Abstract
The investigation of regular and irregular patterns in nonlinear oscillators is an outstanding problem in physics and in all natural sciences. In general, regularity is understood as tantamount to periodicity. However, there is now a flurry of works proving the existence of “antiperiodicity”, an unfamiliar type of regularity. Here we report the experimental observation and numerical corroboration of antiperiodic oscillations. In contrast to the isolated solutions presently known, we report infinite hierarchies of antiperiodic waveforms that can be tuned continuously and that form wide spiralshaped stability phases in the control parameter plane. The waveform complexity increases towards the focal point common to all spirals, a key hub interconnecting them all.
Introduction
Since the realization that the centuryold causal determinism of Boskovic and Laplace needed amendment due to the discovery of deterministic chaos, much effort was devoted during the last few decades to study the intricacies involving the interplay between regular and irregular oscillations produced by nonlinear systems. Traditionally, the emphasis has been in the study of the irregular chaotic oscillations. However, there is a whole class of remarkably regular oscillations that has so far escaped attention, namely antiperiodic oscillations. A quantity x(t) is said to evolve periodically when x(t + T) = x(t), where T is the period between repetitions. The less familiar class of antiperiodic oscillations that we study here obeys the relation x(t + T) = −x(t). Clearly, every antiperiodic pattern with antiperiod T is necessarily a periodic pattern with period 2T. Trivial examples of antiperiodicity are the trigonometric solutions of the harmonic oscillator , , which satisfy the textbook identities sin(t + π) = −sin t, or cos(t + π) = −cos t, where π is the antiperiod and 2π is the period of the oscillations. The system of differential equations defining these trivial solutions is linear and too simple to be flexible enough for a number of applications: it generates only a single wave pattern and allows no changes to it other than rather uninteresting amplitude and/or frequency changes.
Antiperiodicity is known in physics. For instance, Matsubara^{1,2} used this concept in the 1950s when calculating expectation values of physical observables of a quantum field theory at finite temperature, in the requirement that all bosonic and fermionic fields be periodic and antiperiodic, respectively. During the last two decades, antiperiodic problems were spotted and studied extensively in a number of fields. For example, for firstorder ordinary differential equations, the classic criterion of Massera^{3} for periodicity was extended for antiperiodic boundary value problems by Y. Chen^{4,5}. From antiperiodic boundary conditions, the interest shifted to the study of antiperiodic oscillations. Antiperiodicity was investigated for the heat equation^{6}, for secondorder Duffinglike^{7} and pendulumlike^{8} oscillators, and several other systems^{9,10}. Antiperiodic wavelets were discussed by T. Chen^{11}. Antiperiodic solutions for higherorder nonlinear ordinary differential equations are known but for a few specific systems^{8,12}. Smooth antiperiodic solutions are also known for quasilinear partial differential equations^{13}. These works contain references to additional papers dealing with antiperiodic solutions discovered for a plethora of nonlinear equations.
Results
So far, the knowledge accumulated about antiperiodic oscillations dealt substantially with providing existence proofs of isolated solutions for loworder equations under specific conditions, or for higherorder equations with somewhat contrived adhoc forms. Furthermore, the majority of flows studied involve driven (i.e. nonautonomous) systems. All this means that the study of antiperiodicity is still in its infancy and only a few sparse antiperiodic solutions are known for some particular equations.
Here, we report the experimental observation and numerical corroboration of apparently infinite sequences of such elusive antiperiodic oscillations in an autonomous electronic circuit (Fig. 1). Our key discovery is that the complexification of currents and voltages in the circuit occurs mediated by infinite families of selfsustained antiperiodic oscillations that can be tuned continuously as a function of the physical reactances involved. Nowadays, periodic waveforms are the rule in nonlinear systems while oscillators capable of supporting families of tunable antiperiodic waveforms with an unbounded number of peaks within an oscillation are completely unheard of. We detected tunable antiperiodicity while studying the complicated mechanisms underlying the progressive wave pattern complexification generated by the electronic circuit during perioddoubling and periodadding cascades of bifurcations.
As depicted in Fig. 1, our circuit involves two active elements, a nonlinear resistor R and a negative conductance G. It descends from a circuit considered by Chua and Lin^{14} and Stoupoulos et al.^{15}. Our implementation contains a slight variation introduced to account for saturation effects of the real operational amplifier used in G. All phenomena observed with our modified circuit can be also observed in a circuit with ideal elements. For more details about the circuit, see Methods, below.
