1D piecewise smooth map: exploring a model of investment dynamics under (cid:133)nancial frictions with three types of investment projects

We consider a 1D continuous piecewise smooth map which depends on seven parameters, and depending on parameter values it can have up to six branches. This map, proposed by Matsuyama [21; Section 5], describes the macroeconomic dynamics of investment and credit (cid:135)uctuations, in which three types of investment projects compete in the (cid:133)nancial market. Introducing a partitioning of the parameter space according to di⁄erent branch con(cid:133)gurations of the map, we illustrate this partitioning for a speci(cid:133)c parameter setting. Then we present an example of the bifurcation structure in a parameter plane, which includes periodicity regions related to superstable cycles. Several bifurcation curves are obtained analytically, in particular, border-collision bifurcation curves of the (cid:133)xed points. We show that the intersection point of two such curves is an organizing center from which in(cid:133)nitely many other bifurcation curves issue.


Introduction
Nonsmooth maps often appear in applied models when some sharp transition in the state space is modelled by means of piecewise smooth functions (see, e.g. the monographs [6], [7], [33] and references therein).Mathematical tools and methods for studying the dynamics of these maps are currently quite well developed, including those used for smooth systems (see, e.g., [27], [9], [12]) and speci…c for nonsmooth ones ( [10], [2]).A characteristic property of piecewise smooth maps is related to the existence of a border point(s) (or switching manifold(s) in higher dimensions) that separates the regions of di¤erent de…nitions of the map.By varying some parameter, a border-collision bifurcation (BCB for short) can occur when an invariant set of the map (e.g., an attracting …xed point or cycle) collides with a border point, leading to a qualitative change in the dynamics.Typical examples are transitions which cannot occur in smooth maps when, for instance, a BCB of an attracting …xed point leads to an attracting cycle of any period or directly to a chaotic attractor.Since [25], where the notion of border-collision bifurcation was introduced (see also [13], [26]), these bifurcations and related problems are quite actively studied from both theoretical and applied points of view ( [4], [8], [28], to cite a few).An advantage of one-dimensional (1D for short) continuous piecewise smooth maps is a possibility to use skew tent map, which is a 1D piecewise linear map with one border point (see, e.g., [15], [16], [24], [18]), as a border-collision normal form.We refer to [2], where it is explained in detail how to apply the skew tent map to classify possible outcomes of a BCB in a 1D continuous piecewise smooth map.
A special case of this model, in which two di¤erent types of investment projects compete against each other in the presence of …nancial frictions, described in Sections 2-4 of [21], has already been studied in detail in [30], [31], [22], where the corresponding map is de…ned by three branches: increasing, decreasing and ‡at.In the cited papers, we distinguish between two cases, depending on whether a ‡at branch is involved in the asymptotic dynamics or not.In the …rst case, dominant attractors are superstable cycles, and in [30] we introduce a modi…ed U-sequence (see [23]) ordering these cycles using their symbolic sequences.In the second case, the resulting map is unimodal, and its dynamics is characterized by not only standard smooth bifurcations, but also BCBs, leading to speci…c bifurcation structure in the parameter space, including open regions associated with chaotic attractors (this phenomenon is known as robust chaos, see [5]).
In this paper, we deal with the model of Section 5 of [21], which features three di¤erent types of investment projects competing against one another, where the corresponding map can have up to six branches.This leads to a greater variety of possible BCBs and thus to more interesting bifurcation structures.In particular, we describe an organizing center de…ned as an intersection point of two BCB curves, from which in…nitely many other bifurcation curves issue.Recall that such organizing centers are often observed in discontinuous maps, e.g., in Lorenz maps (several examples can be found in [2], see also [3]).
The paper is organized as follows.In Sec. 2 we …rst introduce the map and then some preliminary results follow, which are needed to classify the possible cases related to di¤erent branch con…gurations of the map.The de…nition of the map in each case is given in Appendix.In Sec. 3 we discuss the bifurcation structure of the parameter space of the map illustrating it by several numerical examples.Sec. 4 concludes.

