Normal forms of planar switching systems

In this paper we study normal forms of planar differential systems with a non-degenerate equilibrium 
on a single switching line, 
i.e., the equilibrium is a non-degenerate equilibrium of both the upper system and the lower one. 
In the sense of $C^0$ conjugation 
we find all normal forms for linear switching systems 
and use them together with switching near-identity transformations 
to normalize second order terms, showing the reduction of normal forms. 
We prove that only one of those 19 types of linear normal form decides if the system is monodromic. 
With the monodromic linear normal form, we compute the second order monodromic normal form, 
which gives a condition under which exactly one limit cycle arises from a Hopf bifurcation.

While discussing on bifurcations of a smooth system, one usually reduces the system to its Poincaré normal form ( [9,20]) in the equivalent sense of conjugacy. The idea is to utilize the Jordan form of the linear part, simplified by an invertible linear transformation, to make a near-identity transformation so that the system is changed into the simplest form containing resonant terms only. A similar theory of normal forms is also needed for switching systems but difficulties come from even the linear part, not mentioning the nonlinear terms. In fact, a switching system has more than one linear parts, which exist separately in different regions divided by switching lines. We need to simplify those matrices by either the same invertible linear transformation or a continuous invertible transformation of different linear representations in those regions.
In this paper we study normal forms for planar piecewise-smooth differential system (ẋ,ẏ) = X + (x, y), Y + (x, y) , if y > 0, with a single switching line y = 0, where X ± (x, y), Y ± (x, y) are analytic and vanish at (0, 0). This system with an equilibrium O : (0, 0), which lies persistently on the switching line, is an important class of switching systems such as the car brake system studied in [28], where continuous piecewise-smooth ODEs were considered and the existence and stability of a cycle were established but no explicit formula was given for the direction of the cycle bifurcation. Some interesting works (see, e.g., [18,24]) about Hopf bifurcations for piecewise-smooth systems were done in the case that the equilibrium moves apart from the origin as parameters vary. Here we always restrict the equilibrium at the origin and call the first (second) equation, defined for y > 0 (for y < 0), the upper (lower) system. Flows of the switching system are defined piecewise at y > 0, = 0, < 0 known as the Filippov structure without sliding on the switching line in [4,12,19]. We expand the system as A + x y + a + 20 x 2 + a + 11 xy + a + 02 y 2 b + 20 x 2 + b + 11 xy + b + 02 y 2 + O(|(x, y)| 3 ), if y > 0, near the origin O and show our simplification to the linear terms and the second order terms in the case that the equilibrium O is non-degenerate, i.e., the matrices A ± are both non-singular. We construct continuous piecewise-linear transformations for conjugation and prove in section 2 that system (1) has 19 types of linear normal forms totally. All orbit structures are persistent in normal forms including the possible sliding motions because of topological equivalence. Further, we prove that only one of those 19 types of linear normal form decides system (1) to be monodromic ( [1]), i.e., no orbits approaching O in a definite direction as t → ±∞. In section 3 we show our normalization with second order terms and normalized linear terms, constructing a near-identity transformation with switching to eliminate some second order terms for the second order normal form of system (1). In section 4 we show how the monodromic linear normal form can be used to simplify the above obtained second order normal form further and give the second order monodromic normal form. Since our reductions made in sections 2, 3 and 4 are given with homeomorphisms preserving the switching line invariant, the Filippov structure without sliding on the switching line remains in the reduced normal forms. Such a reduction of normal forms is applied in section 5 to a general monodromic switching system, which was investigated in [28,Theorem 4.4] for existence of a limit cycle. We give a quantity the sign of which determines the rise of exactly one limit cycle and the direction of the Hopf bifurcation. In section 6 we give an example to display the reduction of normal form and illustrate our discussion of Hopf bifurcations, and another example to show the critical performance of the second order normal form.
2. Normalization of linear terms. We need to simplify the linear part at first, i.e., to give a 'linear normal form' for system (1). For this purpose we need to simplify the two matrices A ± on the two sides of the switching line y = 0 by a continuous and invertible piecewise-linear transformation Without loss of generality, we assume that the transformation P preserves the switching line y = 0 invariant. The continuity at points on the line y = 0 and the invertibility imply the following: (2) is a homeomorphism on R 2 if and only if P ± 4 = 0, P + 1 = P − 1 = 0 and P + 3 P − 3 > 0. Proof. P is continuous at any point (x, 0) on the switching line y = 0 if and only if P ± satisfy that for a certain u ∈ R. It follows that P ± 4 x = 0 and P + 1 x = P − 1 x for all x, implying that P ± 4 = 0 and P + 1 = P − 1 respectively. Therefore, where P 1 = P ± 1 and P 1 P + 3 P − 3 = 0 because of the invertibility. Further, the above expression of P ± shows that v = P + 3 y when y > 0 and v = P − 3 y when y < 0. If P + 3 P − 3 < 0, then points on the upper half-plane y > 0 and points on the lower half-plane y < 0 are both mapped by P into the same half-plane, a contradiction to the homeomorphism of P . Hence, P + 3 P − 3 > 0.