Figure 2 presents typical experimental signals obtained for the voltage v_{1}(t) on the capacitor C_{1} as a function of the resistance R_{1} while maintaining all other parameters constant. From this figure we recognize the characteristic signature of antiperiodic oscillations, namely where T/2 is the antiperiod and T is the period of the oscillation. From Fig. 2 it is easy to recognize that an antiperiodic function with antiperiod T is necessarily a periodic function with period 2T. Identical antiperiodicity is detected in measurements of v_{2}, i_{1}, or i_{2} (not shown). For all variables, we could follow the signal up to quite large number of spikes.
Figure 3 shows for v_{1}, v_{2}, i_{1}, i_{2} the first few of an infinite sequence of antiperiodic oscillations. Such patterns were obtained from numerical integration when varying two parameters (given in the leftmost column) simultaneously. These oscillations have an odd number of spikes. Furthermore, the amplitude of the temporal evolutions of v_{2} (in the second column from the left) labeled s_{0}, s_{2}, is slightly smaller than the ones labeled s_{1} and s_{3}. The same is true for i_{2} in the rightmost column.
To understand how antiperiodic patterns depend on R_{1} and R_{2} we performed an additional numerical experiment, studying the variation of the number of peaks systematically on a 2400 × 2400 = 5.76 × 10^{6} rectangular grid of equally spaced parameter points. The circuit equations were integrated with a standard fourthorder RungeKutta algorithm with fixed timestep h = 10^{−6} s, starting computations always from a fixed initial condition v_{1} = 8 V, v_{2} = −5 V, i_{1} = −1 mA, i_{2} = 3 mA. The first 80 × 10^{5} integration steps were discarded as transient. The chaotic/periodic/antiperiodic nature of solutions was determined and recorded in socalled isospike diagrams^{16}: after the transient we integrated for an additional 80 × 10^{5} timesteps and recorded extrema (maxima and minima) of a given variable of interest, up to 800 extrema, counting the number of peaks and checking whether for repetitions. Such highresolution computations are numerically very demanding and, therefore, were performed on a SGI Altix cluster of 1536 highperformance processors running during a period of several weeks to compute many stability diagrams, three of them presented in Fig. 4.
Figure 4 shows stability diagrams indicating how the number of peaks within one period of v_{2}(t) selforganize in control space. As indicated by the colorbars, a palette of 17 colors is used to represent the number of peaks in one period of the oscillations. Patterns with more than 17 peaks are plotted by recycling the 17 basic colors modulo 17, namely assigning to them a colorindex given by the remainder of the integer division of the number of peaks by 17. Multiples of 17 are given the index 17. Black represents “chaos” (i.e. lack of numerically detectable periodicity/antiperiodicity), white and gold colors mark constant (i.e. nonoscillatory) solutions, if any, having respectively nonzero or zero amplitudes of the variable under consideration.
The stability diagrams in Fig. 4 show that selfsustained nonchaotic (i.e. periodic or antiperiodic) oscillations manifest themselves by forming a main spiral phase converging to a focal hub and paving the control space with a multitude of colors. The colors indicate how the number of peaks increases and where exactly do they change along the spiral. From Fig. 4 one also sees that the number of peaks in v_{2}(t) increases steadily by 2 after every turn towards the focal hub. Furthermore, while the period seems to accumulate to a definite limiting value, the number of peaks grows apparently without bound. From Fig. 4 it is also possible to recognize the presence of several additional secondary spirals sandwiched between every turn of the main spiral. From additional magnifications (not shown here) it is possible to recognize unambiguously an apparently unbounded hierarchy of such secondary spirals, that get thinner and thinner as one approaches more and more the focal hub. This hierarchical organization of spirals is similar to the one found recently in other physical oscillators^{17}.
In Fig. 4(b) one sees that the edges, or legs, composing the main spiral display a certain angularity that becomes smoother and smoother near the hub, as it is clear from Fig. 4(c). This nonuniformity has to do, we believe, with the highdimensionality of the parameter hypersurface defined by the flow: although motivated by experiments^{15}, the parameters which were held fixed simply do not produce an optimal section of the hypersurface so as to reflect more regular and symmetric spirals. An optimization of all parameters involved would consume enormous amount of time and, therefore, was not attempted. It is important to mention, however, that in addition to the R_{2} × R_{1} control parameter plane, we also observed antiperiodic oscillations to induce similar spirals in other control planes, e.g. C_{1} × R_{1} and C_{1} × C_{2}. Since resistances are easier to control experimentally than capacitances we preferred to focus here on the R_{2} × R_{1} plane. Antiperiodic patterns evolve continuously when parameters are suitably tuned along spirals. Furthermore, not only the period and the number of peaks but also the amplitude of the oscillations vary regularly when spiraling towards the hub.
