Antiresonance in switched systems with only unstable modes

Maurizio Porri∗ Department of Mechanical and Aerospace Engineering and Department of Biomedical Engineering, New York University, Tandon School of Engineering, Brooklyn, New York 11201, USA

Russell Jeter and Igor Belykh† Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia, 30302-410, USA (Dated: October 10, 2020) Antiresonance is a key property of dynamical systems that leads to the suppression of oscillations at select frequencies. We present the surprising example of a switched system that alternates between unstable modes, but exhibits antiresonance response for a wide range of switching frequencies. Through mathematical analysis, we elucidate the stabilization mechanism and characterize the range of antiresonant frequencies for periodic and stochastic switching. The demonstration of this new physical phenomenon opens the door for a new paradigm in the study and design of switched systems.

PACS numbers: 05.45.-a, 46.40.Ff, 02.50.Ey, 45.30.+s

Introduction. Switched dynamics are pervasive in the- this question by oering the rst example of a switched oretical physics, neuroscience, and engineering [1]. For system composed of unstable modes that displays a stable example, the temporal patterning of interactions within response in a nite range of switching frequencies. Our active matter systems discontinuously evolves in time system has an unstable average that hinders stability in as their comprising units change their spatial organiza- the fast-switching limit, and the instability of both of the tion [25]. Likewise, synchronization in brain networks modes hampers stability for slow-switching frequencies. emerges from sporadic, on-o synaptic interactions be- Governing equations. We consider a two-dimensional tween spiking and bursting neurons [6]. Just as the func- nonlinear switched system with state vector z(t) = tion of several natural systems is based on switched dy- (x(t), y(t)), alternating in time between two possible namics, so too switch logic is ubiquitous in the design of modes, namely, engineering systems, such as converters and communica- tion networks [7, 8]. z˙(t) = Fs(t)(z) (1) Stability is arguably the most fundamental property where the binary signal s(t) switches between 0 and 1 to of these systems [9, 10]. Over the last two decades, a select either of the nonlinear functions F0 or F1. Initially, number of analytical studies have documented a strong we focus on a periodic signal of period T with a duty cycle sensitivity of stability to variations in the switching fre- δ, such that the switch is on for δT units of time and is quency [1115], which has also been registered in applica- o for (1 − δ)T units of time; later, we explore stochastic tions to complex systems [1621]. Switched systems that switching. In order to achieve the desired response, we alternate between stable modes can display an unstable design the modes as behavior at selected switching frequencies [22, 23]. This phenomenon can be viewed as an internal resonance [24],  x2y  x˙ = x − 2c wherein the switching frequency does not allow the dy-  2 2 s = 0 : x + y , namics in each of the stable modes to decay before the x3 − xy2  y˙ = y + c  2 2 onset of a new switch. x + y (2)  y3 − x2y Heraclitus' theory of the unity of opposites has sel-  x˙ = x − c  2 2 dom defeated our intuition of physical processes, from the s = 1 : x + y , xy2 particle-wave duality in quantum mechanics [25] to the  y˙ = y + 2c matter/anti-matter asymmetry problem [26]. Antireso-  x2 + y2 nance continues to be extensively investigated in acous- where c is a positive constant that determines the cou- tics [27], optics [28], biology [29], and quantum physics pling between the two state components; when possible, [30], with a focus on smooth, stable systems. On the we omit the dependence on time. other hand, antiresonance of switched systems with un- For c = 0, each of the modes in (2) reduces to an un- stable modes has never been documented. stable node with decoupled state components. As c in- Is it possible to induce antiresonance in the form of sta- creases, the two components become nonlinearly coupled, ble response of a system switching between two unstable forming a degenerate xed point, shown in Fig. 1. The modes? In this Letter, we provide a positive answer to lack of dierentiability of the vector eld at the origin 2

2 1.0 1.0 1.0

0.5 0.5 1 0.5

y 0.0 y 0.0 y y 0 0.0 -0.5 -0.5 -0.5 -1 -1.0 -1.0 -1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -2 -2 -1 0 1 2 -1.0 -0.5 0.0 0.5 1.0 x x x x

