Piecewise smooth dynamical systems
A piecewise-smooth dynamical system (PWS) is a discrete- or continuous-time dynamical system whose phase space is partitioned in different regions, each associated to a different functional form of the system vector field.
A piecewise-smooth map is described by a finite set of smooth maps \[x \mapsto F_i(x,\mu), \quad \mbox{for} \quad x \in S_i, \quad x \in \mathbb{R}^{n}, \mu \in \mathbb{R}^{p}\] where \(\cap_i S_i = {D} \subset \mathbb{R}^n \) and each \(S_i\) has a non-empty interior. The intersection \(\Sigma_{ij}\) between the closure (set plus its boundary) of the sets \(S_i\) and \(S_j\) (that is, \(\Sigma_{ij}:=\bar{S}_i \cap \bar{S}_j\)) is either an \(\mathbb{R}^{(n-1)}\)-dimensional manifold included in the boundaries \(\partial S_j\) and \(\partial S_i\ ,\) or is the empty set. Each function \(F_i\) is smooth in both the state \(x\) and the parameter \(\mu\) for any open subset \(U \subset S_i\ .\)
A set \(\Sigma_{ij}\) for a piecewise-smooth map is usually termed a border or discontinuity boundary that separates regions of phase space where different smooth maps apply. Examples of piecewise-smooth one-dimensional maps are given inFigure 1.
A piecewise-smooth flow is given by a finite set of ODEs
\( \dot{x} = F_i(x,\mu), \quad \mbox{for} \quad x \in S_i, \quad x \in \mathbb{R}^{n}, \mu \in \mathbb{R}^{p} \)
where \(\cup_i S_i = {D} \subset \mathbb{R}^n\) and each \(S_i\) has a non-empty interior. The intersection \(\Sigma_{ij}:=\bar{S}_i \cap \bar{S}_j\) is either an \(\mathbb{R}^{(n-1)}\)-dimensional manifold included in the boundaries \(\partial S_j\) and \(\partial S_i\ ,\) or is the empty set. Each vector field \(F_i\) is smooth in both the state \(x\) and the parameter \(\mu\ ,\) and defines a smooth flow \(\Phi_i(x,t)\) within \(S_i\ .\) In particular, each flow \(\Phi_i\) is well defined on both sides of the boundary \(\partial S_j\ .\)
A non-empty border between two regions \(\Sigma_{ij}\) will be called a discontinuity set, discontinuity boundary or, sometimes, a switching manifold.
Different types of PWS systems
Piecewise-smooth dynamical flows can be classified according to their degree of smoothness, defined as follows.
The degree of smoothness (DoS) at a point \(x_0\) in a switching set \(\Sigma_{ij}\) of a piecewise-smooth ODE is the highest order \(r\) such the Taylor series expansions of \(\Phi_i(x_0,t)\) and \(\Phi_j(x_0,t)\) with respect to \(t\ ,\) evaluated at \(t=0\ ,\) agree up to terms of \(O(t^{r-1})\ .\) That is, the first non-zero partial derivative with respect to \(t\) of the difference \([\Phi_i(x_0,t)-\Phi_j(x_0,t)]|_{t=0}\) is of order \(r\ .\)
- Systems with DoS equal to zero have discontinuous states across the discontinuity boundaries in phase space. They are typically termed as impacting systems or piecewise-smooth hybrid systems. A classical example is that of an impact oscillator in mechanics (see Example 1 below).
- Systems with DoS equal to 1 have discontinuous vector fields and are usually known as Filippov systems (see Example 2 and related article in Scholarpedia) [Filippov, 1988][Leine et al., 2004]
- Systems with DoS equal to 2 (or larger) have vector fields with discontinuous higher derivatives and are therefore termed as piecewise-smooth continuous systems.
