4, we can select appropriate values of
and
such that
and
can be represented as shown in Figure 5 and Figure 6 for sufficiently large values of k and n.
Remark. Recalling Figure 3, under the mapping
, the right (resp. left) boundary of
is continuously mapped to the outer (resp. inner) boundary of an annulus-like object. Since the maps
and
are affine maps, the inner and outer boundaries of
correspond to the inner and outer boundaries of
, respectively.
Figure 5. Geometric structure of
and
.
Figure 6. Geometric structure of
and
.
Similarly, the inner and outer boundaries of
correspond to the respective inner and outer boundaries of
.
Statement 1. Consider
for fixed k large enough, under the conditions of Theorem 2, there exists a positive integer n such that the inner boundary of
intersects the upper boundary of
in two points, moreover, the inner boundary of
intersects the upper boundary of
in two points.
Proof: For fixed k large enough, any point in
can be expressed as
(54)
For the points in
expressed above, there exists a constant
such that the minimum value of
satisfies
Let n be the integer part of the number
According to the definition of
, the points in
can be expressed using the same formula as (54). Therefore, the maximum value of
for points in
can be determined
As a result, we can roughly draw in Figure 7 that the inner boundary of
intersects with the upper boundary of
at two points.
Figure 7. Geometric structure of
and
.
Consider points in
, any point in
can be represented as follows
(55)
for the smallest
, there exists a constant
,
On the other hand, the points
can be expressed in the same way as (55), and the largest absolute value of z
satisfies the following:
Hence, if we can prove that
and we can roughly draw in Figure 7 that the inner boundary of
intersects with the upper boundary of
at two points.
After performing straightforward calculations, we have
(56)
where
Consider the condition in Theorem 2:
, then from (56), we get
(57)
Consequently, for k large enough,
.
Statement 2. For sufficiently large values of k, under the conditions of Corollary 2,
contains an invariant Cantor set on which the Poincaré map P is topologically semiconjugate to a full shift on four symbols.
Proof: For sufficiently large values of k, according to Statement 1, there exists a positive integer n such that the intersection of
and
consists of two small disjoint vertical strips in
, denoted as
and
. In other words, we have
and we can draw it as Figure 8. Based on Statement 1, we can conclude that the intersection of
and
consists of two small disjoint vertical strips in
, which are denoted by
and
. In other words:
for
.
According to Remark 6, the top and bottom edges of
represent the inner and outer edges of
, respectively. Similarly, the top and bottom edges of
represent the inner and outer edges of
, respectively.
So, with respect to
, the primage of the left (resp, right) vertical boundary of
is included in the left (respectively, right) boundary of
. Similarly, the primage of the left (respectively, right) vertical boundary of
is included in the right (respectively, left) boundary of
, where
.
More importantly, in response to
, the primage of the two non-overlapping vertical strips
and
consists of two separate horizontal strips in
. Similarly, the primage of the two disjoint vertical strips
and
comprises two distinct horizontal strips in
.
Let
represent the primage of
, for
, as depicted in Figure 9.
Using a similar approach, we can conclude that the primage of the four horizontal strips associated with
are four horizontal strips that are encompassed by
. Moreover, each horizontal strip intersects the upper and lower boundaries of
at two distinct points, respectively, for
.
Denote by
the primage of
corresponding to
,
,
, which is shown in Figure 9.
Figure 8. Geometric structure of
and
.
Figure 9. Geometric structure of
and
.
Furthermore, it is evident that the left (resp, right) vertical edges of the primage
and
correspond to the left (resp, right) vertical edges of
and
, respectively. Similarly, the left (resp, right) vertical edges of the primage
and
correspond to the right (resp, left) vertical edges of
and
, respectively.
In summary, we can conclude that:
for
,
.
Additionally, it is clear that the left (resp, right, upper, lower) vertical boundaries of the primage
and
correspond to the left (respectively, right, upper, lower) vertical boundaries
and
, respectively. Similarly, the left (resp, right, upper, lower) vertical boundaries of the primage
and
correspond to the right (respectively, left, lower, upper) vertical boundaries
and
, respectively.
Then, by the Horse lemma, the proof is completed.
Remark. Based on the assertions made in Statements 1 and 2, the demonstration of Corollary 2 is finalized. In a manner akin to the Shil’nikov Theorems expounded in [12] [16] [17] , it becomes apparent that system (1) possesses a minimum of an enumerable infinite number of chaotic invariant sets.
6. Conclusions
This paper proposes an analytical method on the existence of heteroclinic cycles in a class of 3-dimensional piecewise affine systems. Under the study of the corresponding chaotic dynamics, it provides a way to construct chaotic systems. Our method also can be applied to piecewise affine systems with more intricate switching planes, enabling the generation of multiple homoclinic or heteroclinic cycles. Additionally, it is feasible to produce multi-scroll chaotic attractors.
T his article conducts an analysis of the geometric structure of these systems, laying the groundwork for understanding how the types and positions of two equilibrium points, as well as changes in the geometric structure of invariant manifolds, can affect the presence of singular cycles and chaos. The research presented in this article provides sufficient evidence to support the case where both equilibrium points are saddle-focus points and
is situated in the
region. However, challenges still remain in studying other types of situations, necessitating further investigation by scientific researchers.
The existence of singular cycles and chaos in such systems can be influenced by various factors, including the types and positions of equilibrium points, as well as the geometric structure of invariant manifolds. To gain a comprehensive understanding and analysis of the behavior of these systems in more general scenarios, additional research is required.
Scientific researchers can continue to explore the dynamics of systems with different types of equilibrium points, investigating how changes in their positions can impact the presence of singular cycles and chaos. This may involve the development of novel mathematical techniques, conducting numerical simulations, or even experimental studies, depending on the specific characteristics of the system under investigation.
By addressing these unresolved challenges and conducting further research, scientists can deepen our comprehension of the dynamics exhibited by such systems, potentially uncovering new insights and phenomena. These advancements will contribute to the advancement of this field, fostering a more holistic understanding of complex dynamical behaviors.