3.1. Case 1: Two One-Point Attractors
In this case, we first make the following assumption:
H1. There exists a bounded and closed region U in the domain of system (2), such that there are precisely two asymptotically stable equilibria in the interior of U, say and , and on the boundary , all orbits run from outside into interior immediately. Usually, researchers only care about the behaviors within some regions or the structure they are interested in. For example, for biological models, only the quadrants where all variables are non-negative are focused, since variables usually represent the number of population or concentration, and only the positive quantities make sense. Sometimes whether the model has a periodic orbit is concerned as well because this orbit can cause the variable to oscillate. However, in this paper, we are only interested in the structure in assumption H1, and there are three main reasons. First, U is an attracting region from the outside view of it, though there are two stable equilibria within U. That makes it possible to bifurcate between monostable and this kind of bistable structure. Second, the region with the bistable structure as a whole can be considered as one of the attractors in another bistable structure. That is, a bistable structure can be embedded into another bistable structure. As a result, we may be able to use the bistable structure as the cornerstone to study the multi-stable structure. Third, the whole region U is attracting, which is based on the fact that, basically, the number of biological objects can neither become negative nor increase indefinitely due to limited environmental carrying capacity, such as species population, and densities of cells or microorganism.
The main results are presented below.
Theorem 4. For system (2), if H1 is satisfied, then we have the following conclusions: (i) There exists other isolated invariant set, denoted by S, besides of and , and its Conley index is .
(ii) There also exists connecting orbits from the invariant set S to attractors, and .
Proof. (i) Firstly, let us construct an isolating neighborhood for invariant set
S. Since
, is asymptotically stable equilibrium, there must exist two small open neighborhood
and
which are disjoint in
, i.e., the interior of
U, such that
and all orbits that pass through the boundary of
will gradually approach the equilibrium
as time goes to infinity,
. Then the region
is an isolating neighborhood, which is shown in
Figure 1.
It is not difficult to find that the exit set
of isolating neighborhood
N is the boundary of
and
, namely,
. Then by Definition 2 of the index pair in
Section 2, we can easily verify that
is an index pair. By forming the cone over the exit set
, we can obtain that the Conley index of index pair
is
(see also Example 3 in [
17]). According to the Ważewski Property 1, the interior of isolating neighborhood
N must contain isolated invariant set
S.
(ii) If we dig
out from region
U, then the remaining region and the corresponding exit set can be constructed as an index pair
, and its Conley index is
. According to the conclusion (i) above, we know that the attractor
and invariant set
S are contained in
. Thus, by Wedge Sum Property 2 in
Section 2, there must exist connecting orbit from
S to attractor
. Otherwise, their Conley index will not
but
, which is a contradiction. Similarly, there must be connecting orbit from
S to the attractor
. So the proof is completed. □
As mentioned in the introduction, more additional information would be necessary if one wants to obtain more details about an invariant set. So some of the prerequisites or limitations of the following study are based on this idea. Here, we make the following assumption:
H2. The dynamical system is structurally stable.
This assumption is mainly based on the fact that most biological objects, such as human metabolism or species population, are insensitive to small disturbances in the environment unless some massive changes are encountered. Corresponding to the mathematical model, it means that the dynamical behavior of the system is unaffected by small perturbations. Namely, the system is structurally stable, as we assumed above.
Before giving detailed results, we present the following lemma.
Lemma 1. For the two-dimensional system (2), there are at most a finite number of hyperbolic equilibria within a bounded and closed region in the domain of the system. Proof. We prove this lemma by contradiction. We first assume that there are an infinite number of hyperbolic equilibria within the bounded and closed region D. By Bolzano–Weierstrass Theorem (sometimes also called Sequential compactness theorem) we know that the subset consisting of all possible hyperbolic equilibria, denoted by E, is sequentially compact due to the compactness of D. That is, for each sequence of points in E it has a convergent subsequence converging to a point in E. For example, for sequence , its subsequence converge to point . In other words, for all there exists an such that when we have .