Bifurcations
Piecewise-smooth dynamical systems can exhibit most of the bifurcations also exhibited by smooth systems such as period-doublings, saddle-nodes, homoclinic tangencies, etc. provided that these occur away from the discontinuity boundaries. In addition to these, they can also exhibit some novel bifurcation phenomena which are unique to piecewise smooth systems or discontinuity-induced bifurcations (DIBs) [diBernardo et al., 2007]. In the Russian literature these novel transitions were given the collective name of \(C\)-bifurcations (the letter \(C\) stands for the first letter of the Russian word for ``sewing) to distinguish them from phenomena also observed in smooth systems [Feigin,1970; 1995].
A DIB in this sense is any transition observed in the system under investigation which can be explained in terms of interactions between its invariant sets and its switching manifolds in phase space. Thus, DIBs include interactions of fixed points, equilibria and limit cycles with the system switching manifolds. (Invariant tori and chaotic sets can also undergo DIBs but these have been little investigated in the scientific literature so far.)
Let us list some of the most commonly occurring types of codimension-one DIBs (see Figure 2).
- Border collisions of maps [Fig. 2(a)]. These are conceptually the simplest kind of DIBs and occur when, at a critical parameter value, a fixed point of a piecewise-smooth map lies precisely on a discontinuity boundary \(\Sigma\ .\) For maps with singularity of order one (i.e., locally piecewise-linear continuous), there is now a mature theory for describing the bifurcation that may result upon varying a parameter through such an event. Remarkably, the unfolding may be quite complex. See Example 4 below.
- Boundary equilibrium bifurcations [Fig. 2(b)]. The simplest kind of DIBs for flows occurs when an equilibrium point lies precisely on a discontinuity boundary \(\Sigma\ .\) In Filippov systems and hybrid systems with sticking regions, there is also the possibility of pseudo-equilibria, which are equilibria of the sliding or sticking flow but are not equilibria of any of the vector fields of the original system. There are thus possibilities where the equilibrium lies precisely on the boundary between a sliding or sticking region and a pseudo-equilibrium turns into a regular equilibrium (either under direct parameter variation or in a fold-like transition where both exist for the same sign of the perturbing parameter). There is also the possibility that a limit cycle may be spawned under parameter perturbation of the boundary equilibrium, in a Hopf-like transition. See Example 5.
- Grazing bifurcations of limit cycles [Fig. 2(c)]. One of the most commonly found DIBs in applications is caused by a limit cycle of a flow becoming tangent to (i.e., grazing) with a discontinuity boundary. One might naively think that this can be completely understood (upon taking an appropriate PoincarĂ© section that contains the grazing point) as a border collision. However, this is not necessarily the case, as one has to analyze carefully what happens to the flow in the neighborhood of the grazing point. In fact, one can derive an associated map (the, so-called, discontinuity map). But, the link between the singularity of the map and the degree of smoothness of the flow is a subtle one and typically leads to maps with square-root or \(o(\|x\|^{3 \over 2})\) terms.
- Sliding and sticking bifurcations [Fig. 2(d)]. There are several ways that an invariant set such as a limit cycle can do something structurally unstable with respect to the boundary of a sliding region in a Filippov system. The PoincarĂ© maps so derived have the property of typically being noninvertible in at least one region of phase space, owing to the loss of information backward in time inherent in sliding motion. See Example 2.
- Boundary intersection crossing/corner collision [Fig. 2(e)]. Another possibility for a codimension-one event in a flow is where an invariant set (e.g., a limit cycle) passes through the \((n-2)\)-dimensional set formed by the intersection of two different discontinuity manifolds \(\Sigma_1\) and \(\Sigma_2\ .\) See Example 3.
