Strange Nonchaotic Attractors Research Articles

On some generalizations of skew-shifts on

In this paper we investigate maps of the two-torus $\mathbb{T}^{2}$ of the form $T(x,y)=(x+\unicode[STIX]{x1D714},g(x)+f(y))$ for Diophantine $\unicode[STIX]{x1D714}\in \mathbb{T}$ and for a class of maps $f,g:\mathbb{T}\rightarrow \mathbb{T}$, where each $g$ is strictly monotone and of degree 2 and each $f$ is an orientation-preserving circle homeomorphism. For our class of $f$ and $g$, we show that $T$ is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two $T$-invariant graphs. One of the graphs is a strange non-chaotic attractor whose basin of attraction consists of (Lebesgue) almost all points in $\mathbb{T}^{2}$. Only a low-regularity assumption (Lipschitz) is needed on the maps $f$ and $g$, and the results are robust with respect to Lipschitz-small perturbations of $f$ and $g$.

Non-smooth saddle-node bifurcations II: Dimensions of strange attractors

We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of a continuous curve and controlling the geometry of the latter, we determine their Hausdorff and box-counting dimension and show that these take distinct values. Moreover, the same approach allows us to describe the topological structure of the attractors and to prove their minimality.

Autonomous strange nonchaotic oscillations in a system of mechanical rotators

We investigate strange nonchaotic self-oscillations in a dissipative system consisting of three mechanical rotators driven by a constant torque applied to one of them. The external driving is nonoscillatory; the incommensurable frequency ratio in vibrational-rotational dynamics arises due to an irrational ratio of diameters of the rotating elements involved. It is shown that, when losing stable equilibrium, the system can demonstrate two- or three-frequency quasi-periodic, chaotic and strange nonchaotic self-oscillations. The conclusions of the work are confirmed by numerical calculations of Lyapunov exponents, fractal dimensions, spectral analysis, and by special methods of detection of a strange nonchaotic attractor (SNA): phase sensitivity and analysis using rational approximation for the frequency ratio. In particular, SNA possesses a zero value of the largest Lyapunov exponent (and negative values of the other exponents), a capacitive dimension close to 2 and a singular continuous power spectrum. In general, the results of this work shed a new light on the occurrence of strange nonchaotic dynamics.

Dynamic bifurcation and strange nonchaos in a two-frequency parametrically driven nonlinear oscillator

A periodically forced series LCR circuit with Chua’s diode as a nonlinear element exhibits slow passage through Hopf bifurcation. This slow passage leads to a delay in the Hopf bifurcation. The delay in this bifurcation is a unique quantity and it can be predicted using various numerical analysis. We find that when an additional periodic force is added to the system, the delay in bifurcation becomes chaotic which leads to an unpredictability in bifurcation delay. Further, we study the bifurcation of the periodic delay to chaotic delay in the slow passage effect through strange nonchaotic delay. We also report the occurrence of strange nonchaotic dynamics while varying the parameter of the additional force included in the system. We observe that the system exhibits a hitherto unknown dynamical transition to a strange nonchaotic attractor. With the help of Lyapunov exponent, we explain the new transition to strange nonchaotic attractor and its mechanism is studied by making use of rational approximation theory. The birth of SNA has also been confirmed numerically, using Poincaré maps, phase sensitivity exponent, the distribution of finite-time Lyapunov exponents and singular continuous spectrum analysis.

A numerical estimate of the regularity of a family of Strange Non-Chaotic Attractors

We estimate numerically the regularities of a family of Strange Non-Chaotic Attractors related with one of the models studied in (Grebogi et al., 1984) (see also Keller, 1996). To estimate these regularities we use wavelet analysis in the spirit of de la Llave and Petrov (2002) together with some ad-hoc techniques that we develop to overcome the theoretical difficulties that arise in the application of the method to the particular family that we consider. These difficulties are mainly due to the facts that we do not have an explicit formula for the attractor and it is discontinuous almost everywhere for some values of the parameters. Concretely we propose an algorithm based on the Fast Wavelet Transform. Also a quality check of the wavelet coefficients and regularity estimates is done.

