THE SNAPBACK REPELLERS FOR CHAOS IN MULTI-DIMENSIONAL MAPS

. The key of Marotto’s theorem on chaos for multi-dimensional maps is the existence of snapback repeller. For practical application of the theory, locating a computable repelling neighborhood of the repelling ﬁxed point has thus become the key issue. For some multi-dimensional maps F , basic informa- tion of F is not suﬃcient to indicate the existence of snapback repeller for F . In this investigation, for a repeller ¯ z of F , we start from estimating the repelling neighborhood of ¯ z under F k for some k ≥ 2, by a theory built on the ﬁrst or second derivative of F k . By employing the Interval Arithmetic computation, we locate a snapback point z 0 in this repelling neighborhood and examine the nonzero determinant condition for the Jacobian of F along the orbit through z 0 . With this new approach, we are able to conclude the existence of snapback repellers under the valid deﬁnition, hence chaotic behaviors, in a discrete-time predator-prey model, a population model, and the FitzHugh nerve model.

1. Introduction. After Li and Yorke [12] proved the celebrated period-threeimplies-chaos theorem for one-dimensional maps, extending analytic theory of chaos to multi-dimensional maps has become an interesting research topic. In 1978, Marotto advocated the notion of snapback repeller to generalize Li-Yorke's theorem on chaos from one-dimension to multi-dimension. Due to a technical flaw on the condition which leads to the repelling neighborhood of a repelling fixed point (repeller), the definition of snapback repeller was modified in 2005 to validate Marotto's theorem.
Let us first make clear the definition of repelling neighborhood.
Definition 1.1. Consider a C 1 map F : R n → R n and denote by B r (x) the closed ball in R n with center at x and radius r > 0 under certain norm on R n . A fixed pointz of F is called repelling if all eigenvalues of DF (z) exceed one in magnitude. Moreover, if there exist a norm · on R n and a constant c > 1 such that for all x, y ∈ B r (z), x = y, (1.1)

KANG-LING LIAO, CHIH-WEN SHIH AND CHI-JER YU
where B r (z) is defined under this norm, then we call B r (z) a repelling neighborhood ofz.
It is known that ifz is a repelling fixed point of F , then there exist a norm and an r > 0 so that B r (z) is a repelling neighborhood ofz. Notice that this property does not necessarily hold for the Euclidean norm in general. Even ifz is a fixed point and B r (z) is a closed ball centered atz, under some norm, such that |λ(z)| > 1, for all eigenvalues λ(z) of DF (z), for all z ∈ B r (z), (1.2) B r (z) still may not be a repelling neighborhood ofz. The reason is that the norm constructed for such a property depends on the matrix DF (z) which varies at different points z, as the mean-value inequality is applied, cf. [5]. Now, we recall the modified definition of snapback repeller [15] and state Marotto's theorem. For any point z 0 ∈ R n , we denote z j = F j (z 0 ), for 0 ≤ j < ∞, and z −j = F (z −j−1 ), for j ≥ 0, if such pre-images of z 0 exist. Definition 1.2. Letz be a repelling fixed point of the C 1 map F . If there exist a point z 0 =z in a repelling neighborhood ofz and a positive integer ≥ 2, such that z =z and det(DF (z j )) = 0 for 1 ≤ j < , thenz is called a snapback repeller of F .
We call the point z 0 in the Definition 1.2 a "snapback point" of F . Under Definition 1.2, the following Marotto's theorem holds [14,15]. Theorem 1.3. (Marotto's Theorem) If a C 1 map F has a snapback repeller, then F is chaotic in the following sense: there exist (i) a positive integer N , and F has periodic points of period p for every integer p ≥ N , (ii) a scrambled set of F , i.e., an uncountable set S containing no periodic points of F , and (a) F (S) ⊂ S, (b) lim sup k→∞ F k (x) − F k (y) > 0, for all x, y ∈ S, with x = y, (c) lim sup k→∞ F k (x) − F k (y) > 0, for all x ∈ S and periodic points y of F , (iii) an uncountable subset S 0 of S, and lim inf k→∞ F k (x) − F k (y) = 0, for every x, y ∈ S 0 .