Having this lemma, we give the following linear normal forms.
A simple computation shows that M 12 = M 21 and M 22 = 0, but, by Lemma 2.1, the restriction to P + does not allow such a choice of M . Therefore, in Theorem 2.2, N ± 2 and N ± 3 are regarded as different normal forms in switching systems. For the same reason, we differ N ± 4 from N ± 5 and N ± 6 from N ± 1 in switching systems.
Proof. By Lemma 2.1, P + 3 P − 3 > 0. Thus, either P + 3 > 0 and P − 3 > 0 or P + 3 < 0 and P − 3 < 0. First, we give those linear normal forms reduced by transformations defined in (2) with P + 3 > 0 and P − 3 > 0, which preserves the upper half plane y > 0 and the lower one y < 0. The transformation (2) with P + 3 > 0 and P − 3 > 0 changes the system the linear part of system (1), into the form In order to simplify B ± further, we focus on the back-diagonal entries and classify all possibilities of matrix A ± as .., 6 and j = 1, ..., 6. When system (3) lies in case Γ 11 , from (4) we get which are of the form N ± 1 (shown on the second row of Table 1) respectively by choosing P ± 2 = 0 in the transformation. When system (3) lies in case Γ 12 , from (4) we get which are of form N ± 1 (shown on the third row of Table 1) respectively by choosing in the transformation. When system (3) lies in case Γ 13 , from (4) we get which are of form N + 1 and N − 2 (shown on the fourth row of Table 1) respectively by choosing P + 2 = 0 and P − 3 = P 1 A − 12 > 0 in the transformation. Similarly, we consider cases Γ ij for all i = 1, 2, 5, 6, j = 1, ..., 6 and for all i = 3, 4, j = 1, 2, 5, 6. The obtained normal forms are summarized in Tables 1 and 2. In contrast, the cases Γ ij where i = 3, 4 and j = 3, 4 need additional conditions. In case Γ 33 , a simple computation shows that the upper system of (3) can be normalized to N + 2 by choosing P + 3 = P 1 A + 12 > 0 and the lower one of (3) can be normalized to N − 2 by choosing P − 3 = P 1 A − 12 > 0 separately (see the second row of Table 3). Thus, system (3) can be reduced to the normal form N 22 , i.e., the upper system is reduced to N + 2 meanwhile the lower system is reduced to N − 2 , if and only if A + 12 A − 12 > 0 because both P ± 3 > 0 are required in the beginning of the proof. To the opposite, i.e., A + 12 A − 12 < 0, N + 2 is not compatible with N − 2 but system (3) can be reduced to the normal form N 24 by choosing P + 3 = P 1 A + 12 > 0 and P − 3 = −P 1 A − 12 > 0 in the transformation (see the third row of Table 3). Similarly to case Γ 33 , in case Γ 34 a simple computation shows that the upper system of (3) can be normalized to N + 2 by choosing P + 3 = P 1 A + 12 > 0 and the lower one of (3) can be normalized to N − 3 by choosing Table 3). Thus, system (3) can be reduced to the normal form N 23 , i.e., the upper system is reduced to N + 2 meanwhile the lower system is reduced to N − 3 , if and only if A + 12 A − 21 > 0 because both P ± 3 > 0 are required in the beginning of the proof. To the opposite, i.e., A + 12 A − 21 < 0, N + 2 is not compatible with N − 3 but system (3) can be reduced to the normal form N 25 by choosing P + 3 = P 1 A + 12 > 0 and P − (see the fifth row of Table 3).
Similarly to case Γ 33 , in case Γ 43 a simple computation shows that the upper system of (3) can be normalized to N + 3 by choosing P + 2 = P 1 (A + 22 −A + 11 )/(2A + 21 ), P + 3 = 6720 XINGWU CHEN AND WEINIAN ZHANG Table 1. Normal forms for Γij, i = 1, 2, 5, 6, j = 1, ..., 6 P 1 /A + 21 > 0 and the lower one of (3) can be normalized to N − 2 by choosing Table 3). Thus, system (3) can be reduced to the normal form N 32 , i.e., the upper system is reduced to N + 3 meanwhile the lower system is reduced to N − 2 , if and only if A + 21 A − 12 > 0 because both P ± 3 > 0 are required in the beginning of the proof. To the opposite, i.e., A + 21 A − 12 < 0, N + 3 is not compatible with N − 2 but system (3) can be reduced to the normal form N 34 by Table 3).
