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Regularization of the Boundary-Saddle-Node Bifurcation.

1. Introduction

The study of Filippov system was motivated by its considerable applications in mechanical systems exhibiting dry friction [1-4], biological systems [5-7], and control systems [8-10]. In general, Filippov systems, or differential equations with discontinuous right-hand side, model physical processes which experience abrupt transitions between different modes. Modelling in such way we often idealize the transition to be instantaneous. The consequence of doing so is that the established discontinuous model focuses on the overall dynamics of the entire physical process while slightly ignoring the detailed dynamics in the transition stage. An efficient method to learn the dynamics in the transition stage is modelling this process in a small time scale such that the dynamics of one mode smoothly or continuously changes to the other mode. For a given Filippov system this can be realized by a smooth or continuous approximation which removes the discontinuities. This method is referred to as regularization. As a consequence, this enables us to make use of the established theories for smooth or continuous systems to study the dynamical properties of Filippov systems.

Different ways have been proposed to regularize a Filippov system; see [11-13]. In this paper we apply the regularization of [12] to a particular class of planar Filippov systems. Such systems consist of two smooth vector fields that are separated by a smooth discontinuity boundary. The vector field on one side undergoes a standard saddle-node bifurcation, while the vector field on the other side intersects the boundary transversally. Here a "standard saddle-node bifurcation" means a saddle-node bifurcation in smooth systems. We refer for a formal definition of Filippov system and the regularization approach of [12] to Sections 2.1 and 2.2, respectively.

Our aim of this work is to describe the local dynamics in the neighborhood of the codimension-2 boundary-saddle-node (BSN) bifurcation where two equilibria of one smooth vector field go through a saddle-node bifurcation while they lie on the boundary. We treat this particular class of Filippov systems by applying both Filippov's convex method and the regularization approach to the BSN point to obtain its local bifurcation diagram, to understand how all the codimension-1 bifurcations interact. As we shall see in Section 3 the BSN point studied by Filippov's convex method acts as organizing center for three families of codimension-1 bifurcations appearing in its neighborhood, which are standard saddle-node bifurcation, equilibrium transition, and nonsmooth fold bifurcation. However, after regularization, some of the discontinuity-induced bifurcations disappear, such as the equilibrium transition. The unique bifurcation that occurs in the regularized system is the saddle-node-like bifurcation; see Section 2.2. Here the extension "like" means that this bifurcation is not the standard one as known from smooth systems. We refer to [15] for a detailed description of the generic bifurcations in piecewise-smooth continuous systems.

We briefly outline the structure of this paper. In the next section we give an overview of Filippov dynamics and the regularization technique of [12]. Moreover, we explain how to transform the regularized system to a singular perturbation problem. In Section 3 we construct a topological normal form for the BSN bifurcation. Subsequently, we study the BSN bifurcation by Filippov's convex method and obtain its bifurcation diagram and phase portraits in Section 4. After that, in Section 5 we apply the regularization approach to the topological normal form and investigate its local dynamics as a piecewise-smooth continuous system. Finally, in Section 6 we summarize the main results.

2. Preliminaries

In this section we introduce Filippov's convex method and the regularization approach of [12].

2.1. Filippov's Convex Method. We first introduce the basic concepts in Filippov systems.

2.1.1. Filippov Dynamics. A triplet (X, Y, Z) defined on a 2dimensional manifold M is called a Filippov system, denoted by Z. Both X and Y are [C.sup.r] (r [greater than or equal to] 1 or r = [infinity]) vector fields, which are extendable over a full neighborhood of the discontinuity boundary [summation]. The boundary is given as follows:

[summation] = {(x,y) [member of] M : f (x,y) = 0}, (1)

where f : M [right arrow] R is a [C.sup.r] function and has 0 as a regular value. Therefore, [summation] is a smooth 1-dimensional submanifold of M. As a consequence, [summation] separates M to two open subsets [M.sub.x] and [M.sub.y]:

[mathematical expression not reproducible]. (2)

For any (x, y) [member of] M\[summation], the dynamics of Z is given by

[mathematical expression not reproducible]. (3)

According to Filippov's convex method [16], [summation] can be divided into the crossing set

[[summation].sub.c] = {(x, y) [member of] [summation] : Xf (x, y) Yf (x, y) > 0} (4a)

and the sliding set

[[summation].sub.s] = {(x, y) [member of] Z: Xf (x, y) Yf (x, y) [less than or equal to] 0}, (4b)

where Xf(x, y) denotes the directional derivative of f with respect to X at the point (x, y). That is, [summation] = [[summation].sub.s] [union] [[summation].sub.c].