Regular Nonchaotic Attractors with Positive Plural

The study of the strange nonchaotic attractors is an interesting topic, where the dynamics are neither regular nor chaotic (the word chaotic means the positive Lyapunov exponents), and the shape of the attractors has complicated geometry structure, or fractal structure. It is found that in a class of planar first-order nonautonomous systems, it is possible that there exist attractors, where the shape of the attractors is regular, the orbits are transitive on the attractors, and the dynamics are not chaotic. We call this type of attractors as regular nonchaotic attractors with positive plural, which are different from the strange nonchaotic attractors, attracting fixed points, or attracting periodic orbits. Several examples with computer simulations are given. The first two examples have annulus-shaped attractors. Another two examples have disk-shaped attractors. The last two examples with externally driven terms at two incommensurate frequencies have regular nonchaotic attractors with positive plural, implying that the existence of externally driven terms at two incommensurate frequencies might not be the sufficient condition to guarantee that the system has strange nonchaotic attractors.

An analytical study on the synchronization of strange non-chaotic attractors

In this paper we present an analytical study on the synchronization dynamics observed in unidirectionally-coupled quasiperiodically-forced systems that exhibit Strange Non-chaotic Attractors (SNA) in their dynamics. The SNA dynamics observed in the uncoupled system is studied analytically through phase portraits and poincare maps. A difference system is obtained by coupling the state equations of similar piecewise linear regions of the drive and response systems. The mechanism of synchronization of the coupled system is realized through the bifurcation of the eigenvalues in one of the piecewise linear regions of the difference system. The analytical solutions obtained for the normalized state equations in each piecewise linear region of the difference system has been used to explain the synchronization dynamics though phase portraits and timeseries analysis. The stability of the synchronized state is confirmed through the Master Stability Function. An explicit analytical solution explaining the synchronization of SNAs is reported in the literature for the first time.

Coexistence of strange nonchaotic attractors and a special mixed attractor caused by a new intermittency in a periodically driven vibro-impact system

We focus on the coexistence of strange nonchaotic attractors (SNAs) and a novel mixed attractor in a periodically driven three-degree-of-freedom vibro-impact system with symmetry. SNAs are characterized by the local largest Lyapunov exponent and the phase sensitivity property. The Poincare map P is the twofold composition of a six-dimensional implicit map Q, implying the symmetry of the vibro-impact system. Since the map Q can capture two conjugate attractors, it is used to investigate the dynamics of the system. With a suitable parameter combination, the Poincare map P of the vibro-impact system exhibits Neimark–Sacker–pitchfork (NS-P) bifurcation. It is shown that dense phase-locking regions exist in a small parameter interval near this NS-P bifurcation point. Three types of attractors alternate in this small interval: two conjugate phase-locked periodic attractors, two conjugate SNAs and a special type of mixed attractor. As the force frequency $$\omega $$ is increased gradually, many phase-locking regions disappear, and the coexistence of two conjugate SNAs takes place instead, which is accompanied by a quick decrease in the width of phase-locking. If two conjugate strange nonchaotic limit sets are suddenly embedded in a chaotic one, a special mixed attractor is caused by a new intermittency accompanied by symmetry restoring bifurcation. This symmetry restoring bifurcation is the result of the collision between two conjugate strange nonchaotic limit sets and a symmetric limit set.

Dynamics of a Class of Nonautonomous Lorenz-Type Systems

The dynamical properties of a kind of Lorenz-type systems with time-varying parameters are studied. The time-varying ultimate bounds are estimated, and a simple method is provided to generate strange attractors, including both the strange nonchaotic attractors (SNAs) and chaotic attractors. The approach is: (i) take an autonomous system with an ultimate bound from the Lorenz family; (ii) add at least two time-varying parameters with incommensurate frequencies satisfying certain conditions; (iii) choose the proper initial position and time by numerical simulations. Three interesting examples are given to illustrate this method with computer simulations. The first one is derived from the classical Lorenz model, and generates an SNA. The second one generates an SNA or a Lorenz-like attractor. The third one exhibits the coexistence of two SNAs and a Lorenz-like attractor. The nonautonomous Lorenz-type systems present more realistic models, which provide further understanding and applications of the numerical analysis in weather and climate predication, synchronization, and other fields.