Similarly to case Γ 33 , in case Γ 44 a simple computation shows that the upper system of (3) can be normalized to N + 3 by choosing P + 2 = P 1 (A + 22 −A + 11 )/(2A + 21 ), P + 3 = P 1 /A + 21 > 0 and the lower one of (3) can be normalized to N − 3 by choosing Table 3). Table 2. Normal forms for Γ3j and Γ4j, j = 1, 2, 5, 6 Thus, system (3) can be reduced to the normal form N 33 , i.e., the upper system is reduced to N + 3 meanwhile the lower system is reduced to Table 3). As shown in Tables 1, 2  Using transformations defined in (2) with P + 3 < 0 and P − 3 < 0, which exchange the upper half plane y > 0 with the lower one y < 0, we obtain the same linear normal forms N ij (for all (i, j) satisfying (5)) as given for P + 3 > 0 and P − 3 > 0. Finally we delete N ji for all j > i because it is equivalent to N ij by the transformation (u, v) := (x, −y). In fact, which is of form N ij . For this reason, the linear part of the switching system (1) has totally 19 independent normal forms N ij , (i, j) ∈ ∆.
The equivalent complex form of system (3) iṡ where z = x + iy and a ± ij ∈ C. Let (α 1 , α 2 ) · (z 1 , z 2 ) denote the inner product of the complex vectors (α 1 , α 2 ) and (z 1 , z 2 ) in C 2 . By Theorem 2.2 we also obtain linear normal forms in the complex form.
Among the linear normal forms given in Theorem 2.2, we can find the monodromic ones, whose orbits near O rotate around O.

Corollary 2. Switching system (1) is monodromic at equilibrium O if and only if its linear normal form is
Proof. It is known (from e.g. [17]) that for a non-degenerate system without switching there are orbits tending to the origin along a definite direction as t → +∞ or t → −∞ if the Jordan form of its Jacobian matrix at the origin is not of the form N ± 7 . Thus, if a non-degenerate switching system is of a linear normal form other than N + 7 on the upper plane or N − 7 on the lower plane, then there is an orbit tending to the origin along a definite direction in upper plane or lower plane as t → +∞ or t → −∞, a contradiction to monodromy. Moreover, it is necessary for O to be monodromic that β + β − > 0; otherwise, O is not an isolated equilibrium.
Corollary 2 is a result for a non-degenerate isolated equilibrium. It shows that O is monodromic of FF type ( [11]), i.e., system (1) is monodromic at O only in the case that both the upper system and the lower one are monodromic, i.e., A ± have a pair of complex eigenvalues separately. The monodromic O can be of neither PP type nor FP type since O is assumed to be an equilibrium of both the upper system and the lower one in the beginning of this paper (just before (1). Note in Corollary 2 that orbits near O rotate clockwise (resp. anti-clockwise) if both β ± > 0 (resp. < 0). In the opposite case that the normal form is N 77 with β + β − < 0, points on the switching line y = 0 near O are all equilibria, i.e., O is not an isolated equilibrium.
3. Normalization of nonlinear terms. By Theorem 2.2 system (1) with normalized linear part can be written as where N + i , N − j are given in Theorem 2.2 and (i, j) ∈ ∆. In this section we show the normalization of nonlinear terms by giving the second order normal forms for (8).
Remark that the condition (λ + − 2µ + ) 2 + (λ − − 2µ − ) 2 = 0 given in Theorem 3.1 implies λ ± = 2µ ± , which can be regarded as the so-called "double resonance", i.e., the upper system and the lower system, being treated as smooth systems separately on their half planes, are both resonant of second order. However, unlike smooth systems, the remaining terms in the normal form (10) of switching system under this condition include not only those resonant terms x 2 and y 2 , determined by rational dependence between eigenvalues as for smooth systems, but also the term xy, determined by the switching line y = 0. Actually, the x-axis both lies in the eigen-direction of λ ± and is the switching line, showing that the eigenvalues λ ± play a role different from eigenvalues µ ± . If we consider the case λ ± = 2µ ± but the switching line is x = 0, then the normal form is where there is not the term xy. (23) is deduced from the system obtained by Theorem 3.1, by the change of variables x → y, y → x. Theorem 3.1 only gives the second order normal forms for the switching system (8) but its proof shows a method to normalize higher order terms. It is worthy mentioning that the normal forms obtained in Theorem 3.1 can be simplified further in some special cases, for example, the monodromy case, which will be discussed in next section for Hopf bifurcations. 4. Monodromic normal forms. Hopf bifurcation ( [20]) is one of the most important topics in differential equations. Recently, efforts ( [7,11,15,28]) have been made to Hopf bifurcations in switching systems. It is proved in [18, Theorem 2.3] that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type), different from the case of smooth systems. For a normal form approach to Hopf bifurcations, we need to work on systems with monodromy ([1]), i.e., no orbits tending to the considered equilibrium in a definite direction as t → ±∞. Thanks to our Corollary 2, which tells that the switching system is monodromic if and only if it has the linear normal form N 77 with β + β − > 0. In what follows, we apply Theorem 3.1 to give the second order normal form for monodromic switching systems and see how the monodromic linear normal form N 77 makes it possible to simplify the obtained second order normal form further.
