Bifurcations of dynamical systems with sliding: derivation of normal-form mappings

Physica D 170 (2002) 175–205

Bifurcations of dynamical systems with sliding: derivation of normal-form mappings M. di Bernardo a , P. Kowalczyk a,∗ , A. Nordmark b a

Department of Engineering Mathematics, University of Bristol, BS8 1TR Bristol, UK b Department of Mechanics, Royal Institute of Technology, Sweden

Received 17 December 2001; received in revised form 28 May 2002; accepted 30 May 2002 Communicated by E. Ott

Abstract This paper is concerned with the analysis of so-called sliding bifurcations in n-dimensional piecewise-smooth dynamical systems with discontinuous vector field. These novel bifurcations occur when the system trajectory interacts with regions on the discontinuity set where sliding is possible. The derivation of appropriate normal-form maps is detailed. It is shown that the leading-order term in the map depends on the particular bifurcation scenario considered. This is in turn related to the possible bifurcation scenarios exhibited by a periodic orbit undergoing one of the sliding bifurcations discussed in the paper. A third-order relay system serves as a numerical example. © 2002 Elsevier Science B.V. All rights reserved. PACS: 05.45−a Keywords: Discontinuous systems; Sliding bifurcations; Normal-form maps

1. Introduction Discontinuous events characterise the behaviour of an increasing number of dynamical systems of relevance in applied science and engineering. Examples include the occurrence of impacts in mechanical systems [1], stick-slip motion in oscillators with friction [2], switchings in electronic circuits [3–6] and hybrid dynamics in control devices [7]. These systems are often modelled by sets of piecewise-smooth (PWS) ordinary differential equations (ODEs). These are smooth in regions Gi of phase space with smoothness being lost as trajectories cross the boundaries Σi,j between adjacent regions, see Fig. 1. Specifically, we have x˙ = F (x, t, µ),

(1.1)

with F : Rn+m+1 → R being a PWS vector function, t the time variable, µ ∈ Rm a parameter vector, and x ∈ Rn the state vector. In each of the phase space regions Gi , the system dynamics are described by a different functional form, Fi , of the system vector field. ∗ Corresponding author. Tel.: +44-117-9289798; fax: +44-117-9251154. E-mail addresses: [email protected] (M. di Bernardo), [email protected] (P. Kowalczyk), [email protected] (A. Nordmark).

0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 2 ) 0 0 5 4 7 - X

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Fig. 1. Schematic representation of the system trajectory crossing the boundary, Σ between adjacent regions of smooth dynamics.

PWS systems have been shown to exhibit a richness of different dynamical behaviours which include several bifurcations and deterministic chaos [8–10]. Many of these phenomena are due to the unique nature of these systems and involve interactions between the system trajectories and its phase space boundaries. For example, a dramatic change of the system behaviour is usually observed when a part of the system trajectory hits tangentially one of the boundaries between different regions in phase space. When this occurs, the system is said to undergo a grazing bifurcation (also known as C-bifurcations in the Russian literature) [11–15]. Grazing phenomena can lead to several dramatic bifurcation scenarios including a sudden transition from stable periodic motion to fully developed chaotic behaviour (e.g. [16,17]). A particularly intriguing type of solution which is unique to PWS systems is the so-called sliding or Filippov solution [18]. This is a solution which lies entirely within the discontinuity set of the system under investigation and can be analysed by means of two alternative methods: Filippov Convex method [18] and Utkin equivalent control [19]. Sliding is only possible if the direction of the system vector field on both sides of a discontinuity set points towards the set itself so that nearby trajectories are constrained to evolve on it (see Section 3 for further details). Physically, sliding motion may be understood as repeated switching between two different system configurations as has been reported in [20–22]. Recently, it has been shown that a novel class of bifurcations can be observed when, as the parameters are varied, the system trajectories cross regions of phase space where sliding is possible (or sliding regions). According to the nature of these intersections and the properties of the system vector field, different scenarios are possible. These bifurcations can explain, e.g. the formation of so-called stick-slip dynamics in friction oscillators [23], double spiral bifurcation diagrams in power electronic converters [5] and fast-switching trajectories in relay feedback systems [20]. The occurrence of complex transitions involving sliding was independently observed in the Russian literature [24,25] and more recently detailed to the case of relay feedback systems in [20]. It was shown that sliding can be associated to four different codimension-1 bifurcation scenarios of limit cycles which are termed: (i) sliding bifurcation type I; (ii) multisliding bifurcation; (iii) grazing–sliding; and (iv) sliding type II (switching sliding) (as will be detailed later in Section 3). Extensive numerical simulations were carried out to understand the nature of these four scenarios and some of the complex behaviour they can organise. It was found, for example, that sliding bifurcations can lead to the formation of chaotic attractors and asymmetric orbits in a class of entirely symmetric relay feedback systems [26,27]. During the revision of this manuscript, we became aware of some related work by Kuznetsov et al. where the possible cases of sliding bifurcations are listed for two-dimensional discontinuous flows [28]. A fundamental question when dealing with bifurcations is to be able to predict and classify all the possible dynamical scenarios following their occurrence. As shown in [16] in the case of C-bifurcations of maps, the same non-smooth bifurcation can be associated to different scenarios, e.g. period-doubling, sudden transition to chaos

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etc. To account for all the possible transitions, a pressing open problem is the derivation of appropriate normal-form maps for the bifurcations under investigation. The aim of this paper is to present for the first time the derivation of such normal-form maps for general n-dimensional PWS dynamical systems with sliding. Using the concept of discontinuity mapping first introduced in [11], we will present the local analysis of the four sliding bifurcation scenarios mentioned. After giving conditions for sliding to occur, we will briefly describe the main characteristics of these novel bifurcations and give precise conditions for each of them to occur. Using these conditions, we will then derive their normal-form maps. In so doing, we will show that sliding bifurcations are indeed novel type of transitions associated to precise functional forms of corresponding normal-form maps. The use of these maps for classification will be published elsewhere [30]. Our aim is to give analytical formulas for the local maps associated with bifurcations involving sliding which can be used to characterise the dynamics of several systems of relevance in applications. As a representative example, we will use relay feedback systems which have been extensively studied experimentally. The paper is outlined as follows. After stating some preliminary hypotheses, the four possible scenarios of sliding bifurcations are presented in Section 3. Analytical conditions which must be satisfied at the bifurcation point for each case are given. These conditions are used later in the paper to carry out the analytical derivation of appropriate normal-form mappings. These are derived by using the concept of the so-called zero-time discontinuity mapping (ZDM) (see Section 3.2). The detailed derivation of such mapping for each case is outlined in Section 4. In Section 5, numerical analysis of a third-order relay feedback system illustrates the theory presented in the previous section. Finally, in Section 6 we briefly discuss the implications of our results to the analysis of periodic orbits undergoing sliding bifurcations, before drawing some conclusions in Section 7. 2. PWS systems and sliding motion In what follows, we consider a sufficiently small region D ⊂ Rn of phase space where we assume that the n-dimensional system (1.1) can be described by the following equation:
F1 (x) if H (x) > 0, x˙ = (2.1) F2 (x) if H (x) < 0, where x ∈ Rn , F1 , F2 : Rn → Rn are sufficiently smooth in D and H : Rn → R is a sufficiently smooth scalar function (at least C 4 ) of the system states. We label Σ the hyperplane defined by Σ := {x ∈ Rn : H (x) = 0},

(2.2)

which we term as switching manifold. Σ divides D into the two regions: G1 := {x ∈ D : H (x) > 0}

(2.3)

G2 := {x ∈ D : H (x) < 0}.

(2.4)

and Moreover, we assume that there exists a subset of the switching manifold Σˆ ⊂ Σ, labelled as sliding region, which is simultaneously attracting from both sides in regions G1 and G2 . Throughout the neighbourhood of this region we shall assume:

∇H, F2  − ∇H, F1  > 0, where ∇H (or equivalently ∂H /∂x) represents the normal to Σ.

(2.5)

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ˆ it is then constrained to evolve Under these assumptions, if the system trajectory crosses the sliding region Σ, within Σˆ until it eventually reaches its boundary [18]. This is the so-called sliding motion which can be described by considering an appropriate vector field Fs , which lies within the convex hull of F1 and F2 , and is tangent to Σ for x ∈ Σˆ [18,19]. According to Utkin’s equivalent control method (see [19] for further details), such vector field is given by F1 + F2 F2 − F 1 Fs = + Hu (x) , (2.6) 2 2 where Hu (x) ∈ [−1, 1] is some scalar function of the system states. Hu (x) can be obtained in terms of F1 and F2 by considering that Fs must be tangential to the switching manifold, i.e. ∇H, Fs  = 0. Using this condition, we then have

∇H, F1  + ∇H, F2  . (2.7) Hu (x) = −

∇H, F2  − ∇H, F1  We can now define the sliding region Σˆ as Σˆ := {x ∈ Σ : −1 ≤ Hu (x) ≤ 1}.

(2.8)

It follows from (2.5) and (2.8) that

∇H, F2  > 0 > ∇H, F1 ,

(2.9)

ˆ Additionally, we define the boundary of the sliding region as throughout the region of interest (i.e. x ∈ Σˆ \ ∂ Σ). + − ˆ ˆ ˆ ∂ Σ = ∂ Σ ∪ ∂ Σ , where ∂ Σˆ + := {x ∈ Σ : Hu (x) = 1},

(2.10)

∂ Σˆ − := {x ∈ Σ : Hu (x) = −1}.