Preliminaries
The dynamics of the considered model is de…ned by a family of 1D continuous piecewise smooth maps f given by where and the parameters satisfy the following conditions: Map f reduces to the one studied in [30] (see also [22], [31]) for m 2 = m; = 1 and B: In the present paper, we …x values of the parameters ; ; B and m 2 = m in the parameter region E de…ned as follows: and study the bifurcation structure of the ( ; )-parameter plane.In the cited papers, the dynamics of map f is studied in detail, in particular, it is shown that for parameter values belonging to E; map f can have (possibly coexisting) stable and superstable cycles of any period as well as cyclic chaotic intervals of any period; outside E the dynamics of f is rather trivial.Later we recall how the boundaries of region E are obtained (see Sec.3).
As an illustrative example, we consider the following case: The region E is this case is de…ned as In the numerical examples we …x = 0:5; m 1 = m 2 = 1:2; B = 2:5; = 0:15: It is easy to check that the parameter values given in (4) belong to the region E. In Fig. 1(e), we present the partitioning of the ( ; )-parameter plane (for other parameter values …xed as in ( 4)) into the regions related to di¤erent branch con…gurations of map f: All the other …gures in Fig. 1 show related examples of map f .We will use these …gures to illustrate our reasoning below.
According to the introduced notations, the functions f M (w) and f R (w) can be de…ned as In the special case (2), the branch f R1 (w) (which in general is an increasing or decreasing function) becomes constant: The functions 1 (w) and 2 (w) can be de…ned as In Fig. 1, besides map f various examples of these functions are also shown.Consider now the possible solutions of the equation 1 (w) = : The branch 1;2 (w) of 1 (w) is a decreasing function, while the branch 1;1 (w) is a unimodal function with an extremum (minimum) at It holds that w > w for > 1

2
; that is, in this case both branches of 1 (w) are decreasing.Thus, a su¢ cient condition to have a unique solution of the equation For ( 2), the equality in (H2) corresponds to (see Fig. 1(e)).
If the assumption (H2) does not hold, that is, if < 1 2 (so that w < w ), then for 1;1 (w ) < < 1;1 (w ) = 1;2 (w ); the equation 1 (w) = has three solutions denoted w 0 < w 00 < w 000 , where w = w 0 ; w = w 00 are two solutions of 1;1 (w) = ; and w = w 000 = 1 1 is a solution of 1;2 (w) = (see an example in Fig. 1(c)); this case occurs for and (the index A2=3 is clari…ed in Appendix, see ( 20));.for > A2=3 ; the unique solution of 1 (w) = is w = w 0 (see an example in Fig. 1(f)); for < T ; the unique solution of 1 (w) = is w = w 000 .For (2), the equalities ( 8) and ( 9) become (see the curves T and A2=3 in Fig. 1(e), where m = 1:2).The solution of 2 (w) = is For < B; as required, it is unique, and it holds that w < w : The de…nition regions of the various branches of f depend also on an intersection point of 1 (w) and 2 (w): Let it be denoted w c when w c w ; i.e., when it is related to the branch 1;2 (w) of 1 (w); i.e., 1;2 (w c ) = 2 (w c ); or (see an example in Fig. 1(d)); b w when b w w ; i.e., when it is related to the branch 1;1 (w) of 1 (w); i.e., 1;1 (see an example in Fig. 1(g)); note that if < 1 2 and T < < A2=3 (when there are two solutions of the equation 1;1 (w) = ), then b w < w is a su¢ cient condition for the inequality 1 (w) > max f 2 (w); g (de…nition condition for the branch f L ; see (1)) to be satis…ed in just one interval; it holds that b w = w ; that is, Let us summarize now the preliminary observations presented above and distinguish between di¤erent branch con…gurations of map f: It is convenient to divide them into two cases, when w < w c (denoted as Case A) and w > w c (Case B), with further division into subcases, A1, A1 0 , A2, A2 0 , A3 and B1, B1 0 , B2, B3, as explained in Appendix.In Fig. 1(e), we present the partitioning of the ( ; )-parameter plane according to these subcases, and in the …gures around Fig. 1(e), related examples of map f are shown (see Appendix for the de…nition of map f in each case).Since w < w c for > 1 wc m1 , the transition from Case A to Case B occurs at For ( 2), this transition occurs at As one can see in Fig. 1(e), above the line H2 only the cases A1, A2 and A3 can occur; in the strip between the lines H1 and only the cases B1, B2 and B3 can occur; and in the strip between the lines and H2 all the cases can be realized.In particular, for the parameter values belonging to the region bounded by the curves T ; A2=3 and B1=2 , there are regions associated with cases A1 0 , A2 0 and B1 0 , whose distinguishing feature is the presence in f of two de…nition intervals of the branch f L : In Fig. 1(h), we show map f at a special parameter point ( ; ) = ( A1=2 ; A=B ) indicated by the red circle in Fig. 1(e), from which the boundaries of several partitions issue.One could expect that this point is a kind of organizing center from which in…nitely many bifurcation curves issue.However, the true organizing center in the ( ; )-parameter plane is an intersection point O (indicated by the blue circle in Fig. 1(e)) of two BCB curves, = BCM 2 and = BCR1 , as we discuss in the next section.