In particular, the dynamics of a point (x, y) at the sliding set is defined by a unique convex linear combination of X(x, y) and Y(x, y), which is tangent to [summation] at (x, y). This dynamics is called the sliding vector field [16], denoted by [Z.sub.s] and given as follows:

[Z.sub.s] (x,y) = Xf (x,y) Y (x,y)-Yf (x,y) X (x,y)/Xf (x,y)-Yf (x,y), (5)

when Xf (x,y) [not equal to] Yf (x,y). We define Z(x,y) = 0 when Xf (x, y) = Yf (x, y) for (x, y) [member of] [[summation].sub.s].

2.1.2. Equilibria and Bifurcations. Here we give a brief description of the discontinuity-induced bifurcations (bifurcations where the discontinuity of the Filippov systems plays an essential role) of equilibria appearing in planar Filippov system.

Definition 1 (equilibria). A point (x, y) [member of] M\[summation] is an ordinary equilibrium of X (resp. Y) if X(x, y) = 0 and (x, y) [member of] [M.sub.x] (resp., Y(x, y) = 0 and (x, y) [member of] [M.sub.Y]). A point (x, y) [member of] [summation] is called a boundary equilibrium of X (resp. Y) if X(x, y) = 0 (resp., Y(x, y) = 0). A point (x, y) [member of] M is an ordinary (resp., boundary) equilibrium of Z if it is an ordinary (resp., boundary) equilibrium of X or Y. Equilibria of X in [M.sub.Y] and of Y in [M.sub.X] are called virtual equilibria.

Definition 2 (pseudoequilibria). A point (x,y) [member of] [[summation].sub.s] is a pseudoequilibrium of Z if (x, y) is not a boundary equilibrium and [Z.sub.s](x,y) = 0.If [Z.sub.s](x, y) = 0 for (x,y) [member of] [[summation].sub.c], then one calls (x, y) a virtual pseudoequilibrium of Z.

When an ordinary equilibrium of Z collides with the discontinuity boundary [summation], it generically gives either an equilibrium transition (ET) or a nonsmooth fold (NSF) bifurcation; compare [17]. For the ET an ordinary equilibrium collides with a virtual pseudoequilibrium and then becomes a virtual ordinary equilibrium and a pseudoequilibrium; see Figure 1(a). For the NSF bifurcation an ordinary equilibrium collides with a pseudoequilibrium and then they both become virtual; see Figure 1(b).

The following theorem [14] gives conditions for the occurrence of an equilibrium transition or nonsmooth fold bifurcation in a Filippov system that smoothly depends on a parameter [beta].

Theorem 3 (equilibrium transition and nonsmooth fold bifurcation [14]). Assume that X(0,0,0) = 0 and that [D.sub.(x,y)]X(0,0,0) is nonsingular. Moreover, assume that for p = 0 an equilibrium branch (x([beta]), y([beta])) of the vector field X(x, y, [beta]) transversally crosses the discontinuity manifold [summation] at (x, y, [beta]) = (0,0,0); that is,

[mathematical expression not reproducible]. (6)

Finally, assume that the nondegeneracy condition

[delta] [??] [D.sub.(x,y)]f (0,0,0) [([D.sub.(x,y)]X (0,0,0)).sup.-1] Y (0,0,0) [not equal to] 0 (7)

is satisfied. Then, at [beta] = 0, there is an equilibrium transition if [delta] > 0 and there is a nonsmooth fold bifurcation if [delta] < 0.

2.1.3. Equivalence between Filippov Systems. Here we define the equivalence that will be used in this work.

Definition 4 (topological equivalence [18]). Two Filippov systems Z and [??] are topologically equivalent if there is a homeomorphism [phi] that sends the orbits of Z to [??] and the sliding set [[summation].sub.s] to [[??].sub.s].