Examples
Example 1: impact oscillator
One of the simplest and most widely used piecewise-smooth models is that of an ideal single degree-of-freedom impact oscillator in nonsmooth mechanics [Brogliato, 1999]. In particular, consider the motion of a body in one spatial dimension, which is completely described by the position \(u(t)\) and velocity \(v(t)=\frac{d u}{d t}\) of its center of mass. Thus we think of this body as a single particle in space. When in free motion, we suppose that there is a linear spring and dashpot that attach this particle to a datum point so that its position satisfies the dimensionless differential equation
\( \frac{d^2 u}{d t^2}+ 2\zeta \frac{d u}{d t} + u = w(t), \quad \mbox{if} \quad u > \sigma. \)
Here, the mass and stiffness have been scaled to unity, \(2\zeta\) measures the viscous damping coefficient, and \(w(t)\) is an applied external force. We assume that motion is free to move in the region \(u > \sigma\ ,\) until some time \(t_0\) at which \(u = \sigma\) where there is an impact with a rigid obstacle. Then, at \(t=t_0\ ,\) we assume that \((u(t_0),v(t_0)):=(u_-,v_-)\) is mapped in zero time to \((u^+,v^+)\) via an impact law of the form
\( u^+ = u^- \quad \mbox{and} \quad v^+ = -r v^- \ ,\)
where \(0 < r < 1\) is Newton's coefficient of restitution. A representative bifurcation diagram of this system is shown in Figure 3 where the oscillator position is plotted stroboscopically against the frequency, \(\omega\ ,\) of the external forcing term (which is assumed to be sinusoidal).
It can be shown that the most fundamental mechanism organizing the observed complex behaviour is the grazing bifurcation of limit cycles causing the appearance and disappearance of various attractors. A typical grazing bifurcation phenomenon is shown in Figure 4(a). It can be shown that locally to the grazing event, the dynamics of the system can be described by a discrete-time mapping characterized by a square-root singularity. Hence, a ball of initial conditions is infinitely stretched in one direction as the impact velocity tends towards zero at grazing. This can cause dramatic changes in the system dynamics as, for example, the sudden transition from a periodic to a chaotic attractor shown in Figure 4(b).
Example 2: relay-feedback systems
A simple example of Filippov dynamics occurs in feedback relay control systems. The idea of using a switching action (or relay) has been widely employed in control engineering since the 1950s. Indeed, relay control has been used, for instance, in pulsed servomechanisms, tuning controllers in the process industry. More generally, relay systems play an important role in the theory of variable structure controllers [Utkin, 1992], of hybrid systems [Van der Schaft et al., 2000].
Although systems with a relay feedback have been studied for a long time (for example, in the work of Andronov and Flugge-Lotz from the 1950s and 1960s), the dynamics of these systems is not fully understood. It has been shown, for example, that even low-order relay feedback systems can exhibit more complex self-oscillations (either periodic or chaotic), which include segments of sliding motion [di Bernardo et al., 2001][Zhusubalyev et al., 2003]. Examples of engineering control systems with relay elements featuring chaotic behavior as well as quasi-periodic solutions are discussed in [Cook, 1985].
Here we consider a simple class of model problems corresponding to single-input--single-output, linear, time-invariant relay control systems with unit negative feedback of the output variable. Such problems can be written in the general form:
\[\tag{1} \dot x = Ax+Bu,\quad y=C^Tx,\quad u = -{\rm sgn}(y), \]
where the \(n\)-dimensional vector \(x \in R^n\) represents the system
state, the scalar \(y \in R\) is a measure of the output of the
system, and the discrete variable \(u \in \{-1,1 \}\) is the control
input. Also, \(A\in \mathbb{R}^{n\times n}\ ,\) \(B\in \mathbb{R}^{n\times
1}\) and \(C^T\in \mathbb{R}^{1\times n}\) are assumed to be constant
matrices. The input \(u\) and output \(y\) of the linear part are scalar
functions, whereas \(x\ ,\) the state vector, has \(n\geq 1\)
components. Furthermore, it is assumed that the system matrices are
given in observer canonical form; i.e.