On the other hand, by Hartman–Grobman Theorem, we know that there exists a small neighborhood
of
, such that the solutions to system (
2) are homeomorphic to that of linearization of (
2) at
as long as it is inside
. System (
2) have the unique equilibrium
inside
, due to the uniqueness of the equilibrium to linearization. That is, there exists
, such that for any other equilibrium
e if it exists we have
. As a result, this contradicts the statement above, so there are at most a finite number of hyperbolic equilibria within
. The proof is completed. □
Then we introduce the concept of ‘loop’ to simplify the description of the following content. By ‘loop’ we mean the invariant set that consists of equilibrium and complete orbits and is homeomorphic to one-sphere, . Some examples are a periodic orbit, and an invariant set that is consist of a homoclinic orbit and the corresponding saddle, and invariant set that is composed of heteroclinic orbits and related equilibria.
Theorem 5. For system (2), if H1 and H2 are satisfied, then (i) There must be a finite and odd number of equilibria in S, namely equilibria, in which equilibria are the saddles, and k equilibria are unstable nodes or focuses;
(ii) There is at most one loop in S, which is an unstable limit cycle or a loop consisting of saddles, unstable nodes or focuses, and the heteroclinic orbits that flow from the latter to the former. Moreover, if the loop does exist, it contains an attractor inside it, and another attractor is outside of it.
Proof. (i) We prove that the invariant set
S contains a finite number of equilibria first. By H1, we know that region
U is bounded and closed. According to H2 and the Andronov–Pontryagin criterion [
18,
19], we obtain all the equilibria in
U are hyperbolic. Then by Lemma 1, we can conclude that there are at most a finite number of hyperbolic equilibria in
U. So the number of equilibria in
S is also finite.
Then, we prove that S contains an odd number of equilibria. The winding numbers both of the boundary and each attractor are +1. Then by the fact that the winding number of a closed curve is equal to the sum of the winding numbers of all isolated equilibria contained in it, we know that S must contain equilibria other than the two attractors, and the sum of the winding number of these equilibria is −1. Also, all equilibria are hyperbolic due to H2. So we can divide these equilibria into two types according to their winding number: saddle, whose index is −1, and non-saddle, whose index is +1. Node and focus belong to the non-saddle class. Then S must contain an odd number of equilibria, where the amount of the saddle is one more than the number of non-saddle. Otherwise, the sum of the winding number of total equilibria in S can not be −1. So in mathematical terms, S contains equilibria, where ones are saddles, and k are non-saddle. Moreover, because of H1, the stable equilibrium cannot be included in S. Thus, the non-saddle can only be unstable nodes or focuses.
(ii) We prove it by the following four steps:
The first step is to show that the possible loop is an unstable limit cycle or a loop consisting of saddles, unstable nodes or focuses, and the heteroclinic orbits that flow from the latter to the former. If there is no equilibrium on the loop, it will be an unstable limit cycle. That is because S does not contain other attractors, including cycle attractor, by H1 and the system is also structurally stable by H2. If the loop contains equilibrium, it will be the loop consisting of saddles, unstable nodes or focuses, and the heteroclinic orbits that flow from the latter to the former. Because the asymptotically stable equilibria cannot appear again on the loop by H1, and both the homoclinic orbits and the heteroclinic orbits that flow from one saddle to another saddle are not present due to their structural instability by H2. So the loop can only be one of these two candidates if any.
The second step is to give the position of the loop if it exists, and we will show that the loop must be around an attractor. In other words, two attractors are located inside and outside the loop, respectively. If it is not the case, then the loop either does not contain any attractor or contains two attractors.
For the first case, we assume that the loop does not contain any attractor. So there must be an infinite number of orbits in any small neighborhood inside the loop, and they flow further into the interior of the loop. Then their -limit sets are either cycles or points. The cycles are impossible because they must be hyperbolic according to structural stability. Stable cycles cannot appear, and unstable cycles cannot be served as -limit sets. Moreover, it is also impossible for their -limit sets to be points. Because an unstable node or focus cannot be served as the limit set as well, and the stable equilibrium is not contained inside this loop. The saddle is also impossible because the amount of saddles is finite by conclusion (i), and they cannot be the -limit sets of an infinite number of orbits. Therefore, the assumption is not correct, and the loop must contain at least one attractor.
For the second case, we assume that there are two attractors inside the loop. Then there must be an infinite number of orbits in the small neighborhood outside the loop, which are further away from this loop but still stay inside the region U. Their -limit sets are either cycles or points, but both are impossible based on the same analysis as the first case. So the loop cannot contain two attractors either.