FIG. 1. Streamlines of the two unstable modes of the switched FIG. 2. Trajectories of the periodically-switched system in (1) system given in (2) for , with color indicating the inten- c = 4 for c = 4 and δ = 0.5 and dierent values of T (left): T = 0.1 sity of the vector eld for (left) and (right). s = 0 s = 1 (solid red), T = 2 (green), and T = 5 (dashed red). Initial condition is (1, 0) for all cases. Streamlines of the average system in (1) for c = 4 and δ = 0.5, with color indicating the does not allow one to characterize the xed point follow- intensity of the vector eld (right). ing standard practice. Although unstable, the xed point has features of an unstable node and a stable focus. A the nonlinear interplay between the intrinsic dynamics of typical trajectory will rst rotate about the origin, reduc- each mode at intermediate switching frequencies. ing its distance from it like a stable focus, and then it will Antiresonance at intermediate switching. Surprisingly, escape to innity along the principal unstable direction there could be a wide range of intermediate switching like an unstable node. The principal unstable directions periods that induce the stability of the switched system are the - and -axis for the rst ( ) and second y x s = 0 so that the origin becomes a stable focus, as shown in ( ) modes, respectively. The interplay between these s = 1 Fig. 2. The stabilization mechanism relies on switching contrasting features is the key for the antiresonance of slow enough to be attracted toward the origin during the the switched system. rotation, but fast enough to avoid reaching the unstable Slow- and fast-switching limits. In the limit case T  1 principal directions. For and , numerical the time-scale for each mode the dynamics of the c = 4 δ = 0.5 simulations suggest that the range of that guarantees switched system is unstable, as it would spend a large T convergence toward the origin is The fraction of time in one of the unstable modes before it T ∈ (1.18, 4.26). emergence of this antiresonant response is completely could re-switch to the other [12]. After spiraling toward open-loop, whereby switching takes place without any the origin for almost an entire quadrant, each trajectory knowledge of the present state of the system. In the fol- will approach the principal unstable direction of the mode lowing, we provide a mathematical proof for the existence and travel away from the origin. Switching will cause the of this antiresonance window. trajectory to experience a sudden turn and after each pe- Mathematical analysis of stability. System (1) with riod, the distance from the origin will increase, as demon- modes (2) can be conveniently written in polar coordi- strated in Fig. 2. nates, by dening x = r cos θ and y = r sin θ, so that An unstable response is also registered for the fast- switching case, in which T  1, as shown in Fig. 2. This r˙ = r + cr (2s − 1) sin θ cos θ, (4a) result can be explained by examining the average system [13], which is obtained by averaging (2) θ˙ = c (1 − s) cos2 θ + cs sin2 θ. (4b)

(3δ − 2)x2y − δy3 This representation indicates that the nonlinearity is x˙ = x + c , limited to the angular coordinate and that the system x2 + y2 (3) (3δ − 1)xy2 + (1 − δ)x3 has a triangular structure, with the angular coordinate y˙ = y + c . evolving independent of the radial one. This structure x2 + y2 is amenable to a closed-form solution, which we present This system has an unstable focus, without any principal after examining stability under periodic switching to gain unstable direction. Except for the origin, all the trajec- insight into more complex switching processes. tories spiral out to innity. To study the stability of the periodically-switched sys- Although often used in the analytical treatment of tem, we perform the change of variable ρ = 1/r, so that switched systems [12, 13], the slow- and fast-switching (4a) becomes limits oer a close representation of the switched system ρ˙ = −ρ − cρ (2s − 1) sin θ cos θ. (5) when the switching period is much larger and smaller than the time-scale of the individual modes. Predict- This change of variables maps the origin to innity and ing the onset of an antiresonance requires the study of vice-versa, thereby turning the study of antiresonance 3 into the study of resonance. Most importantly, a Carte- 1 sian representation of the transformed system, through 0 , yields the following linear switched ζ = (ρ cos θ, ρ sin θ) -1 system: Re ( λ ) -2  −1 −s(t)c -3 ζ˙(t) = A(t)ζ(t), with A(t) = , (6) 0 1 2 3 4 5 6 7 (1 − s(t))c −1 T System (6) periodically switches between two linear sys- 1 0 tems determined by two Hurwitz matrices, A0 when the switch is o (s = 0) and A1 when the switch is on (s = 1), -1 such that Re ( λ ) -2 −1 0  −1 −c -3 A = , and A = . (7) 0 1 2 3 4 5 6 7 0 c −1 1 0 −1 T These matrices have repeated eigenvalues λ = −1, which 1 yield a stable degenerate node in both modes. 0 The stability of a linear periodically-switched system -1 Re ( λ ) (6) can be examined using classical Floquet theory [31]. -2 More specically, given the transition matrix Φ(t, τ), we -3 can compute the T -periodic function P (t) and the con- 0 1 2 3 4 5 6 7 stant (possibly complex) matrix R T