(2.11)

Note that if x ∈ ∂ Σˆ + , from (2.6) we obtain Fs = F2 , while if x ∈ ∂ Σˆ − , Fs = F1 . Also, it is worth mentioning here that Hu (x) = ±1 defines the equivalent control manifold in Rn whose intersecˆ The analysis which is carried on later in the paper tion with Σ determines the boundary of the sliding region ∂ Σ. assumes that a bifurcation point x ∗ = 0 lies on ∂ Σˆ − . It should be noted that this assumption places no constraints on the theory presented further on in the paper. By ∇Hu (or equivalently ∂Hu /∂x) we indicate the normal vector to ∂ Σˆ ± , which can be expressed as     ∂H ∂H ∂F1 ∂H ∂F2 ∂H ∂ 2H ∂ 2H F + F − + + F F ∇Hu (x) = − 1 2 2 1 ∂x ∂x ∂x ∂x ∂x ∂x ∂x 2 ∂x 2      2  ∂H ∂H ∂H ∂F1 ∂H ∂H ∂ 2H ∂ 2H ∂H ∂F2 F2 − F1 + − F1 + F2 F2 − F1 . + ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x 2 ∂x 2 (2.12) The dynamics of the system while sliding, which are given by the sliding flow, φs (x, t), generated by Fs , will be moving towards the boundary of the sliding strip, say ∂ Σˆ − , if ∇Hu , Fs  < 0. Without loss of generality, we assume that both Σ and ∂ Σˆ − can be flattened by making a series of appropriate near-identity transformations (the same is true if Σ and ∂ Σˆ + are considered). In this case, (2.12) becomes     ∂H ∂F2 ∂H ∂H ∂F1 ∂H + F2 − F1 ∇Hu (x) = − ∂x ∂x ∂x ∂x ∂x ∂x    2   ∂H ∂H ∂H ∂H ∂F1 ∂H ∂H ∂F2 (2.13) − F1 + F2 F2 − F1 . + ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x

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3. Bifurcations involving sliding 3.1. The four possible cases We now introduce the possible bifurcation scenarios involving interactions between trajectories of the system ˆ We give a description of all the possible cases, which we will generically indicate as and the sliding region Σ. sliding bifurcations. We will then give appropriate analytical conditions characterising each of them in the next section. According to the results presented in [20,26,27] and independently in [24], we can identify four possible cases of interactions between the system flow and the sliding section. These can be generalised to the case of n-dimensional PWS dynamical systems of the form (2.1). A three-dimensional schematic representation is given in Fig. 2. For the sake of clarity, we can assume that such local bifurcations involve sections of trajectories belonging to some periodic orbit of the system. Fig. 2(a) depicts the scenario we term as sliding bifurcation of type I. Here, under parameter variations, a part of the system trajectory crosses transversally the boundary of the sliding strip at the bifurcation point (trajectory labelled b in Fig. 2(a)). Further variations of the parameter cause the trajectory to enter the sliding ˆ leading to the onset of sliding motion. Note, that the sliding trajectory then moves locally towards the region Σ, ˆ At the boundary Fs = F1 , the trajectory leaves the switching manifold tangentially. boundary of Σ. In the case presented in Fig. 2(b), instead, a section of trajectory lying in region G1 or G2 grazes the boundary of the sliding region from above (or below). Again, this causes the formation of a section of sliding motion which ˆ We term this transition as a grazing–sliding bifurcation. We note that this transition is the locally tends to leave Σ. immediate generalisation of so-called grazing bifurcations [11] to dynamical systems with sliding.

Fig. 2. The four possible bifurcation scenarios involving collision of a segment of the trajectory with the boundary of the sliding region ∂ Σˆ − .

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A different bifurcation event, which we shall call sliding bifurcation of type II or switching–sliding, is depicted in Fig. 2(c). This scenario is similar to the sliding bifurcation of type I shown in Fig. 2(a). We see a section of the trajectory crossing transversally the boundary of the sliding region. Now, though, the trajectory stays locally within the sliding region instead of zooming off the switching manifold Σ. The fourth and last case is the so-called multisliding bifurcation, shown in Fig. 2(d). It differs from the scenarios presented above since the segment of the trajectory which undergoes the bifurcation lies entirely within the sliding ˆ Namely, as parameters are varied, a sliding section of the system trajectory hits tangentially (grazes) the region Σ. boundary of the sliding region. Further variations of the parameter cause the formation of an additional segment of trajectory lying above or below the switching manifold, i.e. in region G1 or G2 . As shown in [20], this mechanism can give rise to an interesting sliding adding scenario where the accumulation of multisliding bifurcations causes the formation of periodic orbits characterised by an increasing number of sliding sections. Note that a complete list of the possible bifurcation scenarios in systems with sliding involving not only limit cycles but also equilibria and so-called pseudo-equilibria (equilibria of the sliding flow) can be given in two-dimensional systems with sliding (see, e.g. [28]). We focus on the n-dimensional case, aiming at deriving a completely general theory for at least the four cases of sliding bifurcations involving limit cycles described above. In particular, our aim is to understand and predict, hence classify, the possible dynamical scenarios associated to each of these bifurcations. As shown in [29] and further detailed in [30], an invaluable tool for this purpose is represented by so-called discontinuity mappings which were first introduced in [31]. We briefly outline here the main concept of these maps leaving to Section 4 the derivation of the local normal-form mappings associated with each of the four sliding bifurcation events for n-dimensional PWS dynamical systems. Further details on the application of such mappings for the classification of sliding bifurcations will be published elsewhere [29]. 3.2. The discontinuity mapping The discontinuity map can be defined as the correction to be made to trajectories of a PWS system in order to account for the presence of the switching manifold (see Fig. 3). It can be used to construct the Poincaré map of a limit cycle interacting with the switching manifold of a non-smooth system, locally to the bifurcation point. Specifically, consider a periodic orbit, of period T , such as the one shown in Fig. 3 and suppose one wishes to construct an appropriate fixed time T -map, say Γ . To do so, say T1 is the time taken by the orbit to go from some point A to its first intersection, B, with the discontinuity boundary, Σ. Then label T2 the time spent by the orbit on the other side of the boundary and T3 that needed to get from the exit point C back to A. The trajectory from A

Fig. 3. A periodic orbit in a PWS system.

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back to itself would then be described by the composition of flow Φ1 and Φ2 . Namely, one would follow flow Φ1 for time T1 , then switch to flow Φ2 for time T2 and finally get back to A by considering flow Φ1 for time T3 . Starting from nearby points, one would need to adjust the times in order to take into account correctly the position of the discontinuity boundary. Alternatively, one could keep the flow times unchanged but consider appropriate corrections. In particular, such correction mappings should be applied between flows Φ1 and Φ2 and again between flows Φ2 and Φ1 . Namely, the map from a point, x, close to A would then be given by Γ (x) = Φ1 (·, T3 ) ◦ DM21 ◦ Φ2 (·, T2 ) ◦ DM12 ◦ Φ1 (x, T1 ), where DM12 and DM21 are labelled as zero-time discontinuity maps (ZDM). In this sense, the discontinuity map represents the correction brought about by the presence of the switching manifold. Similarly, if one wants to construct a Poincaré map for a given orbit from some Poincaré section back to itself, similar maps can be introduced which take the name of Poincaré discontinuity maps (PDM). Instead of considering fixed times, in this case PDMs introduce appropriate corrections to flows between starting and ending transversal sections (for further details see [32]). Note that a particular ZDM is applied between a given inflow and a given outflow. It consists, as will be detailed in the rest of the paper, of consecutive flow lines starting with the inflow and ending with the outflow so that the total time is zero. For a PDM, we would start from a section transversal to the inflow and end on a section transversal to the outflow. In particular, a PDM can be constructed out of a ZDM by restricting the initial points to the inflow-section and considering an appropriate projection of the endpoints to the outflow-section. This in turn can be also interpreted as changing the time spent in the outflow, i.e. instead of achieving zero time (ZDM), we flow until hitting given section (PDM). These mappings have been shown to be an invaluable tool in characterising bifurcations in PWS and discontinuous dynamical systems [31–34]. In the rest of the paper, we shall show that there is a fundamental difference between normal-form maps associated with different sliding events.

4. Normal-form maps for sliding bifurcations We now present, for the sake of completeness, the detailed derivation of the discontinuity map associated to each of the cases depicted in Fig. 2. Our results (see Eqs. (4.31), (4.59), (4.80) and (4.112)) indicate that the functional form of these maps changes according to the scenario considered. Specifically, the leading-order term of the perturbation accounted for by the discontinuity map varies from case to case (see derivations below for further details). A summarising table is reported in Table 1. To carry out the derivation, we will assume that the vector fields F1 , F2 , Fs are well defined over the entire phase space region of interest. Therefore, we will suppose that the corresponding flows can be expanded as a Taylor series about the bifurcation point x ∗ = 0, t ∗ = 0 as φi (x, t) = x + Fi t + ai t 2 + bi xt + ci t 3 + di x 2 t + ei xt2 + fi t 4 + gi x 3 t + hi x 2 t 2 + ji xt3 + O(5),

(4.1)

Table 1 Summarising table of the differences between DMs for the cases considereda Bifurcation type

DM leading-order term

Equation no.

Sliding type I Grazing–sliding Switching–sliding Multisliding

ε2

(4.31) (4.59) (4.80) (4.112)

a

+ O(ε 3 )

ε + O(ε 3/2 ) ε 3 + O(ε 4 ) ε 2 + O(ε 5/2 )

The last column indicate the equation number of the final expression of the DM for the corresponding sliding bifurcation.