2.2. Regularization Approach. In this section we introduce the regularization approach of [12] and present its properties about equilibria.

According to [12] piecewise-smooth continuous regularization [R.sub.[epsilon]] of the Filippov system Z is given in the following way. Let

[M.sub.R] = [M.sub.X] [union] S [union] [M.sub.Y] (8)

be the extended manifold, where S = [-[epsilon],[epsilon]] x [summation]. That is, [M.sub.R] is obtained by cutting the manifold M along [summation] and then glueing the two pieces together by adding the "strip" S. The regularized vector field is defined on [M.sub.R] as

[mathematical expression not reproducible]. (9)

Here [alpha] is a strictly increasing, smooth function defined in [-1,1] with [alpha](-1) = 0, [alpha](1) = 1. For simplicity we assume [alpha](0) = 1/2.

In this paper we are interested in local dynamics of the Filippov system. Since [summation] is a submanifold of M, there are local coordinates (x, y) in a neighborhood of p such that locally [summation] = {(x, y) | x = 0}. In this case the regularized system can be written as

[mathematical expression not reproducible]. (10)

For the regularized system [R.sub.[epsilon]] we have the following theorem about its equilibria and stability type.

Theorem 5. The ordinary equilibria and pseudoequilibria of Z correspond to equilibria of [R.sub.[epsilon]] with the same type of stability. Boundary equilibria of either X or Y (but not both) correspond to equilibria of [R.sub.[epsilon]]. Common boundary equilibria of X and Y give rise to a curve of equilibria of [R.sub.[epsilon]] in S.

2.3. Regularization as a Singular Perturbation Problem. In this section we briefly review singular perturbation theory. Consider a general singular perturbation problem with the slow time system

[mathematical expression not reproducible] (11)

or equivalently the fast time system, after time rescaling [tau] = t/[epsilon],

[mathematical expression not reproducible], (12)

where x [member of] [R.sup.k], y [member of] [R.sup.m], 0 < [epsilon] << 1.

The slow manifold is the set of equilibria of the fast time system for [epsilon] = 0 and is given by

M ={(x,y) : f(x,y,0) = 0}. (13)

Then M is said to be normally hyperbolic if for every (x, y) [member of] M the matrix [D.sub.x]f has no eigenvalues on the imaginary axis.

For [epsilon] = 0 the slow time system (11) is reduced to the following system on M:

[mathematical expression not reproducible], (14)

called the reduced problem.

The following result of Fenichel (see [19]) on the preservation of normally hyperbolic invariant manifolds plays a central role in geometric singular perturbation theory.

Theorem 6 (Fenichel's invariant manifold theorem). If M is a compact set and normally hyperbolic, then, for [epsilon] > 0 and sufficiently small, there exists a locally invariant manifold [M.sub.[epsilon]] close to M in the [C.sup.1] topology. The manifold Mt is diffeomorphic to M, and thefastflow on [M.sub.[epsilon]] is close to the flow of the reduced equation on M.

Now consider the Filippov system Z = (X,Y, Z) on [R.sup.2]. Recall that given a point p [member of] Z we can find a local coordinate system (x,y) such that [summation] is given by x = 0 and the regularized system is given by (10). We now focus on the region S = [-[epsilon], [epsilon]] x [summation] and we perform the scaling transformation u = x/[epsilon]. This scaling induces in S' = [-1,1] x [summation] the system

[mathematical expression not reproducible]. (15)

Rescaling time by [tau] = t/[epsilon] we get

[mathematical expression not reproducible]. (16)

System (15) is the slow time system while (16) is the fast time system. The slow manifold is given by

M = {(u,y) : (1 - [alpha](u)) [X.sub.1] (0,y) + [alpha] (u) [Y.sub.1] (0,y)

= 0}. (17)

The slow time system (15) for [epsilon] = 0 defines a dynamical system on M with dynamics given by

dy/dt = (1 - [alpha](u)) [X.sub.2] (0, y) + [alpha] (u) [Y.sub.2] (0, y). (18)

Accordingly the slow time system (15) for [epsilon] = 0; that is,

[mathematical expression not reproducible] (19)

is called the reduced problem of the singularly perturbed system.