Marotto's theorem provides an analytic method to detect chaos, which is effective in applications in finding the chaotic regimes (parameter ranges) for dynamical systems. However, in respecting this new definition, locating the snapback point in a repelling neighborhood of the repeller becomes the key issue in applying this theorem. If the norm in (1.1) is not Euclidean, then computing the repelling neighborhood is quite inaccessible. In addition, because of the unstable nature for chaotic behaviors, confirming the extent of repelling neighborhood and finding a snapback point in this neighborhood have been a nontrivial task, from computational view point. In order to overcome such a difficulty, developing skillful computation technique, combined with analytical properties of the maps, is an important research task. While a general approach to confirming snapback repeller has been lacking, certain piecewise-linear maps or maps in specific forms were considered in [7,18].
Notably, in [8], the condition det(DF (z j )) = 0, 1 ≤ j < , is not included in the definition of snapback repeller. Instead, a homoclinic orbit to a repeller was called nondegenerate therein if such condition also holds, and the investigation focussed on bifurcation from degenerate homoclinic orbit to nondegenerate homoclinic orbit.
In this paper, we establish a new approach to judge whether a repeller is a snapback repeller. This approach then leads to effective applications of Marotto's theorem to conclude chaotic behaviors in multi-dimensional maps. The chaotic behaviors for the maps considered in [4,6,9,17,19,20] were studied under the original invalid definition of snapback repeller [14]. With the present approach, we are able to investigate the snapback repellers, under the valid Definition 1.2, and hence chaotic behaviors, for those maps.
We first propose a methodology to exploit the repelling neighborhood of a repeller. In section 2, we extend a previous result on finding repelling neighborhood of a repeller in [13]. Instead of considering the repelling neighborhood of a repeller z for map F , herein, we locate a repelling neighborhood U ofz for map F k , for some integer k ≥ 2. Moreover, if there exists a point y 0 ∈ U , y 0 =z, such that (F k ) (y 0 ) =z, for some integer ≥ 2, and det(DF (y j )) = 0 for −∞ < j < k , thenz is a snapback repeller of F k . Subsequently, we can further argue thatz is also a snapback repeller of F .
In section 3, we introduce our computational approach which is based on the Interval Arithmetic (IA). Interval Arithmetic is an arithmetic defined on sets of intervals, rather than sets of real numbers. Modern development of IA can be traced back to Moore's dissertation [16]; further notion and details for IA can be found in [1,2]. Indeed, IA is an arithmetic that contains all numerical errors of each numerical computation step, and thus can provide a rigorous justification via numerical computation. We shall perform IA computation to find a repelling neighborhoodŨ of repellerz for map F k , for some k ≥ 1, under the Euclidean norm. The above-mentioned two approaches lead to two neighborhoods U andŨ ofz for map F k . The existence of true pre-images ofz under F can be confirmed by IA computation. By justifying det(DF (z)) = 0 for z ∈ U orŨ , we can then confirm thatz is a snapback repeller of map F .
Computer-assisted approaches have played crucial roles in investigating complex dynamics in state-of-the-art research. For example, computer-assisted Morse decomposition and Conley index were performed to study the recurrent dynamics, and the global dynamics in [3]. On the other hand, IA computation is a handy tool with convenient programming packages, and is rather powerful for investigating and confirming local dynamical objects.
The rest of this presentation is organized as follows. In section 2, we present an estimation on the radius of repelling neighborhood of a repeller for map F k , k ≥ 1, under the Euclidean norm. Moreover, we explore the relation of snapback repeller between map F and map F k , k ≥ 2. In section 3, we briefly introduce the notion of IA and apply IA computation to find a repelling neighborhood of repeller z for map F k . We construct in this region a backward orbit ofz under map F k to confirm thatz is a snapback repeller of F k for some k ≥ 2, and hence F . We demonstrate the present theory and IA computation technique in a discrete-time predator-prey system in section 3. In section 4, we apply the present methodologies to study snapback repellers and chaotic behaviors in the discrete-time FitzHugh nerve model and a discrete-time population model.

2.
Estimating repelling neighborhood. Consider a C 1 map F with fixed point z. An eigenvalue condition on DF (z), the linearization of F atz, which leads to the existence of a repelling neighborhood ofz, was given in [10]: If λ > 1, for all eigenvalues λ of (DF (z)) T DF (z); (2.1) then there exist an c > 1 and an r > 0 such that all eigenvalues of (DF (z)) T DF (z) are greater than one for all z ∈ B r (z) (under Euclidean norm), and F (x) − F (y) 2 > c · x − y 2 , for all x, y ∈ B r (z), x = y. Note that the value for the radius r is not specified, and thus this result only indicates the existence of a repelling neighborhood ofz. Condition (2.1) was actually adopted in formulating a disparate definition of snapback repeller in [10]. Later, Marotto [15] pointed out that a repeller can be a snapback repeller under Definition 1.2, without satisfying (2.1). On the other hand, for practical application of Marotto's theorem, to confirm that there exists a snapback point in a repelling neighborhood ofz, one needs to locate this neighborhood beforehand, in respecting Definition 1.