Applicability of the permutation largest slope entropy to strange nonchaotic attractors

We extend the permutation largest slope entropy (PLSE) to the detection of strange nonchaotic attractors (SNA). The initial time series derived from a quasi-periodically forced chaotic map is first transformed into a series of symbols using the order relation. These symbols are thereafter combined into m-length words to obtain quantized ordinal matrices (QOM). Finally, the PLSE is applied to the series of QOM for detecting changes in the behavior of the system. Simulation results show that the QOM transform allows to control the complexity of the time series, hence to reduce the entropy of tori and SNA and to increase that of chaotic motions. This result allows to highlight transitions between the three types of motions, thus showing a nonzero entropy for SNA which nevertheless is smaller than that of chaotic dynamics. The approach is effective for the detection of different routes to SNA from time series.

Effects of quasiperiodic forcing in epidemic models

We study changes in the bifurcations of seasonally driven compartmental epidemic models, where the transmission rate is modulated temporally. In the presence of periodic modulation of the transmission rate, the dynamics varies from periodic to chaotic. The route to chaos is typically through period doubling bifurcation. There are coexisting attractors for some sets of parameters. However in the presence of quasiperiodic modulation, tori are created in place of periodic orbits and chaos appears via finite torus doublings. Strange nonchaotic attractors (SNAs) are created at the boundary of chaotic and torus dynamics. Multistability is found to be reduced as a function of quasiperiodic modulation strength. It is argued that occurrence of SNAs gives an opportunity of asymptotic predictability of epidemic growth even when the underlying dynamics is strange.

Strange nonchaotic self-oscillator

An example of strange nonchaotic attractor (SNA) is discussed in a dissipative system of mechanical nature driven by a constant torque applied to one of the elements of the construction. So the external force is not oscillatory, and the system is autonomous. Components of the motion with incommensurable frequencies emerge due to the irrational ratio of the sizes of the involved rotating elements. We regard the phenomenon as strange nonchaotic self-oscillations, and its existence sheds new light on the question of feasibility of SNA in autonomous systems.

Synchronization of strange non-chaotic attractors via unidirectional coupling of quasiperiodically-forced systems

In this paper, we present a numerical investigation on the robust synchronization phenomenon observed in a unidirectionally-coupled quasiperiodically-forced simple nonlinear electronic circuit system exhibiting strange non-chaotic attractors (SNAs) in its dynamics. The SNA obtained in the simple quasiperiodic system is characterized for its SNA behavior. Then, we studied the nature of the synchronized state in unidirectionally coupled SNAs by using the Master-Slave approach. The stability of the synchronized state is studied through the master stability functions (MSF) obtained for coupling different state variables of the drive and response system. The property of robust synchronization is analyzed for one type of coupling of the state variables through phase portraits, conditional lyapunov exponents and the Kaplan-Yorke dimension. The phenomenon of complete synchronization of SNAs via a unidirectional coupling scheme is reported for the first time.

Non-smooth saddle-node bifurcations III: Strange attractors in continuous time

Non-smooth saddle-node bifurcations give rise to minimal sets of interesting geometry built of so-called strange non-chaotic attractors. We show that certain families of quasiperiodically driven logistic differential equations undergo a non-smooth bifurcation. By a previous result on the occurrence of non-smooth bifurcations in forced discrete time dynamical systems, this yields that within the class of families of quasiperiodically driven differential equations, non-smooth saddle-node bifurcations occur in a set with non-empty C2-interior.

Amorphic complexity

We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For instance, it gives positive value to Denjoy examples on the circle and Sturmian subshifts, while being zero for all isometries and Morse–Smale systems.After discussing basic properties and examples, we show that amorphic complexity and the underlying asymptotic separation numbers can be used to distinguish almost automorphic minimal systems from equicontinuous ones. For symbolic systems, amorphic complexity equals the box dimension of the associated Besicovitch space. In this context, we concentrate on regular Toeplitz flows and give a detailed description of the relation to the scaling behaviour of the densities of the p-skeletons. Finally, we take a look at strange non-chaotic attractors appearing in so-called pinched skew product systems. Continuous-time systems, more general group actions and the application to cut and project quasicrystals will be treated in subsequent work.

A note on the fractalization of saddle invariant curves in quasiperiodic systems

The purpose of this paper is to describe a new mechanism of destruction of saddle invariant curves in quasiperiodically forced systems, in which an invariant curve experiments a process of fractalization, that is, the curve gets increasingly wrinkled until it breaks down. The phenomenon resembles the one described for attracting invariant curves in a number of quasiperiodically forced dissipative systems, and that has received the attention in the literature for its connections with the so-called Strange Non-Chaotic Attractors. We present a general conceptual framework that provides a simple unifying mathematical picture for fractalization routes in dissipative and conservative systems.