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where i = 1, s, O(5) indicates terms of order equal or higher than five and     1 ∂Fi ∂Fi 1 ∂ 2 Fi 2 ∂Fi 2 Fi , bi = ai = , ci = F + Fi , 2 ∂x ∂x 6 ∂x 2 i ∂x 1 ei = 2



1 fi = 24

∂ 2 Fi Fi + ∂x 2





∂Fi ∂x

di =

1 ∂ 2 Fi , 2 ∂x 2

2 

∂ 3 Fi 3 ∂ 2 F i F + ∂x 3 i ∂x 2

, 

  2    ∂Fi ∂ Fi ∂Fi ∂Fi 3 ∂Fi ∂Fi ∂ 2 Fi Fi Fi + Fi , + Fi + + ∂x ∂x ∂x ∂x 2 ∂x ∂x 2 ∂x

  1 ∂ 3 Fi 1 ∂ 3 Fi ∂Fi ∂ 2 Fi , , h = F + i i 6 ∂x 3 4 ∂x 3 ∂x ∂x 2

1 ∂ 3 Fi 2 ∂ 2 Fi ∂Fi ∂ 2 Fi ∂Fi ∂Fi 3 ∂Fi ∂ 2 Fi ji = F + . Fi + F i + 2 2 Fi + 6 ∂x 3 i ∂x ∂x ∂x 2 ∂x ∂x 2 ∂x ∂x

gi =

(4.2)

Note that we have used a shorthand notation here for the higher-order derivative terms, e.g. ∂ 3 Fi 3 x = ∂x 3

 i,j,k=1,2,3

∂ 3 Fi xi xj xk . ∂xi xj xk

In what follows, we shall continue to use this shorthand, with care taken to correctly evaluate the derivative tensors when required. For each case, we shall consider ε-perturbations of the bifurcating trajectory, dividing the derivation in different steps. We start by making rigorous the bifurcation scenarios described in the previous section, giving analytical conditions for their occurrence. 4.1. Analytical conditions In all the cases presented in the previous section, the bifurcation events involve a part of the system trajectory, ˆ At the bifurcation point, say x = x ∗ , t = t ∗ , the following general crossing the boundary of the sliding region ∂ Σ. conditions must be satisfied for all cases. Specifically, we must have 1. H (x ∗ ) = 0, ∇H (x ∗ ) = 0; 2. Hu (x ∗ ) = −1 ⇔ Fs = F1 ⇔ ∇H, F1  = 0 at x ∗ . These conditions state that the bifurcation point: (1) belongs to the switching manifold, which is well defined; (2) is located on the boundary of the sliding region (w.l.o.g. we assume it to belong to ∂ Σˆ − ). In what follows we assume, w.l.o.g., that the bifurcation point is located at the origin, i.e. x ∗ = 0, t ∗ = 0. Unless stated otherwise, it is assumed that all the quantities in the equations below are evaluated at the origin. Using (2.13) and the general conditions for sliding bifurcations, we can now express ∇Hu∗ in terms of ∇H, F1 , F2 as ∇Hu∗ = −

∗ 2 ∗ ∂F1 ∇H , ∗

∇H ∗ , F2  ∂x

(4.3)

where the superscript ∗ denotes quantities evaluated at the bifurcation point x ∗ . In the derivations for the sake of brevity we omit the superscript ∗. Note that the denominator of (4.3) is positive, according to (2.9).

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4.1.1. Case I: sliding bifurcation type I As shown in Fig. 2(a), in this case, the sliding flow moves locally towards the boundary of the sliding region, when perturbed from the bifurcation point. Thus, at the bifurcation point, we must have dHu (φ1 (0, t)) < 0, (4.4) dt t=0 which yields the additional condition

∇Hu , F1  < 0. After substituting (4.3) for ∇Hu into (4.5), we get 

d2 H (φ1 (0, t)) ∂F1 = ∇H, F1 > 0. ∂x dt 2 t=0

(4.5)

(4.6)

4.1.2. Case II: grazing–sliding bifurcation This scenario is equivalent locally to Case I (see Fig. 2(b)). When grazing–sliding occurs, the sliding flow also moves towards the edge of the sliding strip. Thus, condition (4.6) holds in this case as well. 4.1.3. Case III: sliding bifurcation type II (or switching–sliding) In this case, contrary to what assumed for Cases I and II (see Fig. 2(c)), the vector field Fs must point away from ˆ Fs = F1 , we require the the boundary of the sliding region at the bifurcation point. Thus, recalling that along ∂ Σ, extra condition 

dHu (φ1 (0, t)) d2 H (φ1 (0, t)) ∂F1 (4.7) = ∇Hu , F1  > 0 ⇒ = ∇H, F1 < 0. dt ∂x dt 2 t=0 t=0 4.1.4. Case IV: multisliding bifurcation When a multisliding bifurcation occurs (see Fig. 2(d)), the sliding flow is tangential to the boundary of the sliding strip at the bifurcation point. Hence, we must have dHu (φs (0, t)) =0 (4.8) dt t=0 and since Fs = F1 along ∂ Σˆ − , we then have

∇Hu , Fs  = ∇Hu , F1  = 0. Applying (4.3) for ∇Hu into (4.9) yields 

d2 H (φ1 (0, t)) ∂F1 F = ∇H, 1 = 0. ∂x dt 2 t=0

(4.9)

(4.10)

Moreover, assuming w.l.o.g. that the sliding flow has a local minimum at the bifurcation point, we also require d2 Hu (φs (0, t)) > 0, dt 2 t=0 i.e.

 2 

d2 Hu (φs (0, t)) ∂ Hu 2 ∂Hu ∂F1 ∂ 2 Hu 2 ∂F1 + = + F = ∇H , , F F F 1 u 1 1 > 0. ∂x ∂x ∂x dt 2 ∂x 2 1 ∂x 2 t=0

(4.11)

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Under the assumption that ∂ Σˆ − is flat (4.11) becomes

 d2 Hu (φs (0, t)) ∂F1 = ∇Hu , F1 > 0. ∂x dt 2 t=0

(4.12)

Using (4.3) for ∇Hu , we obtain     ∂F1 2 d3 H (φ1 (0, t)) = ∇H, F1 < 0. ∂x dt 3 t=0

(4.13)

Equipped with analytical conditions for each of the cases considered, we can now present the derivation of the normal-form maps associated to each of them. For the sake of clarity, we choose to detail the derivation separately for each of the four cases. 4.2. Case I: sliding bifurcation type I The bifurcation scenario corresponding to this case together with a schematic representation of the map derivation is shown in Fig. 4. Here we can see the bifurcating trajectory φ1 (0, t) crossing Σˆ at the point x ∗ = 0 lying on its boundary ∂ Σˆ − . In order to derive the ZDM, we consider a perturbation of the bifurcating trajectory such that the ˆ (Typically, the system motion will be taken new trajectory hits the sliding region Σˆ at some point, say εx0 ∈ Σ. to this point under parameter variations by a segment of trajectory lying below Σ, generated by flow Φ2 .) The trajectory is then constrained to evolve within Σˆ following the sliding flow φs (εx0 , t). Under condition (4.6), the system evolution hits the boundary of the sliding region Σˆ after some time, say δ, at the point x¯ := φs (εx0 , δ). The system evolution then leaves the plane following φ1 (x, ¯ t). The ZDM in this context is the correction that needs to be applied to the flow at point εx0 in order to account for the presence of the sliding region. Specifically, this correction must be such that the system evolution across the

Fig. 4. A schematic representation of the ZDM derivation for sliding bifurcations of type I.

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surface may be described entirely by applying the discontinuity map and using flow φ1 . As depicted in Fig. 4, the ZDM maps εx0 to some point, say xf . From xf , the trajectory then evolves through x¯ following flow φ1 . Thus, all the important information concerning the presence of the switching manifold Σˆ and its influence is indeed captured by the ZDM. Our aim is to get an analytical expression for the ZDM. To do so we shall proceed in two steps: (i) we evaluate the trajectory from the point εx0 to the point x¯ following φs for time δ; (ii) starting from x¯ we follow φ1 backward in time for the same amount of time as in (i), reaching the final point xf . Note that the elapsed time from εx0 to xf equals 0. 4.2.1. First step Let xm (t) = φ1 (0, t) be the bifurcating trajectory and let us consider perturbations of xm of size ε: x(t) = φs (εx0 , t)

(4.14)

for some x0 which we assume to be such that

∇Hu , x0  > 0.

(4.15)

The condition above ensures that for ε > 0, we analyse the trajectory which crosses the switching manifold within ˆ If ε > 0, then at some time t1 = δ the perturbed trajectory, x(t), will cross the boundary of the sliding region Σ. the sliding region Σˆ at x = x¯ given by x¯ = φs (εx0 , δ) ≈ εx0 + δFs + δ 2 as + εδbs x0 + δ 3 cs + ε 2 δds x02 + εδ 2 es x0 + δ 4 fs + ε 3 δgs x03 + ε2 δ 2 hs x02 + εδ 3 js x0 .

(4.16)

We wish to define δ to be the time such that Hu (x) ¯ = 0 which, since ∂ Σˆ − and Σ are flat to leading-order, implies

∇Hu , x ¯ = 0.

(4.17)

Using (4.16) for x, ¯ (4.17) becomes εx0Hu + δFsHu + δ 2 asHu + εδ(bs x0 )Hu + δ 3 csHu + ε 2 δ(ds x02 )Hu + εδ 2 (es x0 )Hu + δ 4 fsHu + ε3 δ(gs x03 )Hu + ε 2 δ 2 (hs x02 )Hu + εδ 3 (js x0 )Hu ≈ 0,

(4.18)

where the subscript Hu denotes the component of a vector quantity along ∇Hu , i.e. YHu = ∇Hu , Y .

(4.19)

We solve (4.18) for δ as an asymptotic expansion in ε. To establish a leading-order term of the asymptotic expansion, we will balance the first and the second term of (4.18). This gives the asymptotic expansion with the leading-order term of O(ε). Solving (4.18) for δ as an asymptotic expansion in ε with the lowest term of order O(ε) gives δ = γ1 ε + γ2 ε 2 + γ3 ε 3 + O(ε 4 ).