2. An approach to estimate this neighborhood was reported in [13]. The treatment therein can be extended: For a given map F , we consider its k-fold composition G := F k , where k is a positive integer. Letz be a repelling fixed point of F , and hence G. We denote For an r > 0, on the closed ball under Euclidean norm, B r (z), we set 3) The following proposition provides an estimation for the repelling neighborhood of repellerz under map G = F k .
Proposition 2.1. Consider a C 1 map F with fixed pointz. Let s k , η k,r be defined as in (2.2)-(2.3). Then B r (z) is a repelling neighborhood ofz for map F k , under the Euclidean norm, provided that there exist an k ∈ N and an r > 0 such that If F is C 2 , then through the eigenvalues of (DG(x)) T DG(x) and Hessian matrix [∂ k ∂ l G i (x)] k×l , we can also obtain a different estimation for the repelling neighborhood ofz for map G. Such an estimation for the repelling neighborhood has been proposed for a map F in [13]. Herein, we extend it to an estimation for map F k . Notice that when (2.4) fails to hold for map F (i.e., k = 1), it may still hold for map F k for some k ≥ 2. We plan to use the properties when (2.4) holds for map F k for some k ≥ 2, to justify the existence of snapback repeller for map F .
Next, we discuss the relation between the existence of snapback repeller for map F and map F k , k ≥ 2.
(ii) There is an entropy relationship between maps F and F k , namely, is the topological entropy for map F k , see Proposition 2.8 of [11].
In practical application, finding a snapback point in the repelling neighborhood and examining condition (2.6) still require some computation technique which can control numerical error and lead to a rigorous justification on the dynamics. We shall employ Interval Arithmetic computation for such a task in the next section.
3. Computation via Interval Arithmetic. We introduce the Interval Arithmetic in subsection 3.1. In subsection 3.2, we demonstrate our approach and computation to a map F arising from predator-prey models. We hope to verify that the map G := F 2 (i.e., k = 2) admits a snapback repeller, and then deduce that this repeller is also a snapback repeller of map F .
3.1. Interval arithmetic. In this subsection, we introduce the chief notion and theory of the interval arithmetic. First, we denote by IR the collection of all bounded closed intervals in R, and by IR n the collection of all bounded closed n-dimensional boxes in R n , namely, A real interval vector X ∈ IR n represents a box in R n : where operator • ∈ {+, −, ×, ÷}. For division, the assumption 0 / ∈ Y is required. This definition can be easily extended to arithmetic on vectors and matrices with interval entries. Note that a real number a can be treated as a degenerate interval a := [a, a]. The interval arithmetic can therefore be regarded as an extension from the usual arithmetic of real numbers. For a function f = f (x), x ∈ R, its "natural extension " of interval function f (X) is evaluated on interval X, and is a closed interval enclosing {f (x)| x ∈ X}. Such interval functions admit the inclusion monotonicity, i.e., where X and Y lie in the domain of f . Similar notation and consideration apply to functions f = f (x), x ∈ R n and f (X) on interval vector X. One of the most important virtues of IA is that computation is executed in terms of the endpoints of intervals, in the outward rounding mode 1 . One can therefore obtain rigorous bounds for complicated inequalities or inclusions with only machine operations. Furthermore, in combining with the following Interval Newton's method, it is feasible to perform qualitative mathematical proofs through numerical computation.

3.2.
Application to a predator-prey system. Our approach is to locate the repelling neighborhood of a repellerz for F k under the Euclidean norm, for some k ≥ 2, and then find a snapback point y 0 in this neighborhood under F k . We then employ multi-shooting method to locate the homoclinic orbit ofz under F , via IA computation. The last step is to examine condition (2.6) or (2.8) by IA computation and apply Theorem 2.2 to confirm thatz is a snapback repeller of F . This process will conclude the existence of snapback repeller for a C 1 map F through rigorous justification via numerical computation based on IA.