Bifurcation structures in the parameter space
Before we proceed with a description of the bifurcation structure of the ( ; )-parameter plane, let us recall in short what is known about the dynamics of map f in case A1 (see (16)).These results are summarized in Fig. 2 (for details, see [30], [22], [31]).Namely, in Fig. 2(a) we show the bifurcation structure of the ( ; B)-parameter plane for = 0:5; m = 1:2 (as in ( 4)).Here the region E (see (3)) is bounded by the bifurcation curves of the …xed points w L ; w R2 and w R3 associated with the branches f L ; f R2 and f R3 ; respectively: the curve de…ned by ) is related to a degenerate ‡ip bifurcation of w R2 (see [29], where degenerate bifurcations are described); the curve B = 1 m( 1) (denoted BC R3 ) corresponds to a BCB at which w R3 = w = w R2 : Other curves shown in Fig. 2(a) are F B 2 (subcritical ‡ip bifurcation of 2-cycle denoted 2 ), H 2 (homoclinic bifurcation of 2 ), H 1 (homoclinic bifurcation of w R2 ), BC 3 (fold BCB leading to a pair of 3-cycles, attracting 3 and repelling 0 3 ), F B 3 (subcritical ‡ip bifurcation of 3 ), H 3 (homoclinic bifurcation of 3 ), H 0 3 (homoclinic bifurcation of 0 3 ), BC J (contact of the absorbing interval J = [f 2 (w c ); f (w c )] with the ‡at branch f R3 ; occurring when f (w c ) = w ; below BC J the ‡at branch f R3 is involved into asymptotic dynamics, so that the dominant dynamics of map f are superstable cycles).White regions in Fig. 2(a) are related to n-cyclic chaotic intervals C n : For parameter values outside E map f has globally attracting …xed points.
To illustrate the bifurcations mentioned above we show in Fig. 2(b) a 1D bifurcation diagram versus w, where 0:05 < < 0:35; B = 2:5 (the corresponding parameter path is marked in Fig. 2(a) by red arrow).It can be seen, in particular, that for = 0:15 (as in ( 4)), map f has a one-piece chaotic attractor, C 1 = [f 2 (w c ); f (w c )]: This means that for parameter values belonging to the region marked A1 in Fig. 1(e), an attractor of map f is the chaotic interval C 1 .Now let us turn to the bifurcation structure of the ( ; )-parameter plane.We …rst obtain conditions of the simplest bifurcations related to the …xed points of map f : a BCB of the …xed point w R1 of f R1 (w), which is a solution of occurs when w R1 collides with the border point w ; that is, when w R1 = (1 )m 1 : For (2), we have w R1 = B ; so that the BCB curve is given by and for (2) it occurs when In case (2), we have and a fold bifurcation occurs when the two points are merging, i.e., at The bifurcation curves BCR1 ; BCM 2 and F M 1 , obtained above are shown in Fig. 1(e), as well as in Fig. 3(a).In Fig. 3(a) we present bifurcation structure of the ( ; )-parameter plane (an enlarged window of Fig. 1(e)), obtained numerically, where periodicity regions related to attracting cycles of di¤erent periods are shown by di¤erent colors.Since map f in the considered parameter region may have up to six branches, it is a challenging task to give a complete description of this bifurcation structure.However, the presence of ‡at branches in the de…nition of f simpli…es such a description, given that the dominant dynamics in maps with ‡at branches are associated with superstable cycles and their BCBs.As we already mentioned, it occurs in region E below the curve BC J in Fig. 2(a), related to map f in case A1, when the ‡at branch f R3 is involved into asymptotic dynamics.We refer to [30] for details, where in particular so-called modi…ed U-sequence is introduced, which orders the superstable cycles using their symbolic sequences.Similar structures are observed also in Fig. 3(a), however, here more border points are involved into BCBs.
To clarify possible bifurcation sequences, we present in Fig. 3(b) a 1D bifurcation diagram for …xed = 0:36 and 1:115 < < 1:22 (the related parameter path is indicated in Fig. 3(a) by red arrow).It is convenient to comment this diagram for decreasing values of : Our starting point is in the region related to Case A3 (see (21)), below the curve F M 1 ; when a superstable …xed point w M 2 = f M 2 = 1= coexists with an attracting …xed point w M 1 (this point is outside the window shown in Fig. 3(b)).See an example of map f and its attractors in this case in Fig. 4(a).For decreasing ; a ‡ip BCB1 occurs at which w M 2 collides with border point w , leading to a superstable 2-cycle f1= ; f R2 (1= )g (see an example in Fig. 4(b)).Note that using the skew tent map as a border-collision normal form, it is easy to show (see, e.g., [30]) that a superstable …xed point (or cycle) can undergo either a ‡ip BCB, or a fold BCB, or a persistence border collision (leading to an attracting …xed point or cycle).Next BCB occurs when this 2-cycle collides with border point w leading to a 4-cycle which also includes the point w = 1= (in fact, all the cycles of map f in Fig. 3(b) for > A1=2 consist of the point w = 1= and its images).A cascade of ‡ip BCB follows, with alternating border points w and w , which accumulates, similar to the 'smooth' period-doubling cascade, to a speci…c parameter point, an analog of the Feigenbaum accumulation point.One more bifurcation, which is clearly seen in Fig. 3(b), is a fold BCB with border point w leading to a superstable 3-cycle (see Fig. 4(c)).For further decreasing ; the parameter point enters the region related to Case A2 0 (see (19)).Next BCB occurs when the 3-cycle collides with border point w = w 000 ; leading to a superstable 6-cycle, followed by a ‡ip BCB cascade.One more bifurcation indicated in Fig. 3(b) occurs at = A1=2 ; at which the chaotic interval C 1 = [f 2 R2 (w c ); f R2 (w c )] (an example is shown in Fig. 4(d)) for increasing disappears due to the appearance of the ‡at branch f M 2 .
Consider now an intersection point point of two BCB curves, = BCM 2 and = BCR1 (see point O in Fig. 3(a)).It is an organizing center from which in…nitely many other bifurcation curves issue which are BCB boundaries of the periodicity regions related to superstable cycles of map f .To see this, consider a neighborhood of O overlapping with region B3, where map f has ‡at branch f M 2 ; see (25) (it has also the ‡at branch f R3 ; but for the considered parameter values this branch is not involved into asymptotic dynamics).Any superstable cycle of map f includes point w = f M 2 = 1= and its images, thus two superstable cycles cannot coexist, so that their periodicity regions are not overlapping.Approaching point O; two BCB boundaries of a periodicity region (one related to the collision of a periodic point with w = w and the other one with w = w ) tend to each other merging at point O at which w = w = w R1 = w M 2 : Similar bifurcation structure is observed in a neighborhood of O overlapping with region B2, where map f has ‡at branch f R1 ; see (24).All the periodicity regions (with blocks of joined regions related to the same ‡ip BCB cascade) can be ordered according to a modi…ed U-sequence in a similar way as it is done in [30] for the periodicity regions of the superstable cycles in the ( ; B)-parameter plane in region E below the curve BC J (see Fig. 2(a)).Indeed, in Fig. 2(a), an intersection point (indicated by blue circle) of the BCB curves BC L and BC R3 ; which is ( ; B) = (1 1=m; 1); is also an organizing center of a similar kind as point O: Note that organizing centers are often observed in the parameter space of discontinuous maps (see e.g., [2] where several kinds of organizing centers in Lorenz maps are described).