Comparing the singular perturbation problem (15) and (16) with the original Filippov system Z, we directly derive the following result.

Lemma 7. Suppose [X.sub.1](x, y) [not equal to] [Y.sub.1](x,y). Then there is a diffeomorphism h : [[summation].sub.s] [right arrow] M, having the form h(0, y) = (a(y),y), where [alpha]'(u) > 0 at u = a(y), that maps the flow of the sliding vector field [Z.sub.s] to the flow of the reduced problem on M.

Remark 8. The slow manifold and the sliding segment of the boundary manifold are homeomorphic and carry the same dynamics but their geometry is different.

3. The Boundary-Saddle-Node Bifurcation

Here we first define the boundary-saddle-node (BSN) bifurcation.

Definition 9 (boundary-saddle-node bifurcation). Consider a planar Filippov system Z = (X, Y, [summation]) which smoothly depends on parameters [mu] [member of] [R.sup.2]. In particular, for [mu] = [[mu].sub.0] the vector field X has an equilibrium ([x.sub.0],[y.sub.0]). Then the Filippov system Z undergoes a boundary-saddle-node (BSN) bifurcation at [mu] = [[mu].sub.0] when the following conditions hold true:

(1) the equilibrium ([x.sub.0],[y.sub.0]) is located on the boundary; that is, f([x.sub.0],[y.sub.0];[[mu].sub.0]) = 0;

(2) [D.sub.x][X.sub.1]([x.sub.0],[y.sub.0];[[mu].sub.0]) = 0 [D.sub.xx][X.sub.1]([x.sub.0],[y.sub.0];[[mu].sub.0]) [not equal to] 0 and [D.sub.[mu]][X.sub.1]([x.sub.0],[y.sub.0];[[mu].sub.0]) = 0, where [X.sub.1] is the first component of X;

(3) Y([x.sub.0],[y.sub.0];[[mu].sub.0]) is transversal to [mathematical expression not reproducible] at ([x.sub.0],[y.sub.0]); that is, Yf([x.sub.0],[y.sub.0];[[mu].sub.0]) [not equal to] 0;

(4) [delta]([[mu].sub.0]) = [D.sub.(x,y)]f([x.sub.0],[y.sub.0];[[mu].sub.0])[([D.sub.(x,y)]X([x.sub.0],[y.sub.0]; [[mu].sub.0])).sup.-1] Y([x.sub.0,[y.sub.0];[[mu].sub.0] = 0.

Remark 10. Condition (2) implies that the vector field X goes through a saddle-node bifurcation at [mu] = [[mu].sub.0]; condition (4) defines a degenerate situation for the boundary equilibrium ([x.sub.0], [y.sub.0]) at [mu] = [[mu].sub.0]. From Theorem 3 we know that the system goes through a nonsmooth fold bifurcation for [delta] < 0, while it goes through an equilibrium transition for [delta] > 0.

Theorem 11 (topological normal form). Consider a general Filippov system Z which undergoes a BSN bifurcation at [[mu].sub.0]. Then at the equilibrium point ([x.sub.0],[y.sub.0]) there exist a local homeomorphism [phi] depending on parameters [mu] and a smooth invertible reparameterization [mu] [??] [lambda] = ([rho],v) such that Z is topologically equivalent to the following system:

[mathematical expression not reproducible], (20a)


[mathematical expression not reproducible], (20b)

[mathematical expression not reproducible], (20c)

f(x, y; [lambda]) = x - [rho], (20d)

where [kappa] = [+ or-] 1 and [sigma] = sgn (Yf([x.sub.0],[y.sub.0];[[mu].sub.0])) = [+ or-] 1. All higher order terms in X, Y, and f smoothly depend on [lambda].

Note that the vector filed X undergoes a saddle-node bifurcation at v = 0. The parameter p determines the passage of the equilibrium of X through the boundary [summation].

Proof. The method of proving this theorem is similar to [5]. Thus we omit it here.

4. Bifurcation Diagram and Phase Portraits

System (20a), (20b), (20c), and (20d) undergoes 3 different codimension-1 bifurcations: saddle-node bifurcation, equilibrium transition, and nonsmooth fold bifurcation. Next we will only describe these three bifurcations in detail for the case [kappa] = [sigma] = -1. The other cases follow in the same way.