Analytical Study and Experimental Confirmation of SNA Through Poincaré Maps in a Quasiperiodically Forced Electronic Circuit

Quasiperiodically forced series LCR circuit with simple nonlinear element is studied analytically and experimentally. To the best of our knowledge, this is the first time that strange nonchaotic attractors (SNAs) are studied analytically. From the explicit analytical solution, the bifurcation process is shown. With a single negative conduction region of the nonlinear element two routes namely, Heagy–Hammel and fractalization routes to the birth of SNA are identified. The analytical analysis are confirmed by laboratory hardware experiments. In addition, for the first time, a detailed stroboscopic Poincaré map is generated experimentally for two different frequencies, for the above two routes, which clearly confirm the presence of SNAs in these two routes. Also, from the experimental data of the corresponding attractors, we quantitatively confirm the presence of SNAs through singular-continuous spectrum analysis. The analytical results as well as experimental observations are characterized qualitatively in terms of phase portraits, Poincaré map, power spectrum, and sensitivity dependance on initial conditions.

Bifurcations and strange nonchaotic attractors in a phase oscillator model of glacial–interglacial cycles

Glacial–interglacial cycles are large variations in continental ice mass and greenhouse gases, which have dominated climate variability over the Quaternary. The dominant periodicity of the cycles is ∼40 kyr before the so-called middle Pleistocene transition between ∼1.2 and ∼0.7 Myr ago, and it is ∼100 kyr after the transition. In this paper, the dynamics of glacial–interglacial cycles are investigated using a phase oscillator model forced by the time-varying incoming solar radiation (insolation). We analyze the bifurcations of the system and show that strange nonchaotic attractors appear through nonsmooth saddle–node bifurcations of tori. The bifurcation analysis indicates that mode-locking is likely to occur for the 41 kyr glacial cycles but not likely for the 100 kyr glacial cycles. The sequence of mode-locked 41 kyr cycles is robust to small parameter changes. However, the sequence of 100 kyr glacial cycles can be sensitive to parameter changes when the system has a strange nonchaotic attractor.

Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with experimental and analytical confirmation

A new route to strange nonchaotic attractor (SNA), known as multilayered bubble route to SNA, has been identified in a quasiperiodically forced series LCR circuit with a simple nonlinear element. Upon increasing the system control parameter, the stable orbits of the torus become unstable, which induces formation of bubbles in the neighborhood of the resonating region of the torus. We have observed three tori with three smooth branches in the Poincaré map which gradually loose their smoothness and ultimately approach bubble formation, and then approach fractal behavior via SNAs before the onset of chaos. The bubbles gradually enlarge and subsequently another three layers of bubbles are formed as a function of the control parameter. The layers get increasingly wrinkled as a function of the control parameter, resulting in the creation of SNAs which are characterized by Poincaré maps. The multilayered bubble route to SNA is then confirmed by experimental Poincaré maps and explicit analytical solution is developed to further confirm it. Numerically observed bubbling route is characterized qualitatively in terms of phase portraits, power spectrum and further characterized quantitatively, by singular-continuous spectrum analysis, phase sensitivity measure, distribution of finite time Lyapunov exponents, largest Lyapunov exponent and its variance.

Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with a simple nonlinear element

A new route to strange nonchaotic attractors (SNAs), known as multilayered bubble route to SNA, has been identified in a quasiperiodically forced series $$\textit{LCR}$$ circuit with a simple nonlinear element. Upon increasing the control parameter, the stable orbits of the torus become unstable, which induces formation of bubbles in the neighbourhood of the resonating region of the torus. We have observed three tori with three smooth branches in the Poincare map which gradually loose their smoothness and ultimately approach bubble formation, and then approach fractal behaviour via SNAs before the onset of chaos. The bubbles gradually enlarge and subsequently another three layers of bubbles are formed as a function of the control parameter. The layers get increasingly wrinkled as a function of the control parameter, resulting in the creation of SNAs which are charaterized by Poincare maps. Apart from the multilayered bubble route to SNA, Heagy–Hammel and fractalization routes to SNA have also been numerically observed and are characterized qualitatively interms of phase portraits, power spectrum. The above mentioned three routes are further characterized quantitatively, by singular-continuous spectrum analysis, phase sensitivity measure, distribution of finite time Lyapunov exponents, largest Lyapunov exponent and its variance.