Discussion
What is the mechanism responsible for the regular addition of peaks observed along the spiral? We find that such complexification occurs through continuous deformations of the wave patterns, analogously as described recently for a CO_{2} laser with feedback^{18} a system that, however, does not show antiperiodicity and has no spirals in its control space. For antiperiodicity to subsist indefinitely along the spirals as patterns get more and more complicated, it is necessary that wave pattern deformations occur in pairs, simultaneously. While oddspiked antiperiodic oscillations were observed along the spiral, not all oddspiked oscillations lead to antiperiodic oscillations. For instance, the wide onespike phase seen on the top right corner of Fig. 4(a) is characterized by periodic oscillations (not by antiperiodicity). The same is true for the infinite peakdoubling cascades k × 2^{m} issuing from a region of oscillations with k peaks.
Thus far our description was based on counting the number of peaks in the voltage v_{2}(t). What happens when other variables are used to count peaks? Do the peaks of all four variables evolve in unison? Additional numerical work (not presented here) shows that, although each variable produces parameter subdivisions, phases, having their own idiosyncrasies, the picture described for v_{2}(t) remains basically unchanged. Changes in the number of peaks may, or not, require a complete turn along the spiral. Furthermore, the precise location where changes occur may vary slightly, depending on the variable considered. An attempt to uncover the systematics behind all possible changes would only make sense after solving the aforementioned parameter optimization problem. This optimization, of course, is not needed for our present purpose of reporting the discovery of infinite families of the elusive antiperiodic oscillations.
In what sort of systems can one expect to find antiperiodicity? The dissipative flow governing our circuit can be written compactly as dx/dt = f(x), where x = (v_{1}, v_{2}, i_{1}, i_{2}), and the four components of f(x) are given explicitly in the caption of Fig. 1. From these components we recognize that the flow is oddsymmetric, namely that f(−x) = −f(x). We have also observed similar antiperiodicity scenarios in another circuit, containing two diodes as a nonlinear resistance, and in a few flows constructed adhoc to display this symmetry. This makes us believe that antiperiodicity should be present for a whole class of nonlinear oscillators having such symmetry. Thus, oddsymmetry of the flow seems to be a key ingredient for the onset of antiperiodic oscillations although, as already mentioned, not every regular oscillation with oddnumber of peaks in oddsymmetric flows is necessarily antiperiodic. General mathematical conditions concerning periodicity are known^{19}. It would be nice to extend them to take antiperiodicity into account, something that does not seem to be completely trivial to do. For a given set of parameters, the ability to predict whether oscillations will be antiperiodic or periodic seems to be a quite hard mathematical problem that needs to be investigated.
In conclusion, we presented experimental and numerical evidence of the existence of infinite families of tunable antiperiodic oscillations in a reallife physical oscillator and extended what is presently known about such remarkably interesting oscillations. We believe tunable families of antiperiodic oscillations to be a generic feature for an extended class of oscillators. Antiperiodicity remains unexplored in nonlinear dynamics, is potentially interesting for applications and certainly deserves further study.
Methods
The active nonlinear elements R and G of the circuit in Fig. 1 are represented by the following odd symmetric vi characteristics
Here, parameters are functions of the electronic components. So, E_{b} depends of the output voltage swing, V_{sat}, of the operational amplifier, and of its input voltage, V_{cc}. The slopes G_{a} and G_{b} also depend on the nonzero forward voltage V_{γ} of the diodes, modeled here as an ideal diode and a battery. Unless otherwise stated, we follow previous works^{14,15} and fix L_{1} = 9.8 mH, L_{2} = 20.6 mH, C_{2} = 2C_{1} = 12 nF, E_{1p} = 2.5 V, E_{2p} = 11 V, E_{b} = 7.5 V, G_{a} = −0.7 mS, G_{b} = −0.5 mS, G_{c} = 3.35 mS, G_{aa} = −0.5 mS, and G_{bb} = 0.5 mS.
Our circuit uses fast commuting 1N4148 diodes and TL084 operational amplifiers. The chip of the opamps consists of four amplifiers such that the circuit could be easily mounted on a board and the nonlinear resistances R and G implemented using nearly identical operational amplifiers. The 1N4148 has a maximum recovery time of 4 ns and is usually employed in highfrequency applications. The input voltage of the operational amplifier was maintained constant along the experiment at V_{cc} = 15.0 ± 0.6 V. The other relevant parameters are V_{γ} = 0.65 ± 0.06 V and V_{sat} = 12.7 ± 0.9 V.
References
 1.
Matsubara, T. A new approach to quantumstatistical mechanics. Prog. Theor. Phys. 14, 351–378 (1955).
 2.
Coleman, P. Many Body Physics (Cambridge University Press, Cambridge, 2013).
 3.
Massera, J. L. The existence of periodic solutions of systems of differential equations. Duke J. Math. 17, 457–475 (1950).
 4.
Chen, Y. Q. On Massera's theorem for antiperiodic solution. Adv. Math. Sci. Appl. 9, 125–128 (1999).