(4.20)

After substituting (4.20) into (4.18) and solving for coefficients: γ1 , γ2 , γ3 , we get γ1 = −

∇Hu , x0  ,

∇Hu , Fs 

(4.21)

γ2 = −

γ12 ∇Hu , as  + γ1 ∇Hu , bs x0  ,

∇Hu , Fs 

(4.22)

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γ3 = −

γ2 ∇Hu , bs x0  + 2γ2 γ1 ∇Hu , as  + γ13 ∇Hu , cs  γ12 ∇Hu , es x0  + γ1 ∇Hu , ds x02  − .

∇Hu , Fs 

∇Hu , Fs 

(4.23)

Since condition (3.16) for sliding type I demands the denominator of (4.21) to be negative and the numerator of (4.21) to be positive (condition (4.15)), γ1 is positive. Thus, our asymptotic expansion is consistent. After substituting (4.20) into (4.16), we get also the following leading-order expression for x: ¯ x¯ = εχ1 + ε 2 χ2 + ε 3 χ3 ,

(4.24)

where χ1 = x0 + γ1 Fs ,

(4.25)

χ2 = γ2 Fs + γ12 as + γ1 bs x0 ,

(4.26)

χ3 = γ3 Fs + γ2 bs x0 + 2γ2 γ1 as + γ13 cs + γ12 es x0 + γ1 ds x02 .

(4.27)

4.2.2. Second step We now have to consider the evolution of the perturbed trajectory backward in time from the point x¯ to some final point xf , defined as ¯ −δ) ≈ x¯ − δF1 + δ 2 a1 − δb1 x¯ − δ 3 c1 − δ x¯ 2 d1 + xδ ¯ 2 e1 + δ 4 f1 − x¯ 3 δg1 + δ 2 h1 x¯ 2 − xδ ¯ 3 j1 . xf = φ1 (x, (4.28) Substituting (4.20) and (4.24) into (4.28), we can express xf in terms of the initial perturbation ε. Collecting terms at subsequent powers of ε yields the following expression: xf = (χ1 − γ1 F1 )ε + (χ2 + γ12 a1 − γ2 F1 − χ1 γ1 b1 )ε 2 + O(ε 3 ).

(4.29)

Let us substitute (4.25) and (4.26) for χ1 and χ2 , respectively. After carrying out simplifications, we get the following expression for xf to leading-order: xf = εx0 −

1 ∇Hu , x0 2 (F2 − F1 )ε 2 . 4 ∇Hu , F1 

Thus, the ZDM for sliding type I can be written as
if ∇Hu , x0  ≤ 0, εx0 D(x0 ) = εx0 + ε 2 v + O(ε 3 ) if ∇Hu , x0  > 0,

(4.30)

(4.31)

where v=−

1 ∇Hu , x0 2 (F2 − F1 ). 4 ∇Hu , F1 

(4.32)

Expression (4.32) is well defined because of condition (4.5) and non-zero, since condition (4.15) yields ∇Hu , x0  > 0 and F2 = F1 . Using (4.3), (4.32) can be expressed solely in terms of vector fields F1 , F2 and ∇H as v=

1

∇H, (∂F1 /∂x)x0 2 (F2 − F1 ). 2 ∇H, F2  ∇H, (∂F1 /∂x)F1 

(4.33)

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Fig. 5. Schematic representation of constructing the ZDM for the case of grazing–sliding bifurcation.

4.3. Case II: grazing–sliding bifurcation As shown schematically in Fig. 5, in this case a section of the bifurcating trajectory, xm (t) = φ1 (0, t), evolving entirely in subspace G1 (above the switching manifold), hits tangentially ∂ Σˆ − at the point x ∗ = 0. Again, we consider the motion of a trajectory perturbed around the origin. We suppose that the new trajectory hits the switching ˆ at some point, say x¯ and is then confined to evolve on Σˆ until reaching its boundary, ∂ Σˆ − , at the manifold, Σ, ˆ t). point x. ˆ Afterwards, the trajectory zooms off the switching plane Σ following φ1 (x, In the case of grazing–sliding bifurcations, the construction of the ZDM requires three different steps: (i) firstly, ˆ (ii) we then we consider the evolution of the trajectory from the point εx0 backward in time to the point x¯ ∈ Σ; − ˆ study the sliding motion from x¯ to the boundary of the sliding region ∂ Σ , (iii) finally, we consider the evolution along φ1 from the point xˆ to some final point xf . In so doing, we require that the elapsed time to get from the point εx0 to xf is equal to 0. 4.3.1. First step Say xm (t) = φ1 (0, t), the trajectory undergoing a grazing–sliding bifurcation and consider the trajectory x(t) = φ1 (εx0 , t)

(4.34)

for some x0 which we assume to be such that

∇H, x0  < 0.

(4.35)

This condition ensures that the perturbed trajectory crosses Σ (twice) close to x ∗ . In fact, this is equivalent to leading-order to requiring that the trajectory has a negative local minimum value near the origin as such a minimum value is given by ε ∇H, x0  + O(ε2 ).

(4.36)

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Under these conditions if ε > 0 is sufficiently small, there exists some time t1 = −δ such that the trajectory, x(t), ˆ crosses the switching manifold Σ at a point x¯ within the sliding region Σ. Specifically, x, ¯ will be given by x¯ = φ1 (εx0 , −δ) ≈ εx0 − δF1 + δ 2 a1 − εδb1 x0 − δ 3 c1 − ε 2 δd1 x02 + εδ 2 e1 x0 + δ 4 f1 − ε 3 δg1 x03 + ε2 δ 2 h1 x02 − εδ 3 j1 x0 .

(4.37)

We wish to define δ to be the time such that H (x) ¯ = 0, which since H (0) = 0 and Σ is flat, implies

∇H, x ¯ = 0.

(4.38)

Using (4.37) for x, ¯ (4.38) yields to leading-order: εx0H − δF1H + δ 2 a1H − εδ(b1 x0 )H − δ 3 c1H − ε 2 δ(d1 x02 )H + εδ 2 (e1 x0 )H + δ 4 f1H − ε 3 δ(g1 x03 )H + ε2 δ 2 (h1 x02 )H − εδ 3 (j1 x0 )H ≈ 0.

(4.39)

We note that second term in (4.39) is nought since condition (4.6) yields ∇H, F1  = 0. Solving (4.39) for δ as an √ asymptotic expansion in ε gives √ δ = γ1 ε + γ2 ε + γ3 ε 3/2 + O(ε 2 ). (4.40) Substituting (4.40) into (4.39) and solving for γ1 , γ2 , γ3 , we get   −x0H

∇H, x0  γ1 = = −2 , a1H

∇H, (∂F1 /∂x)F1 

(4.41)

γ2 =

1 γ12 c1H + (b1 x0 )H , 2 a1H

(4.42)

γ3 =

1 −γ22 a1H + γ2 (b1 x0 )H + 3γ2 γ12 c1H − γ14 f1H − γ12 (x0 e1 )H . 2 γ1 a1H

(4.43)

Finally, substituting (4.40)–(4.43) into (4.37) and collecting terms at subsequent powers of √ x¯ = χ1 ε + χ2 ε + χ3 ε 3/2 + O(ε 2 ),

√

ε, we obtain (4.44)

where χ1 = −γ1 F1 ,

(4.45)

χ2 = x0 − γ2 F1 + γ12 a1 ,

(4.46)

χ3 = −γ3 F1 + 2γ2 γ1 a1 − γ13 c1 − γ1 b1 x0 .

(4.47)

4.3.2. Second step ˆ In particular, Having derived an expression for x, ¯ we need to consider now the subsequent sliding motion along Σ. the system trajectory starting from x¯ will be constrained to evolve along the sliding manifold for some time, say ∆, until reaching its boundary at the point x. ˆ Using again Taylor series expansion, we can get an approximate expression for xˆ = φs (x, ¯ ∆) as xˆ ≈ x¯ + ∆Fs + ∆2 as + ∆bs x¯ + ∆3 cs + ∆ds x¯ 2 + ∆2 es x¯ + ∆4 fs + ∆gs x¯ 3 + ∆2 hs x¯ 2 + ∆3 js x. ¯

(4.48)

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Now, since xˆ lies on the boundary of the sliding region, ∂ Σˆ − , the following condition must hold:

∇Hu , x ˆ = 0.

(4.49)

Applying (4.48) and (4.49) we get x¯Hu + ∆FsHu + ∆2 asHu + ∆(bs x) ¯ Hu + ∆3 csHu + ∆(ds x¯ 2 )Hu + (∆2 es x) ¯ Hu + ∆4 fsHu + ∆(gs x¯ 3 )Hu + ∆2 (hs x¯ 2 )Hu + ∆3 (js x) ¯ Hu ≈ 0. √ Solving (4.50) for ∆ as an asymptotic expansion in ε, ignoring the trivial solution ∆ = 0, we obtain √ ∆ = ν1 ε + ν2 ε + ν3 ε 3/2 ,

(4.50)

(4.51)

where ν 1 = γ1 ,

(4.52)

ν2 = −

χ2Hu + ν1 (bs χ1 )Hu + ν12 asHu , FsHu

(4.53)

ν3 = −

χ3Hu + ν1 (bs χ2 )Hu + ν1 (ds χ12 )Hu + ν2 (bs χ1 )Hu ν 2 (χ1 es )Hu + 2ν2 ν1 asHu + ν13 csHu + 1 . FsHu FsHu

(4.54)

4.3.3. Third step The last step describes the evolution of the trajectory from xˆ to the point xf obtained by considering flow φ1 backward in time. Specifically, we have xf = φ1 (x, ˆ δ − ∆) ≈ xˆ + (δ − ∆)F1 + (δ − ∆)2 a1 + (δ − ∆)b1 xˆ + (δ − ∆)3 c1 + xˆ 2 (δ − ∆)d1 + x(δ ˆ − ∆)2 e1 + (δ − ∆)4 f1 + xˆ 3 (δ − ∆)g1 + (δ − ∆)2 h1 xˆ 2 + x(δ ˆ − ∆)3 j1 .