The vector field X undergoes a saddle-node bifurcation at v = 0. It has a saddle at (- [square root of v], 0) and a stable node at ([square root of v], 0) for v > 0, while it has no equilibrium for v < 0. The sliding set is given by

[[summation].sub.s] ={(x,y) : x = [rho],[[rho].sup.2] - v [less than or equal to] 0}. (21)

Therefore, the sliding set is empty for v < [[rho].sup.2], while [[summation].sub.s] = [summation] for v [greater than or equal to] [[rho].sup.2]. According to (5), at the sliding set [[summation].sub.s], the sliding vector field [Z.sub.s] is defined as

[mathematical expression not reproducible]. (22)

It is direct to check that ([rho], 0) is the unique attractive pseudoequilibrium.

Now we check that system (20a), (20b), (20c), and (20d) satisfies all the conditions of Theorem 3. For (20a), (20b), (20c), and (20d) it is direct to check that the equilibria ([+ or-] [square root of v],0) transversally meets [summation] at [rho] = [+ or-] [square root of v] or equivalently v = [[rho].sup.2]. Moreover,

[mathematical expression not reproducible]. (23)

By Theorem 3, this implies that the equilibrium ([square root of v], 0) goes through an equilibrium transition for [rho] > 0, while the equilibrium (- [square root of v], 0) goes through a nonsmooth fold bifurcation for [rho] < 0. Apart from these two boundary bifurcations, the line v = 0 for [rho] > 0 in the parameter plane corresponds to a standard saddle-node bifurcation; see Figure 2. The phase portraits corresponding to the open regions I, II, and III labelled in Figure 2 are given in Figure 3.

5. Regularization of the BSN Bifurcation

In this section we apply the regularization approach (9) to the normal form of the BSN bifurcation and study the bifurcations of the regularized system.

Following (9) the regularization of (20a), (20b), (20c), an (20d) for [epsilon] > 0 is defined as follows:

[mathematical expression not reproducible], (24)

where the smooth function a is defined in [[rho] - [epsilon], [rho] + [epsilon]] with [alpha]([rho] - [epsilon]) = 0, [alpha]([rho] + [epsilon]) = 1. For our computations we have chosen [alpha](x) = (x - [rho] + 1)/2.

Now we consider the bifurcations that occur in the regularized system (24). Before that we first define the following bifurcation for piecewise-smooth continuous system.

Saddle-Node-Like Bifurcation (SNL). A "saddle" and a "node" collide at a certain parameter value and then both disappear.

We note that the "saddle" and "node" in the definition are not restricted to a standard saddle and a standard node. They can also be pseudosaddle and pseudonode. The extension "like" means that these bifurcations are not the standard ones as known from smooth systems. A detailed description of bifurcations in piecewise-smooth continuous systems is referred to in [15].

5.1. Equilibrium Transition. Recall that ([square root of v], 0) is an ordinary stable node for v < [[rho].sup.2], [rho] > 0. It hits [summation] for v = [[rho].sup.2] and becomes a boundary node. Then it is replaced by a stable pseudonode on [summation] for v > [[rho].sup.2]. By Theorem 5, this equilibrium is still an equilibrium of [R.sub.[epsilon]] after regularization with different position. It is located in [M.sub.x] for v < [[rho].sup.2], on the line x = -[epsilon] for v = [[rho].sup.2], and in the region (-[epsilon], [epsilon]) for v > [[rho].sup.2]. Thus, the regularized system [R.sub.[epsilon]] does not experience any qualitative change as v varies. This type of discontinuity-induced bifurcation disappears after regularization, see Figure 4.