In short, neither case is possible. So if the loop does exist, it must precisely contain one attractor.
The third step is to prove that there is at most one loop around each attractor. Assume there are two or more loops around each attractor. Then one can find two adjacent loops, say , , and is inside . There must be an infinite number of orbits in the small neighborhoods outside of and inside of . These orbits flow away from the two loops, respectively, and remain between these two loops. Then contradictions can be obtained based on the same analysis as the second step once again. Therefore, there is at most one loop around each attractor.
The final step is to show that it is impossible to have a loop for each attractor. We assume that is not true, then the outer neighborhood of each loop has an infinity number of orbits, which, respectively, away from these two loops and are still stay in the region U. Here the contradiction occurs again. So the assumption is not valid.
From the four steps above, we conclude that S contains at most one loop, which is an unstable limit cycle or a loop composed of saddles, unstable nodes or focuses, and heteroclinic orbits. Moreover, if the loop does exist, it must contain an attractor. The proof is completed. □
In the context above, the Conley index of the invariant set S and its possible components were studied. Now, we are going to consider the configuration of the orbits globally outside of the invariant set S.
Here according to whether the invariant set S contains the loop structure, we make the following two mutually exclusive assumptions:
H3. Invariant set S does not contain loop structure;
H3′. Invariant set S contains loop structure;
Furthermore, we also consider two additional cases according to the connectivity of S:
H4. The invariant set S is connected;
H4′. The invariant set S is disconnected, in which it has a finite number of connected components.
Note that it is beyond the capability of the Conley index to judge whether the invariant set S is connected or not, unless with the help of other information. Here, we are trying to study all possible configurations to understand clearly the internal structure of region U.
In the case of connected, we can obtain the following lemma.
Lemma 2. For system (2), if H1–H4 are satisfied, then for invariant set S, there must be precisely two disjoint orbits, such that the points on it will gradually approach S as time goes to infinity. However, on the other orbits nearing S points will be gradually away from S finally. Proof. We prove this lemma by contradiction. We assume that there exist n orbits, on which points gradually approach the invariant set S, but n is not equal to 2. Then these n orbits must be located on the stable manifolds of the saddles, and the orbits nearing these n orbits will away from the invariant set S finally. From H4, we know that S is connected, and from H2 and H3, we know that S does not have the loop structure, which can be thought of as a simple closed curve in a plane. Thus, S is simply connected, and it is homotopic to a point, denoted by . Then, have n stable manifolds, which are separated by n unstable manifold. So the Conley index of this point is the wedge sum of multiple of . Since the Conley index is algebraic topological invariant, the Conley index of original invariant set S is the wedge sum of multiple of as well. This result contradicts the conclusion (i) in Theorem 4, so there must be precisely two orbits considered above. The proof is completed. □
Theorem 6. For system (2), if H1–H4 are satisfied, then there must exist two different points, and , on , such that the orbits through or will approach invariant set S, that is, their ω-limit sets are subset to S. In addition, the orbits passing through other points on will approach the corresponding attractor or . Proof. By Lemma 2, we obtain that near S there precisely exist two different orbits, on which point are gradually approach S, that is to say, their -limit sets are subset to S. We denote these orbits by and , respectively. On the boundary of U, , there must exist two distinct points and , such that and flow into U through and , respectively. Otherwise, we assume that , comes from the interior of U. Thus, either flow from or flow from S, which are both impossible.
We denote the
-limit sets of
by
,
. Thus,
. Since
S is connected, there must exist a path from
to
. Thus, the two orbits,
and
, together with invariant set
S, can divide the region
U into two sub-regions, as shown in
Figure 2. For each sub-region, from H1 and Lemma 2, we obtain that it is a positively invariant set. So its limit set is nonempty. And then from H3, we know that there are no periodic orbits. So by Poincaré–Bendixson Theorem, the limit set must be equilibrium, namely
or
. In this case, the two stable equilibrium must be evenly distributed into two sub-regions. The proof is completed. □
The Theorem above describes the destination of the orbits through the boundary , i.e., their limit sets. Here, we call the connected set that is composed of orbits , , and invariant set S as separatrix. This is due to the fact that inside of region U, the orbits at each side of the connected set will approach different attractors, respectively.
Next, we consider the disconnected case. For this, we assume that S has K connected components and label these components by . Similar to Lemma 2 and Theorem 6, we have the following results.