 5.
Liu, B. An antiperiodic LaSalle oscillation theorem for a class of functional differential equations. J. Comp. Appl. Math. 223, 1081–1086 (2009).
 6.
Okochi, H. On the existence of antiperiodic solutions to a nonlinear evolution equation associated with odd subdifferential operators. J. Func. Anal. 91, 246–258 (1990).
 7.
Pu, H. & Yang, J. Existence of antiperiodic solutions with symmetry for some highorder ordinary differential equations. Bound. Val. Prob. 108 (2012), and references therein.
 8.
Chen, T., Liu, W. & Yang, C. Antiperiodic solutions for Liénardtype differential equation with pLaplacian operator. Bound. Val. Prob. 194824 (2010), and references therein.
 9.
Girardi, M. & Matzeu, M. Existence of periodic solutions for some second order Hamiltonian systems. Rend. Lincei Mat. Appl. 18, 1–9 (2007).
 10.
Cheng, Y., Cong, F. & Hua, H. Antiperiodic solutions for nonlinear evolution equations. Adv. Diff. Eq. 165, (2012), and references therein.
 11.
Chen, H. L. Antiperiodic wavelets. J. Comp. Math. 14, 32–39 (1996).
 12.
Chen, T. & Liu, W. Antiperiodic solutions for higherorder Liénard type differential equation with pLaplacian operator. Bull. Korean Math. Soc. 49, 455–563 (2012), and references therein.
 13.
Nakao, M. & Okochi, H. Antiperiodic solution for t_{tt} − (σ(u_{x}))_{x} − u_{xxt} = f(x, t). J. Math. Anal. Appl. 197, 796–809 (1996).
 14.
Chua, L. & Lin, G. N. Canonical realization of Chua's circuit family. IEEE Trans. Circ. Syst. 37, 885–902 (1990).
 15.
Stoupoulos, I. N., Miliou, A. N., Valaristos, A. P., Kyprianidis, I. M. & Anagnostopoulos, A. N. Crisis induced intermittency in a fourthorder autonomous electric circuit. Chaos Sol. Frac. 33, 1256–1262 (2007), and references therein.
 16.
Freire, J. G., Pöschel, T. & Gallas, J. A. C. SternBrocot trees in spiking and bursting of sigmoidal maps. Europhys. Lett. 100, 48002 (2012).
 17.
Vitolo, R., Glendinning, P. & Gallas, J. A. C. Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows. Phys. Rev. E 84, 016216 (2011), and references therein.
 18.
Junges, L. & Gallas, J. A. C. Frequency and peak discontinuities observed in selfpulsations of a CO_{2} laser with feedback. Opt. Commun. 285, 4500–4506 (2012).
 19.
Gallas, J. A. C. On the origin of periodicity in dynamical systems. Physica A 283, 17–23 (2000).
Acknowledgements
J.G.F. was supported by FCT, Portugal through the PostDoctoral grant SFRH/BPD/43608/2008. C.C. and A.C.M. acknowledge support from CSIC and PEDECIBA, Uruguay. J.A.C.G. thanks support from CNPq, Brazil. This work was supported by the Deutsche Forschungsgemeinschaft through the Cluster of Excellence Engineering of Advanced Materials. All bitmaps were computed at the CESUPUFRGS clusters.
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Affiliations
Institute for Multiscale Simulations, FriedrichAlexanderUniversität, D91052 Erlangen, Germany
 Joana G. Freire
 , Thorsten Pöschel
 & Jason A. C. Gallas
Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, Montevideo, Uruguay
 Cecilia Cabeza
 , Arturo Marti
 & Jason A. C. Gallas
Departamento de Física, Universidade Federal da Paraíba, 58051970 João Pessoa, Brazil
 Joana G. Freire
 , Arturo Marti
 , Thorsten Pöschel
 & Jason A. C. Gallas
Centro de Estruturas Lineares e Combinatórias, Universidade de Lisboa, 1649003 Lisboa, Portugal
 Joana G. Freire
 & Jason A. C. Gallas
Instituto de Altos Estudos da Paraíba, Rua Infante Dom Henrique 1001801, 58039150 João Pessoa, Brazil
 Jason A. C. Gallas
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Contributions
C.C., A.M. and J.A.C.G. conceived and designed the experiments. C.C. performed the experiments. J.G.F. and J.A.C.G. performed the simulations. J.A.C.G. wrote the main manuscript. All authors discussed the results and reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Jason A. C. Gallas.
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Further reading

1.
Periodicity hubs and spirals in an electrochemical oscillator
Journal of Solid State Electrochemistry (2015)

2.
Discontinuous Spirals of Stable Periodic Oscillations
Scientific Reports (2013)
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