In this subsection, we demonstrate our approach by the following discrete-time predator-prey system [4,17] F (x, y) = (f 1 (x, y), f 2 (x, y)) = xe b(1−x)−ay , x(1 − e −ay ) . In [17], shadowing arguments and computer-assisted techniques were employed to show thatz 2 is a snapback repeller of F , under the original (invalid) definition of snapback repeller, and sup-norm on R 2 . Herein, we employ the valid Definition 1.2 of snapback repeller to justify thatz 2 is indeed a snapback repeller of F . Moreover, the approaches in [10,13] can not be applied in this case, as the condition (2.1) fails to hold forz 2 . In addition, by using Proposition 2.1, we find that B 0.009 (z 2 ) is a repelling neighborhood ofz 2 for map F 2 . Hence, there is a potential thatz 2 is a snapback repeller of F 2 . Therefore, we plan to verify thatz 2 is a snapback repeller of map G(x, y) := F 2 (x, y), and hence of F , by Theorem 2.2. Let us abbreviatez 2 asz and adopt the Euclidean norm on R 2 . We shall use the built-in IA tools in Mathematica to perform the computation and rigorous justification. For convenience, definitions of all functions below will be naturally extended to their interval versions without further noting. Denote by N r (z) the box  Therefore, there exists a uniquez ∈ Z 0 such that H(z) = 0, and thus a true fixed pointz of F in Z 0 , by Theorem 3.1.
Remark 3.1. Proposition 2.1 allows us to find a preliminary repelling neighborhood ofz under G (B 0.009 (z) for this predator-prey system). By employing IA computing, it is possible to locate a larger repelling neighborhood (B 0.036 (z) in this case). Basically, it takes less computation to find a repelling neighborhood ofz by using Proposition 2.1, whereas larger repelling neighborhood is more advantageous to finding the snapback point.
(III) Finding a snapback point z 0 in the repelling neighborhood B 0.036 (z) and examining (2.7): We shall construct successive pre-images ofz under G to locate a point y 0,q ∈ B 0.036 (z) with G (y 0,q ) lies in the region B 10 −11 (z), for some ≥ 2. We then apply Theorem 3.1 to confirm the true homoclinic orbit ofz under G.
We use the forward iteration to exhaustively sift the possible points of y 0,q by robust numerical computation. First, we check the forward iterations of numerous sample points in the region B 0.036 (z). If there is a point which is mapped back into the region B 10 −11 (z) after some iterations, then we pick it as a candidate for the snapback point. Second, we use numerical Newton's method to refine the locations of those candidate points and find the numerical snapback points. Herein, through computation, we choose one of the numerical snapback points: y 0,q = (0.3958825716962441, 0.3712349439188911) ∈ B 0.036 (z).
Since IA must enclose all numerical errors in the iterations of functions, overestimation is inevitable. On the other hand, a little deviation of the initial point in the beginning may result in very different terminal point by iterations, and the IA computing will break down due to the enormously accumulated overestimation. To prevent this, we employ the multi-shooting method which decomposes the whole iteration into several shorter sections and restrains the overestimation of IA.
More precisely, instead of solving a single equation G (y) −z = 0, with one node {y}, to determine the entire long orbit, {y = y 0 , G(y 0 ), · · · , G (y 0 ) =z}, we consider a system of m connecting equations, S m (y, v 2 , · · · , v m ) = 0, i.e., with m independent nodes {y, v 2 , · · · , v m }, whose shorter sections of orbits can be lumped together into {y = y 0 , G(y), · · · , G p−1 (y), G p (y) = v 2 , G(v 2 ), · · · , G p (v m−1 ) = v m , With such a treatment, the number of iterations for each node is p which is far less than , and therefore the numerical errors induced by iterations can be controlled. Based on this multi-shooting method, we obtain the following theorem. Proof. For = 11, if we set m = 4, p = 3 and take √ 2 (ỹ 9,q ), then we can justify det (DS 4 ) = 0 on U, by IA computation. Next, define I(S 4 , U) : =ẑ − (DS 4 (U)) −1 S 4 (ẑ), whereẑ = (ỹ 0,q ,ỹ 3,q ,ỹ 6,q ,ỹ 9,q ) ∈ R 8 is the middle point of U. We can then justify I(S 4 , U) ⊆ U, by IA computation. Therefore, in the interval vector U, there exists a root (y 0 , v 0,2 , v 0,3 , v 0,4 ) of S 4 . Accordingly,z = G 11 (y 0 ), and thus y 0 ∈ B 0.036 (z) is a snapback point of G. We can further use IA to verify det (DG(x)) = 0, for any x ∈ ∪ 10 j=1 N 10 −11 / √ 2 (ỹ j,q ). Hence, det(DG(ỹ j )) = 0, for 1 ≤ j ≤ 10, and thusz is a snapback repeller of G. Proof. First, in order to prevent overestimation, we divide the region N 0.036 (z) into finitely many small parts V i , namely, ∪ i V i = N 0.036 (z). For each V i , we firstly use numerical Newton's method to find a region U i ⊆ N 0.072 (z) such that for any v ∈ V i ⊆ N 0.036 (z), there exists a numerical pre-image u q ∈ U i of v under map F . Next, we use IA to prove that there exists a true preimage u ∈ U i of v such that F (u) = v. To this end, we need to examine , and inclusion monotonicity (3.1). Subsequently, we obtain . Therefore, there exists a unique u ∈ U i ⊆ N 0.072 (z) such that h v F (u) = 0, i.e., F (u) = v, by Theorem 3.1.