5.2. Nonsmooth Fold. Recall that for v > [[rho].sup.2], [rho] < 0, an ordinary saddle (- [square root of v],0) and a stable pseudonode ([rho],0) coexist, colliding on [summation] for v = [[rho].sup.2] and then both disappear for v < [[rho].sup.2], [rho] < 0. By Theorem 5 these two equilibria both preserve in their regularization [R.sub.[epsilon]] with the same type of stability. To be precise, the saddle of [R.sub.[epsilon]] is located in [M.sub.X], while the stable node is located in the region (-[epsilon],[epsilon]) for v > [[rho].sup.2]. The unique equilibrium of [R.sub.[epsilon]] is located on the line x = -[epsilon] for v = [[rho].sup.2]. However, there is no equilibrium in the regularized system for v < [[rho].sup.2]. Notice that [R.sub.[epsilon]] is a piecewise-smooth continuous system; then a saddle-node-like bifurcation occurs in [R.sub.[epsilon]]; see Figure 5.

5.3. Standard Saddle-Node Bifurcation. Finally, the discontinuous system (20a), (20b), (20c), and (20d) has a standard saddle at (-[square root of v], 0) and a standard stable node at ([square root of v], 0) for any v > 0, colliding for v = 0, and then both disappear for v < 0. By Theorem 5 the equilibria for different v persist after regularization. Thus a saddle-node-like bifurcation occurs in the regularized system [R.sub.[epsilon]]; see Figure 6.

From the above results we find that the saddle-node-like bifurcation is the unique bifurcation occurring in the regularized system, which has codimension-1; see Figure 7. The phase portraits corresponding to the open regions I and II are presented in Figure 8.

6. Conclusions

This paper investigated the boundary-saddle-node (BSN) bifurcation by Filippov's convex method and the regularization approach. This is a codimension-2 bifurcation of Filippov systems. We derived the topological normal form of the BSN bifurcation and gave a detailed description of its bifurcations and phase portraits. After regularization the Filippov system becomes a piecewise-smooth continuous system. Some of the discontinuity-induced bifurcations disappear, such as equilibrium transition. The unique bifurcation that occurs in the regularized system is the saddle-node-like bifurcation. The BSN point becomes a codimension-1 bifurcation in the piecewise-smooth continuous system, which makes our discussion easier. The regularization approach enables us to make use of the established theories for continuous systems and slow-fast systems to study the local behavior around the BSN bifurcation. We will apply the regularization approach to more complicated Filippov systems involving limit cycles in the future work, where more advantages of our approach in proving the existence and uniqueness of limit cycles will be presented.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.


The author greatly appreciates the fruitful discussions with Ying Gao for her valuable suggestions. This research is supported by the Research Foundation of Education Bureau of Hebei Province under Grant no. QN2016026 and the Doctoral Starting up Foundation of Hebei GEO University under Grant no. B201612.


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Xia Liu (iD)

School of Mathematics and Physics, Hebei GEO University, Shijiazhuang 050031, China

Correspondence should be addressed to Xia Liu;

Received 10 June 2017; Accepted 9 November 2017; Published 4 January 2018

Academic Editor: Andrei D. Mironov

Caption: Figure 1: Equilibrium transition and nonsmooth fold bifurcation. The discontinuity boundary [summation] is displayed as a solid line. The dashed line represents the motion of the equilibrium as a parameter changes. Round points represent ordinary equilibria (filled) or virtual equilibria (hollow). Square points represent pseudoequilibria (filled) or virtual pseudoequilibria (hollow) [14].

Caption: Figure 2: Bifurcation diagram of Z (20a), (20b), (20c), and (20d) in the ([rho], v)-plane when [delta] = [sigma] = -1.

Caption: Figure 3: From left to right phase portraits corresponding to the open regions I, II, and III in Figure 2, respectively.

Caption: Figure 4: (a) Equilibrium transition between an ordinary node and a pseudonode. (b) The corresponding regularization. (c) The corresponding slow-fast flow.

Caption: Figure 5: (a) Nonsmooth fold bifurcation in the case of boundary node. (b) The corresponding regularization. (c) The corresponding slow-fast flow.

Caption: Figure 6: (a) A standard saddle-node bifurcation. (b) The corresponding regularization. (c) The corresponding slow-fast flow.

Caption: Figure 7: Bifurcation diagram of the regularization of system (20a), (20b), (20c), and (20d) in ([rho], v) plane when [delta] = [sigma] = -1.

Caption: Figure 8: From left to right phase portraits corresponding to the open regions I and II in Figure 7, respectively.
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Title Annotation:Research Article
Author:Liu, Xia
Publication:Advances in Mathematical Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
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