1 R = log[Φ(T, 0)], FIG. 3. Real part of the characteristic exponents of the linear T (8) periodically-switching (6) as functions of T for δ = 0.5 and P (t) = Φ(t, 0) exp(−Rt). dierent values of c: c = 2 (top), c = 4 (middle), and c = 6 (bottom). Red regions identify values for which the real parts Through P (t), we dene a Lyapunov transformation that of both the exponents are negative and green regions values converts the original switched system into an ancillary, for which one is positive. time-invariant one with state matrix equal to R [32]. The entire stability analysis of the original switched system (1) reduces to the study of the eigenvalues of R, the so- positive, thereby triggering the instability of the ancil- called characteristic exponents. lary system. Such an instability, in turn, manifests into The linear switched system is (uniformly and exponen- the stable dynamics of the original nonlinear system. tially) asymptotically stable if and only if all the charac- There is a wide range of values of T for which the teristic exponents have negative real parts. The compu- original system is asymptotically stable (uniformly and tation of the characteristic exponents requires the calcu- exponentially) with a basin of attraction that comprises lation of the two-dimensional monodromy matrix Φ(T, 0), the whole phase space, except of a principal instability given by direction. Such a direction corresponds to the eigenvec- tor of R that is associated with the eigenvalue with neg- (9) Φ(T, 0) = exp(δT A1) exp[(1 − δ)TA0]. ative real part. In general, for T > 2/(cpδ(1 − δ)), R has two real eigenvectors that coincide at a phase By taking the logarithm of the matrix and computing its − arctan p(1 − δ)/δ for T = 2/(cpδ(1 − δ)) and then eigenvalues, we analytically determine the two character- separate one from each other as T increases. In the limit istic exponents, as functions of , , and , as shown in δ c T of T → ∞, they approach 0 and −π/2, with the lat- Fig. 3. Specically, we determine ter corresponding to the eigenvalue with the smaller real part, that is, the principal unstable direction of the orig- log 2 1 λ = −1 − + log [2 − cT (δ(1 − δ)cT ± inal system [34]. Within the window of stability of the T T original system, the principal direction will thus vary in i pδ(1 − δ)(−4 + δ(1 − δ)c2T 2) . (10) an interval contained between − arctan pδ/(1 − δ) and −π/2; for example, for c = 2 and δ = 0.5, it varies be- In agreement with simulations in Fig. 2, small and tween −1.06 and −1.45. large values of the switching periods result into an unsta- Extension to stochastic switching. While the proposed ble dynamics, that is, negative characteristic exponents analytical treatment strictly applies to periodic switch- of the ancillary, linear system [33]. For suciently large ing, insight into the antiresonance of system (1) to more values of c, one of the characteristic exponents becomes complex switching processes could be garnered by ex- 4

1.0 study Markovian switching process [22]. We simulate 0.8 (11) and (12) for 1,000 switches and 1,000 realizations 0.6 of the stochastic switching sequences. The initial angle p 0.4 θ(0) is drawn uniformly at random from 0 to 2π and con- 0.2 vergence is ascertained by comparing the initial with the 0.0 nal radius. 0 1 2 3 4 5 6 7 Predictably, for small values of , the coupling between T c the state components does not suce to create the spi- 1.0 0.8 raling trajectories that are required for the stabilization 0.6 of the origin, thereby hindering antiresonance through p 0.4 stochastic switching, as shown in Fig. 4. For larger val- 0.2 ues of c, we observe the onset of antiresonance through 0.0 stochastic switching, but the range of antiresonance pe- 0 1 2 3 4 5 6 7 riods considerably narrows with respect to the periodic T case [35]. This phenomenon is due to the nonlinearity of 1.0 the system, which reduces the set of favorable switching 0.8 sequences as the period increases. 0.6