Conclusion
The present paper can be considered as a starting point for a detailed investigation of the dynamics of the Matsuyama model in a more generic case.We described partitioning of the parameter space of the corresponding map into the regions related to di¤erent branch con…gurations of this map.This partitioning was presented for a speci…c parameter setting which allowed us to get several bifurcation curves analytically.The obtained results were illustrated by 1D and 2D bifurcation diagrams.Since the considered map depends on seven parameters, while in the present work …ve of them were …xed, work is needed to get a complete description of possible bifurcation sequences.From the dynamical view point, we expect to observe new interesting bifurcation structures associated with the interplay of several (up to …ve) border points.The detailed investigation of possible organizing centers related to the collisions with di¤erent border points is also left for a future work.

Case B: w > w c
Let now w > w c ; i.e., < A=B where A=B is given in (14) (see the region below the line A=B in Fig. 1(e)).Again, we need to distinguish between several subcases depending on the value of : The case B1 occurs when < 1 ( b w) = 2 ( b w) =: B1=2 ; < or < T ; > ; where is de…ned in (13) and T in (8).The corresponding map is given by (B1) f (w) = The branch f R1 (w) is decreasing if m 1 > m 2 ; increasing if m 1 < m 2 ; and it can be convex or concave.For m 1 = m 2 it is ‡at, f R1 (w) = B .See an example of map f in case B1 in Fig. 1(g).If < B1=2 ; > and > T , then we have case B1 0 when map f is given by: ) and b w < w for < ; for (2), we have = r m 3 B = (see Fig.1(e)).

Figure 2 :
Figure 2: (a) Bifurcation structure of the ( ; B)-parameter plane of map f in case A1 (see (16)).The region E (see (3)) is bounded by the curves BC L ; BC R3 and F B R2 ; other parameters are …xed as in (4); white regions are related to n-cyclic chaotic intervals C n and colored regions to attracting cycles (some regions are marked by numbers which are periods of the related cycles); (b) 1D bifurcation diagram corresponding to the cross-section at B = 2:5 of the 2D diagram shown in (a) (the related parameter path is indicated in (a) by the red arrow).