Lemma 3. For system (2), if H2 and H3 are satisfied, and the connected invariant set is composed of saddle, unstable nodes or focus and connecting orbits, in addition, its Conley index is , then there must be precisely one orbit, such that the points on it will gradually approach as time goes to infinity. However, on the other orbits neighboring , points will run away from finally. Proof. The proof is similar to the one for Lemma 2, and we prove it by contradiction as well. Since is connected and from H2 and H3, we know that it does not have a loop structure and is simply connected. So can be homotopically shrunk to a point, denoted by . We assume that there exist precisely n orbits, on which point will gradually approach as time goes to infinity, and n is not equal to 1. Then the Conley index of is the wedge sum of the multiple of . Since the Conley index is an algebraic topological invariant, the Conley index of the original invariant set is also the wedge sum of the multiple of . This contradicts , so there must be one orbit as required. Thus the points on other orbits will be away from finally. The proof is completed. □
Theorem 7. For system (2), if H1, H2, H3, H4 are satisfied, then there must be disjoint points, , on , such that the orbits through will approach invariant set S. That is to say, their limit set is a subset of S. In addition, orbits passing through other points on will approach corresponding attractor, or . Proof. From H4
and Property 2 in
Section 2, one has
Due to the conclusion that
in Theorem 4, for invariant set
S, there must be one connected component whose Conley index is
and all the Conley index of other
connected component is the additive identity [
8], namely
. Otherwise, the Conley index of
S will be the Wedge sum of some items, which contradicts
.
To make it easier to describe, we re-label K connected components by exchanging the label of the first component and the component whose Conley index is . In the process, if the Conley index of component is exactly , then the label method remains the same. After this process, the Conley index of the first component will be , and the Conley index of others will be .
For connected component , by Lemma 2 above, we know that there must be two orbits whose limit sets are subset to , and we label them by . Similar to the Theorem 6, , must come from outside of region U. We denote the intersection points of these two orbits and boundary by , .
For each of other components, say . By Lemma 3 above we also know that there must be one and only one orbit whose limit set is subset to . This orbit must come from outside of U as well, and we denote the corresponding intersection point by .
Finally, we obtain different points on the boundary , and the orbits through them all approach the corresponding connected component of invariant set S. For other points on the boundary, the orbits that through them finally approach one of the two attractors. □
Similar to the connected case, there is also a separatrix in the disconnected case. The difference is that this separatrix is composed of the connected component , which Conley index is , and the two orbits, and . On the boundary , the points on the same side of this separatrix will approach the same attractor except a finite number of points, namely in the proof above.
To summarize, the separatrix acts as a role of the threshold, and the point on the boundary can be considered as an initial condition. Except for a finite set of initial boundary points that tend to the invariant set S, almost all the initial points tend to one of the two attractors. More importantly, those initial points located on the different sides of separatrix will have different destinies.
Theorem 8. For system (2), if H1, H2, H3, H4 are satisfied, then there must be one and only one point on the boundary of U, denoted by , such that its limit set is a subset of S, and within the region bounded by S some attractor, say , is contained. Moreover, all the other points on the boundary flow to another attractor, . Proof. First, by Theorem 5(ii) and H3
, there is one and only one loop in
S, which is an unstable limit cycle or a loop consisting of saddles, unstable nodes or focuses, and heteroclinic orbits flowing from the latter to the former. Moreover, this loop must contain an attractor, say
, and another attractor
is outside this loop, as shown in
Figure 3.
Next, we can obtain that S and the region bounded by S, where there are attractor and heteroclinic orbits flowing to , form a simply connected domain. Its Conley index is because its isolating neighborhood can be obtained by digging out a small open neighborhood of , like in Theorem 4, and the exit set is precisely the boundary .
Next, we prove that there is only one orbit that flows to S from the outside of S. There is only one loop in S, as shown in Theorem 5 above, so that we can deform homotopically the region bounded by this loop into a point. Then this point is asymptotically unstable since the original loop is unstable. As a result, the region bounded by S becomes a new invariant set, denoted by , which is consists of unstable nodes or focuses, saddles, and heteroclinic orbits, and its Conley index is still . Therefore, it can be obtained from Lemma 3 that there is only one orbit that tends to . So there only exists an orbit that flows to invariant S as well, and the limit set of this orbit must be a saddle, which is a subset of S but does not on the loop in S.