(4.55)

Substituting (4.40), (4.44), (4.48) and (4.51) into (4.55), we can express xf in terms of the initial perturbation ε. √ Namely, collecting terms at subsequent powers of ε yields √ xf = (χ1 + ν1 Fs + (γ1 − ν1 )F1 ) ε + (χ2 + as ν12 + ν2 Fs + (γ1 − ν1 )b1 (χ1 + ν1 Fs ) + (γ1 − ν1 )2 a1 + (γ2 − ν2 )F1 + bs χ1 ν1 )ε + O(ε 3/2 ). Since, according to (4.45) and (4.52), ν1 = γ1 and χ1 = −γ1 F1 , then the term at using (4.45), (4.46) and (4.52), we get xf ≈ (γ12 a1 + x0 − γ12 bs F1 + γ12 as )ε, which after some further algebraic manipulations, yields to leading-order:  

∇H, x0  xf = εx0 − (F2 − F1 ) ε + O(ε 3/2 ).

∇H, F2 

(4.56) √

ε simplifies to 0. Moreover, (4.57)

(4.58)

Under our assumptions, F1 = F2 , ∇H, F2  > 0 and, according to (4.35), ∇H, x0  < 0. Hence, (4.58) is non-zero and well defined. In conclusion, the local normal-form map associated to a grazing–sliding bifurcation is piecewise-linear and can be written in the form
εx0 if ∇H, x0  ≥ 0, D(x0 ) = (4.59) εx0 + εv + O(ε 3/2 ) if ∇H, x0  < 0,

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Fig. 6. A schematic representation of constructing the ZDM for the sliding bifurcation type II.

where v=−

∇H, x0  (F2 − F1 ).

∇H, F2 

(4.60)

4.4. Case III: sliding bifurcation type II In the switching–sliding bifurcation scenario, the bifurcating trajectory is characterised by hitting the sliding region on its boundary and then sliding away from it along the switching manifold (see Fig. 6). Let us consider a new trajectory hitting the switching manifold outside the sliding region in the neighbourhood of the point x ∗ . Let’s say that εx0 is the point at which the new trajectory is rooted. The ZDM in this case represents the correction that must be taken into account in order to describe locally the entire trajectory using just the sliding flow φs . The analytical construction of the ZDM proceeds in two steps: (i) firstly we consider the system evolution from εx0 using flow φ1 until it reaches the sliding region at the point x¯ after some time δ; (ii) then we flow backwards in time for the same amount of time from the point x¯ using the sliding flow φs . As usual, the ZDM is then the mapping from εx0 to xf . 4.4.1. First step Let xm (t) = φs (0, t) be the trajectory which undergoes a sliding bifurcation type II. We consider ε-perturbations of xm (t) of the form x(t) = φs (εx0 , t)

(4.61)

for some x0 which we assume to be such that

∇Hu , x0  < 0.

(4.62)

ˆ Condition (4.62) ensures that, for ε > 0, the trajectory crosses the switching manifold outside the sliding region Σ.

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Firstly, we evaluate the flow φ1 from the point εx0 until its first intersection with the sliding region, say x. ¯ Assume such intersection to take place after some time δ, we have x¯ = φ1 (εx0 , δ) ≈ εx0 + δF1 + δ 2 a1 + εδb1 x0 + δ 3 c1 + ε 2 δd1 x02 + εδ 2 e1 x0 + δ 4 f1 + ε3 δg1 x03 + ε 2 δ 2 h1 x02 + εδ 3 j1 x0 .

(4.63)

We wish to define δ to be the time such that H (x) ¯ = 0, which since H (0) = 0 and Σ is flat, implies

∇H, x ¯ = 0.

(4.64)

Using (4.63) for x, ¯ (4.64) yields that to leading-order: εx0H + δF1H + δ 2 a1H + εδ(b1 x0 )H + δ 3 c1H + ε 2 δ(d1 x02 )H + εδ 2 (e1 x0 )H + δ 4 f1H + ε 3 δ(g1 x03 )H + ε2 δ 2 (h1 x02 )H + εδ 3 (j1 x0 )H = 0.

(4.65)

The first term in (4.65) disappears since the point x0 lies on the switching manifold by definition. Moreover, at the bifurcation point, we also have ∇H, F1  = 0, thus the second term of (4.65) vanishes. Solving (4.65) for δ as an asymptotic expansion in ε with the lowest term of O(ε) gives δ = γ1 ε + γ2 ε 2 + γ3 ε 3 + O(ε 4 ),

(4.66)

where γ1 = −2 γ2 = −

∇Hu , x0  ,

∇Hu , Fs 

γ1 (γ12 c1H + γ1 (e1 x0 )H + (d1 x02 )H ) , 2γ1 a1 + (b1 x0 )H

γ22 a1H + γ2 (d1 x02 )H + 3γ2 γ12 c1H + 2γ2 γ1 (e1 x0 )H + γ14 f1H + γ13 (x0 j1 )H + g1 x03 (b1 x0 )H + 2γ1 a1H 3 2 2 γ (h1 x0 )H + γ1 g1 x0H − 1 . (b1 x0 )H + 2γ1 a1H

(4.67)

(4.68)

γ3 = −

(4.69)

Note that the analytical conditions for the sliding bifurcation type II, Eq. (4.7), guarantee that such asymptotic expansion is consistent. Finally, substituting (4.66) into (4.63), we get the following expression for x¯ (we shall consider terms up to third-order): x¯ = εχ1 + ε 2 χ2 + ε 3 χ3 ,

(4.70)

where χ1 = x0 + γ1 F1 ,

(4.71)

χ2 = γ2 F1 + γ12 a1 + γ1 b1 x0 ,

(4.72)

χ3 = γ3 F1 + γ2 b1 x0 + 2γ2 γ1 a1 + γ13 c1 + γ12 e1 x0 + γ1 d1 x02 .

(4.73)

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4.4.2. Second step To obtain the ZDM, we now have to solve the system backward in time using flow φs , starting from x¯ for time −δ. Using again appropriate Taylor series expansions, we get 2 ¯ −δ)≈x−δF ¯ xf = φs (x, ¯ 3 cs − δ x¯ 2 ds + xδ ¯ 2 es + δ 4 fs − x¯ 3 δgs + δ 2 hs x¯ 2 − xδ ¯ 3 js . s +δ as −δbs x−δ

(4.74)

Using (4.66) and (4.70), we can then express xf in terms of the initial perturbation ε. Namely, collecting terms up to third-order yields xf = (χ1 − γ1 Fs )ε + (χ2 + γ12 as − γ2 Fs − γ1 bs χ1 )ε 2 + (χ3 − bs (γ1 χ2 + γ2 χ1 ) − γ1 ds χ12 + γ12 es χ1 − γ3 Fs + 2γ2 γ1 as − γ13 cs )ε 3 + O(4).

(4.75)

Using definitions (4.71) and (4.72) for χ1 and χ2 , respectively, we get after some algebraic manipulation: xf = εx0 + vε 3 + O(ε 4 ),

(4.76)

where v = γ13 (c1 − cs + bs as − bs a1 ) + γ12 (es x0 + e1 x0 − bs b1 x0 − ds Fx0 − ds x0 ) + γ1 (d1 x02 − ds x02 ) + 21 γ2 (F2 − F1 ) ∇Hu , x0 .

(4.77)

An explicit expression for v in terms of the vector fields F1 , F2 and Fs is not reported here for the sake of brevity. In the special case of linear vector fields, F1 , F2 , Fs , v, can be simplified to take the form   1 1 ∇Hu , x0 3 ∂F1 + ( ∇Hu , F2  − ∇Hu , F1 ) (F2 − F1 ). v=− (4.78) 3 ∇Hu , F1 2 ∂x 2 Substituting (4.3) for ∇Hu (x) allows to express the equation above as   

∂F1 ∂F1

∇H, (∂F1 /∂x)x0 3 2 1 ∂F1 v= + ∇H, F1 − ∇H, F2 (F2 −F1 ). (4.79) 3 ∇H, F2  ∇H, (∂F1 /∂x)F1 2 ∂x ∇H, F2  ∂x ∂x In conclusion, the normal-form map for a sliding bifurcation of type II can be written as
if ∇Hu , x0  ≤ 0, εx0 D(x) = εx0 + ε 3 v + O(ε 4 ) if ∇Hu , x0  > 0,

(4.80)

where v is given by (4.77). 4.5. Case IV: multisliding bifurcation We have come now to the last of the four sliding bifurcation scenarios presented in Section 3. As shown in Fig. 7, ˆ hits tangentially the boundary of the sliding in this case a segment of the system trajectory lying entirely on Σ, − ∗ region, ∂ Σˆ at the bifurcation point x = 0. In order to construct the ZDM, we consider, as usual, the trajectory obtained by applying a perturbation of size ε to it, with ε sufficiently small. Such a trajectory is characterised by crossing the boundary ∂ Σˆ at some point, say x. ¯ The system then switches to flow F1 , evolving above the switching manifold, until reaching it again at a point ˆ labelled xˆ ∈ Σ. The ZDM in this case represents the correction that needs to be applied to the system flow, in order to allow the local description of the trajectory entirely in terms of the sliding flow φs . To derive such a mapping we need

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Fig. 7. A schematic representation of constructing ZDM for multisliding bifurcation case.

to consider three different steps (see Fig. 7): (i) we consider the perturbed trajectory assuming that it crosses the boundary of the sliding region at some time t1 = −δ; (ii) we evolve the system forward using flow φ1 until it reaches the switching manifold, after some time, say ∆; (iii) we flow the system using flow φs so that the total time spent flowing forward and backward is zero. 4.5.1. Step I Let xm (t) = φs (0, t) be the sliding trajectory undergoing a multisliding bifurcation at x ∗ = 0, t ∗ = 0 and let us consider perturbations of xm of size ε given by x(t) = φs (εx0 , t)

(4.81)

for some x0 which we assume to be such that

∇Hu , x0  < 0.