Proposition 3.5 indicates that N 0.036 (z) ⊆ F (N 0.072 (z)). In addition, G −1 is a contraction on B 0.036 (z), as B 0.036 (z) is a repelling neighborhood ofz. Therefore, y j ∈ N 0.072 (z), for − ∞ < j ≤ −1, since y 0 ∈ B 0.036 (z). Moreover, by using IA computation, we can justify det (DF (x)) = 0, for all x ∈ N 0.072 (z). Therefore, (2.6) is met, andz is a snapback repeller of map F . 4. Further applications. In this section, we further apply our approach and IA computation to study the existence of snapback repellers in two two-dimensional maps.
Example 4.1. Consider the two-dimensional map F = (f 1 , f 2 ) defined by The map (4.1) arises from a population model, see [20]. Existence of Marotto's chaos was concluded by employing the original definition of snapback repeller in [14], which is invalid. Let us consider specifically the parameter a = 7, b = 1.3, as an illustration. For system (4.1),z := (x,ȳ) = (1, a−b) = (1, 5.7) is a repelling fixed point. We compute the eigenvalues of (DF (z)) T DF (z) and find λ ≈ 0.864894 so that condition (2.1) is not met. However, we compute the eigenvalues of (DF 2 (z)) T DF 2 (z) and find λ ≈ 15.6542. Therefore, Proposition 2.1 can be applied with k = 2. We compute to find s 2 ≈ 3.95654, and r = 0.08, and hence B 0.08 (z) is a repelling neighborhood ofz for G := F 2 . In addition, the multi-valued inverse of map (4.1) can be expressed explicitly as We compute to find numerical pre-images ofz under G:z . Computation shows that 0 / ∈ det(DG(Z 1 )) and I(G, Z 1 ) ⊂ Z 1 where I is defined as (3.2). Hence, by Theorem 3.1, there exists an unique y 0 ∈ Z 1 such thatG(y 0 ) = 0, i.e., G 4 (y 0 ) =z.
In addition, since G −1 is a contraction on B 0.014 (z), we have det DF (y j ) = 0, for −∞ < j ≤ 39. Thus,z is a snapback repeller of F .

Conclusion.
We have presented an approach which combines analytic property and computation technique to verify the existence of snapback repeller for multidimensional maps. For a given map F , we established two methodologies to locate the repelling neighborhood of a repeller of map F k , for some k ≥ 2, under the Euclidean norm. The first one is to estimate the radius of the repelling neighborhood through computing the first or second derivative of F k , whereas the second is to employ the Interval Arithmetic computation. We have further employed the IA computation to locate the snapback point in the repelling neighborhood and examine the nonzero determinant condition on the Jacobian of F . By exploring and utilizing the relation between the existence of the snapback repeller for map F and map F k , k ≥ 2, we confirmed the existence of snapback repeller for map F . This new finding also serves as an indicator to whether a repeller of F is indeed a snapback repeller. The present approach has relaxed the restriction of the methods reported in [10,13], and provided concrete examples connected to the studies in [11]. Not only that the conditions for the present theory are all computable numerically, but the numerical results through IA computing also provide a rigorous confirmation to the existence of snapback repeller. We have employed the present approach to explore the existence of snapback repellers under the valid definition, hence chaotic behaviors, in a predator-prey model, a discrete-time population model, and the FitzHugh nerve model. This paper has demonstrated that IA computation is effective and handy (with convenient packages) in observing and concluding chaotic dynamics in multidimensional maps, as incorporated with pertinent analytic theory.