\( A=\left(\begin{array}{ccccc} -a_1 & 1 & 0 & \cdots & 0\\ -a_2 & 0 & 1 & \cdots & 0\\ \vdots & & & \ddots&\vdots\\ -a_{n-1} & 0 & 0 & 0 & 1\\ -a_n & 0 & 0 & 0 & 0 \end{array} \right),\quad B=\left(\begin{array}{c} b_1 \\ b_2 \\ \vdots \\ b_4 \\ b_5 \end{array} \right),\quad C^T=\left(\begin{array}{c} 1\\0\\ \vdots\\0\\0\end{array}\right)^T. \)
Relay feedback control systems can exhibit several different types of sliding bifurcations. In particular, it has been shown that four main types of sliding bifurcations of limit cycles can occur in Filippov systems. Figure 5 depicts schematically each of these cases. The crossing-sliding bifurcation of a periodic solution of a three-dimensional relay feedback system of the form (1) is shown in Figure 6. (For more information see the related Scholarpedia article on Filippov systems.)
Example 3: DC/DC converter
DC--DC converters are circuits that are used to change one DC voltage to another. In the past, this was done by converting the DC voltage to an AC one, passing this through a transformer, and then transforming the resulting AC voltage back to a DC one. This procedure results in significant energy loss and rather bulky devices. To convert between voltages with domestic electronic devices, such as laptop computers, something more compact and with less energy loss is needed. The DC--DC converters frequently employed use electronic switches to convert from one DC voltage to another, with negligible energy loss. Significantly, such mechanisms can be implemented using small solid state devices. The use of switches means that DC--DC converters represent inherently non-smooth dynamical systems, which when driven beyond their designed operating limits can give rise to complex dynamics of the form studied in this book. In fact, there is already a rich literature on the many possible forms of dynamics of DC--DC converters, including rapidly switching periodic and chaotic motions (see [Banerjee & Verghese, 2001], [Zhusubalyev et al., 2003] and references therein).
A representative schematic of the well-known ideal DC--DC buck converter is shown in Figure 7. The corresponding mathematical model is a PWS system with degree of smoothness equal to unity (Filippov). DC--DC converters have been shown to exhibit complex behaviour including several types of bifurcations and chaos (see bifurcation diagram in Figure 8). The organizing DIB, causing the sudden jump to a large-amplitude chaotic evolution in this case is a corner-collision bifurcations causing the sudden transition from periodic to chaotic attractors (for more details on this diagram see [Banerjee, Verghese 2001] and references therein).
Example 4: a piecewise-linear continuous map
There is a considerable literature on the border-collision bifurcations of piecewise-linear maps (see [diBernardo et al., 2007] and references therein). Let us focus here on the particular case of one-dimensional maps that, without loss of generality, can be written in the form
\( x \mapsto f(x), \quad i=1,2 \)
where \( f = F_1=\alpha x + \mu\ ,\) if \( x\leq0\ ,\) and \( f=F_2=\beta x + \mu\ ,\) if \( x> 0\ ,\) depending on three real parameters \(\mu\ ,\) \(\alpha\) and \(\beta\ .\) The most interesting dynamics occurs for \(\alpha>0\) and \(\beta<0\ .\) Note that by introducing the rescaling \(\tilde{x}= x/|\mu|\ ,\) we can assume without loss of generality that \(\mu=\pm 1\ .\) The primary DIB in these maps is the border-collision bifurcation that occurs as \(\mu\) varies through zero, for which parameter value there is a trivial fixed point at \(x=0\ .\) Thus, treating \(\mu\) as the bifurcation parameter, we see that the dynamics is scale invariant; that is, all dynamics for \(\mu\) of a certain sign can be mapped trivially into the dynamics for \(|\mu|=1\ .\) Figure 9 shows some examples of possible bifurcation scenarios in this map. In particular, Figure 9(a) shows the transition from a one-periodic to a four-periodic attractor while Figure 9(b) the transition from a one-periodic to a chaotic attractor observed for slightly different parameter values. The most notable cascade is the period-adding shown in Figure 10 where periodic orbits of increasing periodicity alternate with bands of chaotic evolution. This is a typical phenomenon often observed in PWS systems which is clearly different from the period-doubling cascade to chaos often observed in smooth dynamical systems.