The orbit approaching S must flow from the outside of U, and it must intersect the boundary at a point, which is . The other orbits passing through will only eventually flow towards . The proof is completed. □
Theorem 9. For system (2), if H1, H2, H3, H4 are satisfied, then on the boundary of U, there must be K different points, whose limit set is a subset of S, and other points eventually flow to attractor . Proof. It can be obtained from H3 and Theorem 5 that there is only one loop structure in S, and this loop bounds the attractor . Then, the invariant set S and the region it surrounds, containing and related heteroclinic orbits flowing to it, constitute a new invariant set, denoted by . It also has K connected components, and each component is simply connected. The Conley index of this new invariant set is the addition identity, , and the computation method is the same as that of Theorem 8. Then, by Property 2, the Conley index of each component is , as well.
For the new variant set , the structure of the connected component containing the loop is the same as that of the combined structure of S with the region bounded by it in Theorem 8. Therefore, for this component, there must be only one orbit tending to it, denoted by . For the other components, according to Lemma 3, there is only one orbit tending to each component, denoted, respectively, by . Here, K orbits must come from the outside of U, and they intersect with the boundary at K different points. Therefore, for the K orbits passing through these K points, their limit sets are K components, respectively, subset to S, and all of the orbits passing through the remaining boundary points will eventually flow to the attractor . The proof is completed. □
When a loop exists, it does have another kind of separatrix, namely, the cycle separatrix. Different from the separatrix above, cycle separatrix divides the region U into two sub-regions that are not homotopy equivalent. The first one is an annular region, which is outside of the separatrix, and another is simply connected, which is inside of the separatrix. The loop structure in Theorems 8 and 9 just happen to be the cycle separatrix. Inside of it, almost all orbits flow towards attractor , while outside of the loop, almost all orbits, including the orbits coming from outside of U, approach attractor .
In the preceding Theorems or Lemmas, H1 requires orbits must enter the interior of U immediately. This restriction is a little bit harsh, and it can be relaxed slightly as below. Now the outside orbits are allowed to run along the boundary for a while once they touch the boundary, but eventually, they must enter the interior of the U the same as before. In other words, the boundary can contain orbit segments that comes from the outside of U and eventually enters the inside of U. So the new assumption is
H1′. There exists a closed region U in the domain of system (2), such that there are precisely two asymptotically stable equilibria in the interior of U, say and , and on the boundary , orbits run from outside into interior immediately, or firstly run along the boundary for a while then into its interior. With the assumption H1 replaced by H1, the invariant set S within U and its connection with two attractors are the same as those in previous studies. Thus, Lemmas 1–3, Theorems 4 and 5 are still valid, but Theorems 6–9 need to be modified appropriately. That is because the last four Theorems involve the points in boundary . And the expression “disjoint points” in each theorem or lemma should be replaced with “disjoint points or boundary segments”. Here, we do not take these related results as corollaries or new theorems. Accordingly, their proofs are omitted as well since they are almost the same as the proofs in related theorems. No matter whether the boundary is composed of points or orbit segments, as long as they all run into the interior of U, the internal structure of U will not be affected.
3.2. Case 2: A Cycle Attractor and a One-Point Attractor
In this case, we consider the second bistable structure, which contains a cycle attractor and a one-point attractor. Here, we first give the following two assumptions, which have the same boundary conditions as H1, and the relative positions of two attractors are also considered.
H5. There exists a closed region U in the domain of system (2), such that there are precisely two attractors, an asymptotically stable periodic orbit and an asymptotically stable equilibrium , where is inside , and on the boundary , all orbits run from outside into interior immediately; H6. There exists a closed region U in the domain of system (2), such that there are precisely two attractors, an asymptotically stable periodic orbit and an asymptotically stable equilibrium , where is outside , and on the boundary , all orbits run from outside into interior immediately; Similar to Theorem 4, we have the following results.
Theorem 10. For system (2), if H5 is satisfied, then (i) There is other isolated invariant set, denoted by S, in the annular region bounded by the circle and the point , and its Conley index is .
(ii) There are the connecting orbits from S to both attractors, and , as well.
Proof. The proof is similar to that of Theorem 4.