(4.82)

As for the grazing–sliding case presented in Section 4.3, this condition ensures that the perturbed trajectory crosses ∂ Σˆ − (twice) close to x ∗ . At some time t1 = −δ the trajectory, x(t), crosses the boundary of the sliding region at x = x. ¯ Using (4.1) we then obtain x¯ = φs (εx0 , −δ) ≈ εx0 − δFs + δ 2 as − εδbs x0 − δ 3 cs − ε 2 δds x02 + εδ 2 es x0 + δ 4 fs − ε 3 δgs x03 + ε2 δ 2 hs x02 − εδ 3 js x0 .

(4.83)

¯ = 0, thus, we have We defined δ as the time such that Hu (x)

∇Hu , x ¯ = 0.

(4.84)

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Hence, substituting (4.83) in (4.84) we obtain εx0Hu − δFsHu + δ 2 asHu − εδ(bs x0 )Hu − δ 3 csHu − ε 2 δ(ds x02 )Hu + εδ 2 (es x0 )Hu + δ 4 fsHu − ε 3 δ(gs x03 )Hu + ε2 δ 2 (hs x02 )Hu − εδ 3 (js x0 )Hu ≈ 0.

(4.85)

The second term of (4.85), −δ ∇Hu , Fs , is zero because of the general conditions for sliding bifurcations reported √ in Section 3.2. Thus, solving (4.85) for δ as an asymptotic expansion in ε yields √ δ = γ1 ε + γ2 ε + γ3 ε 3/2 + γ4 (ε 2 ) + O(ε 5/2 ), (4.86) where  γ1 =

−x0Hu = asHu

 −2

∇Hu , x0  ,

∇Hu , (∂F1 /∂x)F1 

(4.87)

γ2 =

1 γ12 csHu + (bs x0 )Hu , 2 asHu

(4.88)

γ3 =

1 −γ22 asHu + γ2 (bs x0 )Hu + 3γ2 γ12 csHu − γ14 fsHu − γ12 (x0 es )Hu . 2 γ1 asHu

(4.89)

Note that for the sake of brevity, we omit the explicit expression for γ4 which is particularly cumbersome and can be easily obtained by means of an algebraic manipulation software such as Maple or Mathematica. Conditions (4.12) and (4.82) guarantee that also in this case the asymptotic expansion performed is well defined. Substituting (4.86)–(4.89) into (4.83), we obtain an expression for x¯ of the form √ x¯ = χ1 ε + χ2 ε + χ3 ε 3/2 + χ4 ε 2 + O(ε 5/2 ), (4.90) where χ1 = −γ1 Fs ,

(4.91)

χ2 = x0 − γ2 Fs + γ12 as ,

(4.92)

χ3 = −γ3 Fs + 2γ2 γ1 as − γ13 cs − γ1 bs x0 ,

(4.93)

χ4 = −γ4 Fs + (2γ3 γ1 + γ22 )as − γ2 bs x0 − 3γ2 γ12 cs + γ14 fs x0 γ12 es .

(4.94)

4.5.2. Second step We now study the motion of the trajectory after it leaves the sliding region at the point x. ¯ The trajectory then follows flow φ1 for some time, say ∆, until it reaches the next intersection with the switching manifold at the point xˆ = φ1 (x, ¯ ∆). Expanding the flow as a Taylor series about the origin, we can express xˆ as ¯ xˆ ≈ x¯ + ∆F1 + ∆2 a1 + ∆b1 x¯ + ∆3 c1 + ∆d1 x¯ 2 + ∆2 e1 x¯ + ∆4 f1 + ∆g1 x¯ 3 + ∆2 h1 x¯ 2 + ∆3 j1 x.

(4.95)

Since xˆ lies on the switching manifold we have

∇H, x ˆ = 0.

(4.96)

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Substituting (4.95) into (4.96), we then obtain to leading-order: x¯H + ∆F1H + ∆2 a1H + ∆(b1 x) ¯ H + ∆3 c1H + ∆(d1 x¯ 2 )H + (∆2 e1 x) ¯ H + ∆4 f1H + ∆(g1 x¯ 3 )H + ∆2 (h1 x¯ 2 )H + ∆3 (j1 x) ¯ H = 0.

(4.97)

Using the general conditions for sliding bifurcations reported in Section 3.2 and (4.9) together with the fact that x¯ lies on the switching manifold, it is trivial to show that that the first three terms of (4.97) vanish. Moreover, since x¯ belongs to the boundary of the sliding region, we have ∇Hu , x ¯ = 0. Using (4.3), this implies that the fourth term in (4.97) is also null. √ After substituting for x¯ (4.90) into (4.97), we solve (4.97) for ∆ as an asymptotic expansion in ε. Ignoring the trivial solution ∆ = 0, we obtain √ ∆ = ν1 ε + ν2 ε + ν3 ε 3/2 + ν4 ε 2 , (4.98) where ν1 = ν2 =

√ 3γ1 c1H + γ1 c1H −3 + (4(b1 a1 )H /c1H ) , 2c1H −ν1 (ν1 (e1 χ2 )H + 2(d1 χ2 χ1 )H + (g1 χ13 )H + ν1 (h1 χ12 )H + ν12 (j1 χ1 )H + ν13 f1H ) (d1 χ12 )H + 2ν1 (e1 χ1 )H + 3ν12 c1H

(4.99) .

(4.100)

We omit for the sake of brevity the explicit expression for ν3 and ν4 which are particularly lengthy and do not add any extra information. (As mentioned earlier, they can be obtained by using any algebraic manipulation software.) Note that, as discussed in Remark A.1 in Appendix A, under the assumption that Σ and ∂ Σˆ − can be flattened, applying (A.7) and (4.99) we obtain ν1 = 3γ1 . Finally, after substituting (4.98) into (4.97) yields   √ ε5 3/2 2 xˆ = ψ1 ε + ψ2 ε + ψ3 ε + ψ4 ε + O , 2

(4.101)

(4.102)

where ψ1 = χ1 + ν1 F1 ,

(4.103)

ψ2 = χ2 + ν1 b1 χ1 + ν2 F1 + ν12 a1 ,

(4.104)

ψ3 = χ3 + ν1 b1 χ2 + ν1 d1 χ12 + ν2 b1 χ1 + ν12 e1 χ1 + ν3 F1 + 2ν2 ν1 a1 + ν13 c1 ,

(4.105)

ψ4 = χ4 + ν1 b1 χ3 + ν2 b1 χ2 + ν12 e1 χ2 + d1 (2ν1 χ1 χ2 + ν2 χ12 ν2 ) + ν1 g1 χ13 + ν12 h1 χ12 + ν3 b1 χ1 + 2ν2 ν1 e1 χ1 + ν13 j1 χ1 + ν4 F1 + (2ν1 ν3 + ν22 )a1 + 3ν2 ν12 c1 .

(4.106)

4.5.3. Third step The last step describes the evolution of the trajectory from xˆ to xf = φs (x, ˆ δ −∆). We use again Taylor expansion evaluated at the bifurcation point to obtain an approximate expression for xf . Thus, we have 2 ˆ es +(δ−∆)4 fs xf ≈ xˆ + (δ − ∆)Fs + (δ − ∆)2 as + (δ − ∆)bs xˆ + (δ − ∆)3 cs + xˆ 2 (δ − ∆)ds + x(δ−∆)

+ xˆ 3 (δ − ∆)gs + (δ − ∆)2 hs xˆ 2 + x(δ ˆ − ∆)3 js .

(4.107)

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Substituting (4.86), (4.90), (4.95) and (4.98) into (4.107), we can obtain an expression for xf in terms of the initial perturbation ε. Collecting terms at subsequent powers of ε yields the following expression for xf to leading-order: √ xf = ξ1 ε + ξ2 ε + ξ3 ε 3/2 + ξ4 ε 2 + O(ε)5/2 , (4.108) where ξ1 = χ1 + γ1 F,

(4.109)

ξ2 = χ2 + ν12 a1 + ν2 F + ν1 b1 χ1 + (χ1 + ν1 F )(γ1 − ν1 )bs + (γ1 − ν1 )2 as + (γ2 − ν2 )F

(4.110)

and expressions for ξ3 and ξ4 are reported in Appendix A (Remarks A.3 and A.4). Substituting (4.91), (4.92) and (4.101) into (4.109) and (4.110), we can show that ξ1 = 0 and ξ2 = x0 + γ12 as + γ12 a1 − γ12 b1 Fs .