p Conclusions. Stabilization of unstable modes via pe- 0.4 riodic or stochastic perturbations is a critical problem 0.2 in the study of dynamical systems, with applications in 0.0 0 1 2 3 4 5 6 7 physical, biological, and engineering systems. Just as an oscillating suspension can stabilize the inverted con- T guration of the classical Kapitza's pendulum [36], vi- FIG. 4. Probability of converging to the origin for stochastic brational stabilization is used in hovering insects and versus periodic switching in system (1) as functions of T for apping wing micro-air vehicles [37]. Likewise, syn- δ = 0.5 and dierent values of c: c = 2 (top), c = 4 (middle), chronization in switched networks of discrete- [38] and and c = 6 (bottom). Black dots correspond to stochastic continuous-time oscillators [39] can be stabilized through switching, and solid lines to periodic switching. Color cod- non-fast switching in multiple, disconnected intervals of ing follows Fig. 3: red and green lines identify values of T the switching period, termed as windows of opportunity. for which periodic switching leads to instability and stability, respectively. Common to these examples is the co-existence of stable modes. To date, there was no example of stabilization of amining the polar representation in (4). Equation (4b) a switched system composed of only unstable modes admits the following solution: through periodic or stochastic switching. Here, we have closed this gap, by providing the rst example of antireso- s = 0 : θ(t) = arctan[c(t − ti) + tan θ(ti)], (11) nant response of a two-dimensional switched system with s = 1 : θ(t) = −arccot[c(t − ti) + cot θ(ti)], only two unstable modes. Remarkably, the average dy- namics of the switched system is unstable, thereby pro- where t is the instant of switching. i hibiting stabilization through fast-switching, and the un- With knowledge of the time evolution of the angular stable dynamics of each mode ensures instability at low coordinate in (11), we can solve (4a) for the radial coor- switching frequencies. Yet, the system acquires stability dinate to obtain in a range of intermediate switching frequencies. s 2 The key property of each of the modes that aords r(t) 1 + tan θ(ti) s = 0 : = exp(t − t ) , stabilization is a special, hybrid type of unstable dynam- r(t ) i 1 + (c(t − t ) + tan θ(t ))2 i i i ics, characterized by a xed point that has features of s 2 r(t) 1 + cot θ(ti) an unstable node and a stable focus. The latter enables s = 1 : = exp(t − ti) 2 , r(ti) 1 + (c(t − ti) − cot θ(ti)) the system with a latent stability property that can be (12) eectively explored while switching in the antiresonance where we made evident the linearity by factoring the ini- frequency range. This rich behavior is rooted in spectral tial condition on the left-hand-sides of the equations. properties, which are known to change continuously with Equations (11) and (12) are valid for any choice of the respect to perturbations of the system parameters. As a system parameters and switching process. As an example result, this hybrid dynamics is robustly present across a of stochastic switching, we consider the case where we re- wide parameter range and could be extended to higher- tain the same switching instants as the periodic process, dimensional systems. In loose terms, the stabilization but the value of s(t) is drawn with equal probability be- mechanism can be explained using the analogy of jug- tween 0 and 1; the approach could also be adapted to gling a hot potato between two hands: the system must 5 spend some time in one mode (one hand) to take advan- Nonlinearity 31, 1331 (2018). tage of its latent stability property (the time it takes to [16] M. Frasca, A. Buscarino, A. Rizzo, L. Fortuna, and acquire a rm control of the potato) and then it must S. Boccaletti, Physical Review Letters 100, 044102 leave this mode before its inherent instability manifests (2008). (before the hand gets burned). [17] T. E. Gorochowski, M. di Bernardo, and C. S. Grierson, Physical Review E 81, 056212 (2010). The simplicity and richness of our system make a com- [18] D. J. Stilwell, E. M. Bollt, and D. G. Roberson, SIAM pelling case for it to become a nonlinear dynamics text- Journal on Applied Dynamical Systems 5, 140 (2006). book example of impossible stable dynamics emerg- [19] A. Mondal, S. Sinha, and J. Kurths, Physical Review E ing from antiresonant switching between unstable modes. 78, 066209 (2008). We hope that our system can nd a home close to some [20] P. So, B. C. Cotton, and E. Barreto, Chaos: An In- illustrious, two-dimensional examples of counterintuitive terdisciplinary Journal of Nonlinear Science 18, 037114 nonlinear dynamics, like Peano's paradox in a system (2008). [21] P. C. Bresslo and S. D. Lawley, Physical Review E 96, that is unstable but convergent to the origin [40]. 012129 (2017). This work was supported by the National Science [22] S. D. Lawley, J. C. Mattingly, and M. C. Reed, Com- Foundation (USA) under grants Nos. CMMI-1561134 munications in Mathematical Sciences 12, 1343 (2014). and CMMI-1932187 (to M.P.), Nos. DMS-1909924 (to [23] M. Porri, R. 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