(i) Firstly, we construct an isolating neighborhood of invariant set
S. We can find an open neighborhood
of
since
is asymptotically stable, such that the orbits passing through the boundary
will flow into the interior of
and be attracted by
in the end. Moreover, we can also find an annular neighborhood
of
, which is bounded by two simple closed curves
and
, as shown in
Figure 4, such that the orbits passing through the boundary points will run into the interior of
and be attracted by attractor
. Then the annular region bounded by curves
and
is an isolating neighborhood, and the exit set is precisely composed of two boundary curves
and
. So its Conley index is
computed by collapsing the exit set into a point. By the Ważewski Property 1 of Conley index, we obtain that its isolated invariant set is not empty, and there must exist other isolated invariant set
S except two attractors.
(ii) The region bounded by curve is an isolating neighborhood, and its Conley index is , which does not equal the wedge sum of the two Conley indices of invariant and S, . So, by Wedge Sum Property 2 of Conley index, in addition to S and , there must be at least one connecting orbit from S to . Similarly, we also obtain the existence of connecting orbit from S to . The proof is completed. □
We should note that the system considered in H5 and H6 contains the periodic orbits, which are a kind of loop structure, so we do not consider the assumption H3 (or H3) in the current situation. Nevertheless, considering H2 is still rational. Given this, we have the following result.
Theorem 11. For system (2), if H5 and H2 are satisfied, then (i) There must be a finite and even number of equilibria in S, namely , , in which k equilibria are saddles, and other k equilibria are unstable nodes or focuses.
(ii) There is one and only one loop structure in S, and it contains in its interior.
(iii) Invariant set S is connected.
(iv) Denote the isolated invariant set within the region bounded by cycle attractor and boundary by . Then it must contain a finite and even number of equilibria, which is the same as (i).
Proof. (i) The proof of (i) is omitted here since its strategy and process are analogous to that of Theorem 5.
(ii) The proof of (ii) is divided into the following four steps:
The first step is to give the possible loop structure in S. There are two candidate types, an unstable limit cycle and the loop structure consisting of saddles, unstable nodes or focuses, and heteroclinic orbits. The loop must be an unstable limit cycle if there are no equilibria on it. Moreover, it must be the second type of loop if there are some equilibria on it. That is because there is no possibility of an additional stable equilibrium in S by H5, and also, no homoclinic orbit of the saddle flow to itself and the heteroclinic orbit of saddle flow to another saddle according to the structure stability assumption in H2.
The second step is to illustrate the position of the loop, that is, the loop, if any, must contain inside it. We prove it by contradiction. We first assume that the region bounded by the loop does not contain . Then in any small neighborhood inside either loop mentioned in the previous step, there must be an infinite number of orbits, which further flow into the interior of the loop. Then the limit sets of these infinite orbits are either equilibria or periodic orbits. The periodic orbit is impossible because the stable periodic orbit cannot appear again due to H5, and the unstable periodic orbit cannot be served as a limit set. Moreover, the equilibria are also impossible. Because unstable focus or nodes cannot be used as a limit set as well, and stable equilibria cannot reappear. Saddles are also impossible because their total number is finite, and it is impossible for an infinite number of orbits with a finite number of saddles as their limit set. So the assumption we made is not correct, and the loop must contain inside of itself.
The third step is to prove the existence of the loop by contradiction as well. First, we assume that a loop structure of any kind does not exist. Then the invariant set
S must consist of saddles, unstable nodes or focuses, and heteroclinic orbits according to Theorem 11 (i). Next, freely choose a saddle in
S, such as
, whose two stable manifolds must come from distinct unstable nodes or focuses, say
and
. We continue to check if both equilibria are connected to other saddles. It may assume that
is not, but
is connected to another saddle, denoted by
. Next, we continue to consider the stable manifold of
. Analogously, we can always end this process because the number of equilibria in
S is finite. In the end, we can obtain a simply connected invariant set similar to
Figure 5a, and its Conley index is
due to its asymptotical instability. We continue to count all other saddles, then obtain a finite number of the invariant set like
Figure 5a. If this process does not pick up all the unstable nodes or focuses, they can be considered as unstable invariant set disconnected from each other, and their Conley indices are still
. Finally, we obtain that the invariant sets
S is a collection of invariant set just like
Figure 5a and unstable equilibria. In this view, the Conley index of
S is the wedge sum of a finite number of
, which contradicts the Theorem 10 (i). So, there must be a loop structure.