(4.111)

Eq. (4.111) can be further simplified considering that at the bifurcation point Fs = F1 and thus, as shown in Appendix A (Remark A.2), a1 = as . Moreover, from the definitions of the coefficients of the Taylor expansion (4.1), it follows that bs F = 2as = 2a1 . Thus (4.111) reduces to ξ2 = x0 . To obtain the leading-order term of the local mapping we need to find an expression for the first non-vanishing term of xf . The next term under consideration stands at ε 3/2 in (4.108) and is characterised by the coefficient ξ3 . This can also be shown to vanish to zero (see Remark A.3 in Appendix A). Thus, to leading-order the correction to be made to εx0 in order to account for the presence of the switching manifold (the ZDM) is at least of order O(ε 2 ). Remark A.4 in Appendix A contains an explicit expression for this term and its coefficient ξ4 . Having established the leading-order term of the ZDM for the multisliding bifurcation case, the general form of the map can be written as
if ∇Hu , x0  ≥ 0, εx0 D(x0 ) = (4.112) εx0 + ε 2 v + O(ε 5/2 ) if ∇Hu , x0  < 0, where v = ξ4 and as mentioned above, the explicit expression for ξ4 in the general case is reported in Appendix A. We will present here the resulting expression for ξ4 in the case of piecewise-linear vector fields F1 , F2 and their constant difference. After lengthy algebraic manipulations, the resulting expression yields ξ4 =

9 9

∇Hu , x0 2

∇Hu , x0 2 ∂F1 (F2 − F1 ) +

∇Hu , F2 (F2 − F1 ), 4 ∇Hu , (∂F1 /∂x)F1  ∂x 8 ∇Hu , (∂F1 /∂x)F1 

(4.113)

which is non-zero if no non-standard cancellation occurs. Substituting in (4.113) for ∇Hu (4.3) yields the above equation to take the form   9

∇H, (∂F1 /∂x)x0 2

∇H, (∂F1 /∂x)F2  ∂F1 ξ4 = − (4.114) − (F2 − F1 ). 2 ∇H, F2  ∇H, (∂F1 /∂x)2 F1  ∂x

∇H, F2  5. A representative example: relay feedback systems We use a three-dimensional representative example to give numerical confirmation of the results presented in the paper. Specifically, we study the normal-form maps of sliding bifurcations in the three-dimensional relay feedback system analysed in [20,26,27]. In so doing, we perform a comparison between analytical normal-form maps and numerically computed ones.

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5.1. Third-order system: matrix representation We consider a third-order relay feedback system having the following state-space representation: x˙ = Ax + Bu,

u = −sgn(y), y = Cx,

(5.1)

where 

−(2ζ ω + λ)

1

 2 A=  −(2ζ ωλ + ω ) 0 −λω2

0

0



 1 ,



k



   B=  2kσρ  , kρ 2

0

 T 1    C= 0 .

(5.2)

0

Thus, vector fields F1 and F2 may be written as F1 = Ax − B,

F2 = Ax + B.

Applying Utkin’s equivalent control method, we can express the vector field Fs governing the flow on the switching manifold as Fs = As x, where As can be expressed as  0  2 As =   4σρζ ω + 2σρλ − 2ζ ωλ − ω 2ρ 2 ζ ω + ρ 2 λ − λω2

0 −2σρ −ρ 2

0



 1 .

(5.3)

0

Note, that the first column of matrix As does not influence the value of vector field Fs (by definition x1 = 0 during sliding). In the case considered H (x) = Cx and Hu (x) = −CAx/CB. Hence, the switching manifold is defined as Σ := {x ∈ R 3 : Cx = 0}, where the boundary of the sliding region is given by   CAx ± ∂Σ := x ∈ H : − = ±1 . CB

(5.4)

(5.5)

The dynamics of the system presented has been extensively studied numerically in [20,26,27]. We shall note here that the sliding bifurcation type I, the grazing–sliding bifurcation and the multisliding bifurcation have been detected in this system but no evidence of switching–sliding bifurcation has been found. In what follows we will present the local mappings for these three cases of sliding bifurcations using numerical simulations. We will compare numerical results with analytical expressions obtained using the results presented in Section 4. In general, the bifurcations mentioned above occur in the relay system under investigation at points x ∗ = ∗ (x1 , x2∗ , x3∗ ) ≡ 0. Since, our analytical derivation requires the bifurcation point to be located at x ∗ = 0, in order to compare numerical and analytical results an appropriate transformation of the matrices A, B and C needs to be applied. (Equivalently, we could evaluate the maps obtained analytically at δx = x − x ∗ . We choose to consider a change of coordinates in order to follow as closely as possible the theoretical derivation presented in the paper.)

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After translating the bifurcation point x ∗ to the origin, the matrices A, B, C take the form       −(2ζ ω + λ) 1 0 0 2k       ∗ ∗ 2  A =  B2 =  B1 =   −(2ζ ωλ + ω ) 0 1  ,  −2kσρ + x3  ,  2kσρ + x3  ,  C = 1

−λω2  0 0 .

0

−kρ 2

0

(5.6)

kρ 2 (5.7)

The applied transformation causes matrices B1 , B2 to have a different form for the transformed vector fields F1 and F2 , but the matrices A, C stay the same for both vector fields, i.e. F1 = A x + B1 , F2 = A x + B2 . We need to apply the same transformation to the vector field Fs governing the sliding motion. After that we get a new vector field governing the sliding flow, Fs = As x + Bs , where     0 0 0 0     ∗ 2  As =  (5.8) Bs =   4σρζ ω + 2σρλ − 2ζ ωλ − ω −2σρ 1  ,  −2kσρ + x3  . 2ρ 2 ζ ω + ρ 2 λ − λω2

−ρ 2

0

−kρ 2

Having transformed the system under consideration in such a way that the bifurcation point x ∗ is located at the origin, we can now compare analytical expressions of the ZDM derived in Section 4 with numerical results. In what follows, we will derive analytical forms of the ZDM for all the four cases of sliding bifurcations. We will then validate the analytical results numerically for the three bifurcation scenarios detected in the relay system under investigation. 5.1.1. Sliding bifurcation type I The ZDM in the case of relay feedback system can be obtained from (4.31) after substituting expressions defining vector fields F1 , F2 into (4.33). Thus, the ZDM can be written as  for C  A x ≥ 0,  x (5.9) D(x) = 1 (C  A x)2     A x < 0. x + − B ) + H.O.T. for C (B 2 1 2 (C  B  2 )(C  A B  1 ) Following [26], we consider the sliding bifurcation type I observed in the circuit when ω = λ = k = −σ = 1, ! "T ρ ≈ 2.09885, at the point x ∗ = 0 1 9.924404784 . We can now compare the analytical mapping (5.9) with numerical data. To do so we plot how an arbitrarily chosen coordinate of some final point scales as the initial conditions are perturbed by an amount ε from the origin. It should be noted here that it suffices to vary one coordinate of the initial point so as the condition C  A x < 0 is satisfied. Since the identity term of the ZDM in Fig. 8 has been subtracted, a fortiori, the analytical curve converges to 0 at the bifurcation point. We can see in Fig. 8, that the numerical and analytical curves do converge asymptotically, confirming our analytical predictions. 5.1.2. Grazing–sliding bifurcation We consider now the grazing–sliding detected for −σ = ρ = k = 1, λ = 0.05, ζ = 0.0485, occurring at the ! "T point: x ∗ = 0 1 −1.84481226874003 . Using the results of Section 4.3, the ZDM in this case has the form  for C  x ≥ 0, x  D(x) = (5.10) Cx  x −   (B2 − B1 ) + O(x 3/2 ) for C  x < 0. CB2

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Fig. 8. ZDM for sliding bifurcation type I after subtracting identity term.

As for the previous case, the numerical and analytical data are compared after subtracting the identity from the ZDM. The comparison between numerical and analytical data is presented in Fig. 9. Again we see a good agreement between the analytical predictions and the numerical simulations. Our numerics confirm that the mapping has indeed a leading-order linear behaviour locally to the bifurcation point. 5.1.3. Multisliding bifurcation in the relay feedback system We move now to the case of multisliding bifurcations. As reported in [26], a multisliding bifurcation is observed in the relay system considered, when λ = k = −σ = ρ = 1, ζ = 0.05, ω = 10.24176 and takes place at the point: ! "T x ∗ = 0 1 −2 . Using the results of Section 4.5, the ZDM takes the form

D(x) =

  x 

for C  A x ≤ 0,

  9 (C  A x)2 C  A B  2    (B  2 − B1 ) + O(x 5/2 ) for C  A x > 0. − A x − 2 (C  B2 )(C  A A B  1 ) CB 2

(5.11)

In Fig. 10, we see that the analytical and numerical curves converge asymptotically, confirming the quadratic nature of the multisliding normal-form map. 5.1.4. Sliding bifurcation type II Finally, we come to sliding bifurcations type II. These events have not been observed in relay feedback systems. Thus, we limit our presentation to the derivation of the ZDM for this bifurcation event using the results of Section 4.4.

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Fig. 9. ZDM after subtracting identity term for grazing–sliding bifurcation, plotted x2 coordinate versus x1 coordinate.

Fig. 10. ZDM for multisliding bifurcation (after subtracting identity mapping)—range of perturbation: 10−6 –10−4 .

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Specifically, we get  for C  A x ≤ 0,  x   D(x) = C  A B1 − C  A B2 2 (C  A x)3   x + A (B2 − B1 ) + O(x 4 ) for C  A x > 0. + 3 (C  B2 )(C  A B  1 )2 C  B2 (5.12) The numerical validation of this map is currently under investigation.