Example 5: boundary equilibrium bifurcations
As a final example, we consider the boundary equilibrium bifurcation occurring in the three-dimensional PWL flow of the form\[ \dot{x}= F(x) \]
where \( F(x)=N_1 x + M\mu\ ,\) if \(C^Tx > 0\) or \(F(x)=N_2 x + M\mu\ ,\) if \(C^Tx <0\ .\)
In this case, if
\( N_1=\left(\begin{array}{ccc} -0.8 & 1 & 0 \\ -0.57 & 0 & 1 \\ -0.09 & 0 & 0 \end{array} \right),\quad N_1=\left(\begin{array}{ccc} -0.1 & 1 & 0 \\ -0.2 & 0 & 1 \\ -60 & 0 & 0 \end{array} \right), \)
and
\( M=\left(\begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right),\quad C^T=\left(\begin{array}{c} 1\\0\\ 0\end{array}\right)^T, \)
the transition of the system equilibrium through the discontinuity boundary causes the appearance of a chaotic attractor as a result of the bifurcation at \(\mu=0\ .\) See Figure 11.
References
- S. Banerjee, G. Verghese (2001) Nonlinear Phenomena in Power Electronics: Bifurcations, Chaos, Control, and Applications, Wiley-IEEE Press
- M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk (2007) Piecewise-smooth dynamical systems: Theory and Applications, Springer doi:10.1007/978-1-84628-708-4_2.
- M. di Bernardo, M. I. Feigin, S. J. Hogan, and M. E. Homer (1999) Local Analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems. Chaos, Solitons and Fractals, 10:1881-1908
- M. di Bernardo, K. H. Johansson, and F. Vasca (2001) Self-Oscillations and Sliding in Relay Feedback Systems: Symmetry and Bifurcations. International Journal of Bifurcations and Chaos, 11(4):1121-1140
- B. Brogliato, Nonsmooth Mechanics (1999) Nonsmooth Mechanics, Springer doi:10.1007/978-1-4471-0557-2_7.
- P. A. Cook (1985) Simple feedback systems with chaotic behaviour. Systems & Control Letters, 6:223-227 doi:10.1016/0167-6911(85)90071-4.
- M. I. Feigin (1970) Doubling of the oscillation period with C-bifurcations in piecewise continuous systems, Prikladnaya Matematika i Mekhanika 34:861, in Russian.
- M.I. Feigin (1994) Forced Oscillations in Systems with Discontinuous Non-linearities, in Russian
- M.I. Feigin (1995) The increasingly complex structure of the bifurcation tree of a piecewise-smooth system, J. Appl. Maths. Mechs. 59:853 doi:10.1016/0021-8928(95)00118-2.
- A.F. Filippov (1988) Differential Equations with Discontinuous Righthand sides, Kluwier Academic Press doi:10.1007/978-94-015-7793-9.
- R.I. Leine, H. Nijmeijer (2004) Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Springer doi:10.1007/978-3-540-44398-8.
- M. Oestreich, N. Hinrichs, K. Popp, C. Budd (1997) Analytical and experimental investigation of an impact oscillator, Proc. of DETEC'97 - ASME Design Engineering Technical Conferences
- V. I. Utkin (1992) Sliding Modes in Control Optimization, Springer doi:10.1007/978-3-642-84379-2.
- A. J. Van der Schaft and J. M. Schumacher (2000) An introduction to Hybrid Dynamical Systems, Springer
- Z. Zhusubalyev, E. Mosekilde (2003) Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems, World Scientific
Internal references
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629. doi:10.4249/scholarpedia.1629.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358. doi:10.4249/scholarpedia.1358.
- James Murdock (2006) Unfoldings. Scholarpedia, 1(12):1904. doi:10.4249/scholarpedia.1904.
External links
See also
Bifurcations, Dynamical systems, Mappings, Periodic orbit, Smooth dynamics, Unfoldings