July 30, 2016 12:29 WSPC/S0218-1274 1630019 International Journal of Bifurcation and Chaos, Vol. 26, No. 8 (2016) 1630019 (12 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127416300196 The Simplest Map with Three-Frequency Quasi-Periodicity and Quasi-Periodic Bifurcations Alexander P. Kuznetsov∗ and Yuliya V. Sedova† Kotel’nikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Zelenaya 38, Saratov 410019, Russian Federation ∗[email protected] †[email protected] Received February 16, 2016 We propose a new three-dimensional map that demonstrates the two- and three-frequency quasi- periodicity. For this map, all basic quasi-periodic bifurcations are possible. The study was real- ized by using Lyapunov charts completed by plots of Lyapunov exponents, phase portraits and bifurcation trees illustrating the quasi-periodic bifurcations. The features of the three-parameter structure associated with quasi-periodic Hopf bifurcation are discussed. The comparison with nonautonomous model is carried out. Keywords: Quasi-periodic oscillations; quasi-periodic bifurcations; torus map. 1. Introduction The basic ones are the next three bifurcations: Quasi-periodic oscillations are widespread in nature • the saddle-node bifurcation. Collision of stable and technology [Blekhman, 1988; Landa, 1996; and unstable tori leads to the abrupt birth of Pikovsky et al., 2001; Anishchenko et al., 2007a]. higher-dimensional torus. Examples of quasi-periodic behavior are found in • the quasi-periodic Hopf bifurcation. Torus of Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com electronics [Glazier & Libchaber, 1988], neurody- higher dimension is born softly. by NANYANG TECHNOLOGICAL UNIVERSITY on 08/08/16. For personal use only. namics [Izhikevich, 2000], astrophysics [Lamb et al., • the torus-doubling bifurcation. 1985], physics of lasers [Mel’nikov et al., 1991], geol- ogy [Didenko, 2011] and other areas of science. Currently there are no reliable identification algo- In the simplest case, the quasi-periodic oscil- rithms for such bifurcations and their search is exe- lations are characterized by the presence of two cuted by means of the characteristic dependence incommensurable frequencies. In the phase space, of Lyapunov exponents on parameter [Broer et al., attractors corresponding to such oscillations have 2008a, 2008b; Vitolo et al., 2011]. the form of invariant tori [Anishchenko, 1995; While examining the quasi-periodic bifurcations Shil’nikov et al., 2001; Broer & Takens, 2010]. the selection of the model for studying plays a There are more complicated cases of greater number crucial role. Some aspects can be studied based of incommensurable frequencies — then multifre- on nonautonomous systems. However, such sys- quency oscillations are observed which correspond tems form a separate, special class. As concern- to invariant tori of higher dimensions. Invariant ing the known autonomous models, the number of tori may undergo a variety of bifurcations. In this low-dimensional variants is not very large. Several case we speak about quasi-periodic bifurcations examples of suitable generators have been proposed [Broer et al., 2008a, 2008b; Vitolo et al., 2011]. recently. The first example, obviously, is a Chua’s 1630019-1 July 30, 2016 12:29 WSPC/S0218-1274 1630019 A. P. Kuznetsov & Y. V. Sedova circuit, which is described by specific piecewise- the three-dimensional system in the form of three linear characteristics [Matsumoto et al., 1987]. coupled in a ring of logistic maps with a specific cou- In [Nishiuchi et al., 2006] a generator based pling. Nevertheless, the analysis of the problem in on the modified Bonhoeffer–van der Pol sys- this paper seems inadequate. The noteworthy model tem is offered. Also we know a modification from [Broer et al., 2008a, 2008b; Vitolo et al., 2011] of climate Lorenz model — the Lorenz-84 low- was used for studying quasi-periodic bifurcations, order atmospheric circulation model [Shil’nikov but its construction is a complex formal procedure. et al., 1995]. Another radiophysical example imple- Although the model describes the very important mented as a four-dimensional system is a modified points, the discussed picture is still not quite com- Anishchenko–Astakhov generator [Anishchenko & plete. (The authors carried out only two-parameter Nikolaev, 2005; Anishchenko et al., 2006, 2007a]. analysis, they mainly focused on the description of In [Kuznetsov et al., 2010; Kuznetsov et al., 2013, resonance 1:5.) 