6. General remarks In Section 4, we have derived expressions for the ZDMs in the four cases of sliding bifurcations. We have shown that the discontinuity mappings consist of two different expressions on either side of a boundary, with a degree of continuity across the boundary that varies for the different cases. In [30], we will show how these discontinuities translate into discontinuities in the Poincaré mapping for a given periodic orbit. Here we shall mention that generally the type of discontinuity is preserved under compositions with a smooth mappings. Thus, the type of discontinuity found in the ZDM describing particular bifurcation scenario will characterise the full Poincaré map. Since we have found that in the grazing–sliding case the ZDM has a piecewise linear character, there is a possibility of a sudden bifurcation as a hyperbolic orbit crosses the discontinuity boundary. Since the Poincaré map in this case has a discontinuous Jacobian, the possible bifurcation scenarios following grazing–sliding can be classified using techniques developed for piecewise-linear maps (see [8,16]). In all the other three cases, dramatic bifurcation scenarios should be expected for non-hyperbolic orbits. This would then typically be a co-dimension 2 bifurcation, as one parameter is needed to make the orbit non-hyperbolic, and another to make it encounter the discontinuity. Nevertheless, if the discontinuity in the Poincaré mapping, say in the second derivative, is large in magnitude, sliding bifurcations can organise secondary bifurcations. For example, a saddle-node bifurcation is often observed in numerical simulations as the parameters are further varied from a multisliding bifurcation point (see [26]). These and other issues will be discussed in detail in [30].

7. Conclusions We have discussed the characterisation of novel bifurcations in dynamical systems with sliding. In particular, using so-called discontinuity mappings, we derived normal-form maps for each of the four possible scenarios involving interactions of system trajectories with the sliding region. To obtain analytical leading-order approximations for such mappings, we performed a combination of Taylor series expansions and asymptotics. We showed that the leading-order term of the local map is quadratic for the sliding bifurcation type I and multisliding bifurcation, or cubic in the case of sliding bifurcation type II. In the grazing–sliding bifurcation case, the normal-form map has a piecewise linear character. Thus, as shown in [8,9,16] if grazing–sliding bifurcation occurs in the system a wide range of dynamical behaviour may be observed, including sudden jumps to chaos and period-adding bifurcations. We also discussed how the normal-form mappings can be used to analyse the behaviour of periodic orbits undergoing sliding bifurcations. We should mention here that analysis presented in this paper captures the dynamics associated with trajectories that are in the close vicinity of the bifurcating trajectory. Global analysis of periodic orbits still remains an open issue.

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Further work shall be directed towards a consistent classification of the possible bifurcations exhibited by the maps derived in this paper. Ongoing research is devoted to the analysis of sliding bifurcations in systems of relevance in applications.

Appendix A Remark A.1. An important result follows from the assumption that Σ and ∂ Σˆ can be considered linear. Specifically, assuming Σ to be linear we get ∂ 2 Hu = 0. ∂x 2

(A.1)

Substituting for ∂Hu /∂x (2.13) and differentiating with respect to x yields ∂ 2 Hu f  g − fg = = 0, ∂x 2 g2 where

(A.2)



     ∂H ∂F1 ∂H ∂F2 ∂H ∂H ∂F1 ∂H ∂H ∂F2 ∂H ∂H f =− + F2 − F1 + − F1 + F2 , (A.3) ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x       ∂H ∂H ∂F1 ∂H ∂ 2 F1 ∂H ∂ 2 F2 ∂H ∂H ∂F2 ∂H ∂F2 ∂H ∂F1 f =− − + − F F + − 2 1 ∂x ∂x 2 ∂x ∂x 2 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x        2 2 ∂H ∂ F2 ∂H ∂H ∂F2 ∂H ∂ F1 ∂H ∂H ∂F1 ∂H ∂F1 ∂H ∂F2 + + − + F F − + , 1 2 ∂x ∂x 2 ∂x ∂x 2 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x (A.4) 2  ∂H ∂H F2 − (A.5) F1 , g= ∂x ∂x    ∂H ∂F2 ∂H ∂H ∂H ∂F1  g =2 F2 − F1 − . (A.6) ∂x ∂x ∂x ∂x ∂x ∂x

After substituting into (A.2)–(A.6) for f , f  , g, g  and appropriate rearrangement we obtain ∂H ∂ 2 F1 1 ∂H ∂F1 ∂H ∂F2 1 ∂H ∂F2 ∂H ∂F1 2 ∂H ∂F1 ∂H ∂F1 = + − . ∂x ∂x 2 ∇H, F2  ∂x ∂x ∂x ∂x ∇H, F2  ∂x ∂x ∂x ∂x ∇H, F2  ∂x ∂x ∂x ∂x We should note, that in the multisliding case

 ∂ 2 F1 2 ∇H, F = 0, ∂x 2 1

(A.7)

(A.8)

since every term on the right-hand side of (A.7) contain ∇H, (∂F1 /∂x)F1  = 0. Remark A.2. This remark is valid for the multisliding bifurcation case. The general condition for all the sliding bifurcations ensure that at the bifurcation point Fs = F1 . Thus, in the multisliding bifurcation case terms denoted as a1 , as take the same value. Let us express as as   1 ∂F1 as = (A.9) Fs + 21 (F2 ∇Hu , Fs  − F1 ∇Hu , Fs ) . 2 ∂x

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We can simplify as so that (A.9) takes the form as =

1 ∂F1 Fs . 2 ∂x

(A.10)

Since as = 21 (∂F1 /∂x)Fs , a1 = 21 (∂F1 /∂x)F1 and Fs = F1 , this implies that as = a1 . Remark A.3. In what follows, we present the general expression for the term denoted as ξ3 in Eq. (4.108). To get an expression for ξ3 , we will substitute (4.86), (4.90), (4.95) and (4.98) into (4.107). Since Fs = F1 we shall refer to the vector field F1 when the vector field F without a subscript is used. Thus, ξ3 has the form ξ3 = ((γ3 − ν3 )F + χ3 + 2ν1 ν2 a1 + χ1 ν12 e1 + χ12 ν1 d1 + 2(γ1 − ν1 )(γ2 − ν2 )as + ν13 c1 + ν1 b1 χ2 + ν2 b1 χ1 + (γ1 − ν1 )3 cs + ((χ1 + ν1 F )(γ2 − ν2 ) + (χ2 + ν12 a1 + ν2 F + ν1 b1 χ1 )(γ1 − ν1 ))bs + ν3 F + (χ1 + ν1 F )(γ1 − ν1 )2 es + (χ1 + ν1 F )2 (γ1 − ν1 )ds )ε 3/2 .

(A.11)

We can simplify the equation above using the fact that a1 = as ; and substituting for (4.91), (4.92) and (4.94) χ1 , χ2 , χ3 subsequently. From Taylor expansion, it also follows that: ds F 2 = 3cs − bs as , d1 F 2 = 3c1 − b1 a1 , es F = 3cs , e1 F = 3c1 . Thus, after lengthy algebraic manipulations, (A.11) takes the form ξ3 = (−3b1 γ2 F + 4bs γ2 F + 4as ν2 − 2γ2 as + 3b1 x0 + 6ν2 a1 − 3bs x0 − 4bs F ν2 − F ν2 b1 )γ1 + (6Fbs b1 + 9c1 + 6bs as − 18a1 bs − 9cs − 3b1 a1 + 3as b1 )γ13 .

(A.12)

Finally, after further simplifications and using the fact that ν1 = 3γ1 , bs F = 2a1 , b1 F = 2a1 (A.12) can be rewritten as ξ3 = 9(c1 − cs )γ13 + (3b1 x0 − 3bs x0 )γ1 .

(A.13)

Using the fact that γ12 = −2( ∇Hu , x0 / ∇Hu , (∂F1 /∂x)F1 ) and substituting for cs , c1 , bs , b1 the elements of Taylor expansion (4.1). Then, we get 

 

∇Hu , x0  ∂ 2 F1 2 ∂F1 2

∇Hu , x0  ξ3 = −3γ1 F + F1

∇Hu , (∂F1 /∂x)F1  ∂x 2 1

∇Hu , (∂F1 /∂x)F1  ∂x     

∇Hu , x0  ∂ 2 Fs 2

∇Hu , x0  ∂Fs ∂Fs 2 ∂F1 − x0 − x0 . (A.14) F + Fs +

∇Hu , (∂F1 /∂x)F1  ∂x 2 s ∂x ∂x

∇Hu , (∂F1 /∂x)F1  ∂x Expression (A.14) can be shown to be identically nought, noting that 1. 2. 3.

∂Fs ∂F1 1 1 = + F2 ∇Hu − F1 ∇Hu , ∂x ∂x 2 2   ∂ 2 F1 1 ∂F2 ∂ 2 Hu ∂F1 ∂ 2 Hu ∂ 2 Fs , = + + − − ∇H F ∇H F u 2 u 2 2 ∂x ∂x ∂x ∂x ∂x 2 ∂x 2       ∂F1 ∂F1 ∂F1 2 1 ∂Fs 2 F = F+ F2 ∇Hu F1 − F1 ∇Hu F1 . ∂x ∂x 2 ∂x ∂x

Since the above substitutions simplify (A.14) to nought the local map in the multisliding bifurcation case does not contain terms of O(ε3/2 ).

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Remark A.4. To get an expression for ξ4 , in (4.108) where ξ4 appears, we substitute (4.86), (4.90), (4.95) and (4.98) into (4.107). The resulting expression yields ξ4 = ψ4 + (ψ1 (γ3 − ν3 ) + ψ2 (γ2 − ν2 ) + ψ3 (γ1 − ν1 ))bs + (2(γ1 − ν1 )(γ3 − ν3 ) + (γ2 − ν2 )2 )as + (γ4 − ν4 )F + (2ψ1 (γ1 − ν1 )(γ2 − ν2 ) + ψ2 (γ1 − ν1 )2 )es + (γ1 − ν1 )4 + gs (γ1 − ν1 )ψ13 + 3(γ1 − ν1 )2 (γ2 − ν2 )cs + hs (γ1 − ν1 )2 ψ12 + js ψ1 (γ1 − ν1 )3 .

(A.15)

After substituting the values of χ1 , χ2 , χ3 , χ4 given in (4.91)–(4.94) in (4.103)–(4.107) for ψ1 , ψ2 , ψ3 , ψ4 , we can get the final expression for ξ4 .

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