Proof. By Theorem 2.2 and the last row of Table 1, the linear system of (1) can be changed into N 77 by the transformation where H 1 := (2|A + 21 |) −1 and
Finally, similarly to Lemma 2.1, one can check that the composition of transformations (25) and (12), i.e., x y + p 20 x 2 + p + 11 xy + p + 02 y 2 q + 11 xy + q + 02 y 2 , if y ≥ 0, , and H 1 , H ± 3 are given just below (25), is a homeomorphism of R 2 onto itself. Theorem 4.1 makes a further simplification based on Theorem 3.1. Actually, Theorem 3.1 normalizes system (8), which contains 12 independent coefficients of quadratic terms, into the form (9) of only 4 independent coefficients. Accordingly, in the above proof of Theorem 4.1, system (26) is normalized into (29). One can check that which restricts system (29) to be of the form (9). What Theorem 4.1 does is utilizing the monodromy to simplify system (29) further, i.e., eliminate at least one of parameters γ ± , η ± in (9). For this purpose, we need to append at least one of the following equalities to the linear algebra system (32). The linear algebra system (30), given in the proof of Theorem 4.1, is exactly (32) associated with D − 20 = D − 02 = 0, which proves true above with a choice of the near-identity transformation and eliminates η − from (9), giving (24). Similarly, one can consider (32) associated with another one D + 20 = D + 02 = 0 in (33), which eliminates η + from (9), giving the same (24) with the change y → −y. For the same reason, considering the equality C − 20 = C − 02 = 0 (or the one C + 20 = C + 02 = 0) in (33), we eliminate γ − (or γ + ) from (9) and obtain the form (or the same (34) with the change y → −y). Remark that the above obtained forms (24) and (34) are the simplest ones in the further simplification. In fact, seen from the proof of Theorem 4.1, the uniqueness ofp 20 ,p ± 11 ,p ± 02 ,q ± 11 ,q ± 02 solved from the linear algebra system (30) implies that we can eliminate no more. We further note that (34) is somehow the same as (24) if we exchange x with y, which however moves the switching line y = 0 to be the line x = 0 and converts an upper-lower switching system to a left-right switching one. For convenience, we call (24) the second order monodromic normal form of system (1).

Applications to Hopf bifurcations.
In what follows, we use the second order monodromic normal form (24) to discuss Hopf bifurcations of the switching system ẋ y = parameterized by λ ∈ U ⊆ R C 1 -smoothly, where U is a given small neighborhood of 0. This system is non-degenerate and monodromic at the origin O for all λ ∈ U . By Theorem 4.1, system (35) is conjugate by a transformation of form (31) to the form where α ± , β ± , γ ± , η + are all C 1 real functions of λ and β ± (λ) = 0. Thus, the problem of Hopf bifurcations for (35) is reduced to appearance of limit cycles from the origin of (36).
There have been many discussions (see e.g. [7,11,15,18,28]) on Hopf bifurcations for switching systems. In [28] periodic orbits are discussed by applying an appropriate version of the implicit function theorem for fixed points of the return map without computing the second order Lyapunov quantity. Unlike [18] the above Theorem 5.1 about Hopf bifurcations does not consider the change of the origin O from an equilibrium to a regular point under a perturbation. Although the same form N 77 of linear part is also considered in [7,11,15,28], the second order Lyapunov quantity is computed under the assumption that α ± = 0 and β ± = −1 in [7,15] for some special systems, that is, the upper and lower systems are both of Bautin form and have a center at O in [7] and the upper and lower systems are both Liénard equation in [15]. Under the assumption that β ± = −1, the second order Lyapunov quantity is computed in [11], where all coefficients of quadratic terms are considered but the expressions of the second order Lyapunov quantity are complicated even in the complex form. In our paper a more general form of switching systems is considered and the computation of the second order Lyapunov quantity is simplified by reduction to normal forms. Our method is applicable to monodromic (1) without requiring the linear part to be N 77 .
In this paper we always assume that O is a non-degenerate equilibrium when considering nonlinear terms. In the case that either A + or A − is nilpotent, by Theorem 2.2 one can find 29 possibilities of normal forms for the switching system, much more possibilities than smooth systems, for which only one possibility is nilpotent. So, for degenerate equilibria there will be lots of work in the future.