2015; Kuznetsov & Stankevich, 2015], a family It can be noted that in [Anishchenko et al., of simple generators of quasi-periodic oscillations 1994] the coupled logistic maps with quasi-periodic described by three-dimensional models was sug- forcing are studied. The research concerns bifurca- gested and studied (including experiment). There tions and mechanisms of transition to chaos through have been studied cases of autonomous dynamics the destruction of three-dimensional torus. How- as well as on the generator under external forcing ever, not all the essential points have been inves- and dynamics of coupled oscillators [Anishchenko tigated in detail. Moreover, this model belongs to et al., 2006, 2007b; Kuznetsov et al., 2013; Stanke- the class of nonautonomous systems, i.e. systems vich et al., 2015]. with external forcing. It is known that flow systems (differential equa- Thus, there is a problem to construct tions) are quite difficult for research, especially autonomous models in the form of maps with the when concerning the study of delicate effects like minimum necessary dimension equal to three that quasi-periodic bifurcations. Therefore, it is natu- allows to study the properties of quasi-periodic ral to deal with a simpler model such as discrete dynamics and quasi-periodic bifurcations (we use maps. In this case, the image of two-frequency oscil- the world “simplest” in the paper title to emphasize lations in the phase space is an invariant curve. As the lowest dimension of our autonomous model). for flows, the same object emerges in the Poincar´e In the present paper, such a model is represented section of torus. (Thus frequently such invariant by a discrete analogue of quasi-periodic generator curves are also referred to as tori.) The higher is [Kuznetsov et al., 2013, 2015]. We focus on the the map dimension the higher may be dimension research of the structure of the parameter space of the torus. Most recently, appropriate investiga- focusing on the phenomena associated with quasi- tions of model maps have been undertaken. In paper periodic Hopf bifurcation. [Kuznetsov et al., 2012b] the four-dimensional sys- Int. J. Bifurcation Chaos 2016.26. Downloaded from www.worldscientific.com tem of two coupled universal maps with Neimark– 2. Torus Map Construction by NANYANG TECHNOLOGICAL UNIVERSITY on 08/08/16. For personal use only. Sacker bifurcation is examined. In [Sekikawa et al., 2014; Kamiyama et al., 2014], a four-dimensional Let us consider the generation of quasi-periodic map representing two logistic maps with delay was oscillations [Kuznetsov et al., 2013, 2015]: 2 4 2 studied. The authors of the papers [Hidaka et al., x¨ − (λ + z + x − βx )˙x + ω0x =0, 2015a; Hidaka et al., 2015b] undertook an analo- (1) z b ε − z − kx2. gous study of six-dimensional system in the form ˙ = ( ) ˙ of three logistic maps with delay. In [Adilova et al., The multiplier ahead of the derivativex ˙ contains 2013] the six-dimensional model which represents the parameter λ which characterizes the depth two coupled discrete versions of the R¨ossler system of the positive feedback in the oscillator, the non- is examined. linear term x2 stimulates the excitation of oscilla- However, the dimension of the above-mentioned tions and the term x4 responsible for the saturation systems is too large compared to the dimension of the oscillations at large amplitudes. The dynam- which is formally necessary to observe two- and ics of the variable z can occur linearly at a speed three-frequency quasi-periodicity. An exception is b or undergo the nonlinear saturation due to the the paper [Dementyeva et al., 2014] that studied term kx˙ 2. 1630019-2 July 30, 2016 12:29 WSPC/S0218-1274 1630019 Three-Frequency Quasi-Periodicity and Quasi-Periodic Bifurcations Here h is a discrete time step. Note that for the first equation we use a semi-explicit Euler scheme, i.e. we take the value of the variable y in (n + 1)th moment. This discretization usually leads to more physically correct models [Morozov, 2005]. 3. Properties of Torus Map with Quasi-Periodic Dynamics Let us study the obtained map. We use the same set of parameters as for Fig. 1 and will gradually increase the discretization parameter h. In the cen- ter of Fig. 2 a numerically calculated Lyapunov Fig. 1. Lyapunov chart for generating of quasi-periodic chart for system (3)atthevalueh =0.05 is shown. oscillations (1), b =1,ε =4,k =0.02, ω0 =2π. Different colors in the chart denote the following regions defined in accordance with the spectrum of Lyapunov exponents Λ1, Λ2, Λ3: The model (1) has an equilibrium x0 = y0 =0, z0 = ε. It is easily verified that this equilibrium (a) P — periodic regimes (cycles), Λ1 < 0, Λ2 < 0, undergoes Andronov–Hopf bifurcation AH at λ = Λ3 < 0; −ε leading to the birth of a limit cycle. By increas- (b) T2 — two-frequency quasi-periodicity, Λ1 =0, ing the parameter λ, the Neimark–Sacker bifurca- Λ2 < 0, Λ3 < 0; tion NS consisting in the birth of a two-dimensional (c) T3 — three-frequency quasi-periodicity, Λ1 =0, torus is possible. In Fig. 1, we demonstrate the Λ2 =0,Λ3 < 0; Lyapunov chart for system (1) where different col- (d) C — chaotic regimes, Λ1 > 0, Λ2 < 0, Λ3 < 0; ors indicate the areas of periodic regimes P ,the (e) HC — hyperchaotic regimes, Λ1 > Λ2 > 0, two-frequency quasi-periodicity T2 and chaos C.We Λ3 < 0; can see the curve of Neimark–Sacker bifurcation NS (f) D — a divergence of trajectories.