The final step is to prove the uniqueness of the loop. We assume this is not true. That means there are two or more loops. From the second step, we know that
must be contained in the smallest loop, which is then contained in the second smallest loop again. And so forth, just like the Russian doll. These loops are finite due to the compactness of region
U and the structural stability of the system. Next, let us consider the two outermost loops, as shown in
Figure 5b (other cases are similar). Any small inner neighborhood of the outer loop must have an infinite number of orbits that further flow into the interior of this loop. Their
limit sets are either equilibria or loops. However, neither is possible through the same analysis as the second step. So the assumption is not valid, and the loop structure is unique.
(iii) We assume that the invariant set S is not connected. We know that S contains a loop, and the Conley index both of this loop and S is . Therefore, if there are other invariant sets, which are not connecting to the loop and also disconnected from each other, then their Conley index must be by Property 2. From (i), we know that S only can contain unstable nodes or focuses, saddles, and heteroclinic orbits. Also, Lemma 3 shows that there must be an orbit that is approaching this invariant set. Because the invariant sets are disconnected from each other, this orbit cannot come from S, and it cannot come from other invariant sets and as well. So, the contradiction occurs, and S must be connected.
(iv) We omit the proof of (iv) again because it can be proved via the winding number and the same idea as Theorem 5. The proof is completed. □
Similar to the case of two one-point attractors with the loop structure appears, the loop in Theorem 11 is the cycle separatrix. Almost all the orbits outside and inside of the separatrix flow towards the cycle attractor and the one-point attractor , respectively.
Theorem 12. For system (2), if H6 is satisfied, then (i) There is an isolated invariant set, denoted by , in the region bounded by cycle attractor , and its Conley index is .
(ii) There is also an isolated invariant set, denoted by , outside the cycle attractor but except the one-point attractor , and its Conley index is . Moreover, there are connecting orbits from to and , respectively.
Proof. (i) Known from H6,
is an asymptotically stable periodic orbit. Therefore, there must be a simple closed curve, denoted by
, in the small inner neighborhood of
, so that the orbits passing through
will flow to its outside and eventually approach
, as shown in
Figure 6. Thus, the closed region bounded by
together with the curve
form an index pair, and its Conley index is
, which is not equal to
. So, according to Property 1, there must be an isolated invariant set,
, in the region bounded by
, and it is also within the region bounded by
.
(ii) Similar to the above, we can also find a simple closed curve in the small outer neighborhood of so that the orbits passing through will flow to . Moreover, for , we can find an open neighborhood as well, similar to Theorem 4, so that the orbits passing through the boundary will flow to . Next, we dig out the interior of along with , resulting in a region with two holes. As a result, this region, together with the boundary of these two holes, constitutes an index pair, whose Conley index is not but . So, there is also an isolated invariant set between and .
Finally, we can prove the existence of connecting orbits by the same idea as Theorem 4. Firstly, digging out the region bounded by forms the index pair , which proves the existence of connecting orbit from to . Similarly, digging out the region shows the presence of connecting orbit from to . The proof is completed. □
When H6 and H2 are satisfied, we can refer to the situation of two one-point attractors. That is because if we see the cycle attractor from its outside, the cycle attractor and the one-point attractor have the same dynamics. Moreover, if we contract homotopically the region bounded by the cycle attractor, it would be a one-point attractor. In this case, there can be different separatrices. When the unstable loop structure appears, which must around or , this loop structure would be the cycle separatrix. However, when the unstable loop structure does not appear, the separatrix would be similar to the situations discussed below of Theorems 6 and 7.
Corresponding to the previous subsection, the assumption H5 and H6 can also be relaxed to the followings, respectively.
H5′. There exists a closed region U in the domain of system (2), such that there are precisely two attractors, an asymptotically stable periodic orbit and an asymptotically stable equilibrium , where is inside , and on the boundary , all orbits run from outside into interior either immediately or firstly run along the boundary for a while then into its interior. H6′. There exists a closed region U in the domain of system (2), such that there are precisely two attractors, an asymptotically stable periodic orbit and an asymptotically stable equilibrium , where is outside , and on the boundary , all orbits run from outside into interior either immediately or firstly run along the boundary for a while then into its interior.