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Background and Objectives: The Josephson effect is widely used for both generating and receiving very high frequency signals. The approaches and methods of nonlinear dynamics are commonly used: construction of phase portraits, bifurcation analysis, Kuramoto model approach, etc. Usually large arrays of identical junctions are considered. However, the non-identity of junctions leads to new and interesting effects. A very popular model is a chain of junctions connected via an RLC circuit. In this case, two types of non-identity are possible – in terms of critical currents IN through the junctions and in terms of the value rN of junction resistances. An increase in the number of junctions from two to three leads to the possibility of quasiperiodic dynamics with invariant tori of dimensions both two and three and to a complex structure of the parameter space. Materials and Methods: In this paper, we will mainly consider three junctions that are not identical in terms of resistance. As the main research tool, we will use the method of construction of Lyapunov exponent charts. Within the framework of this method, the type of dynamics of the system is determined by the signature of the spectrum of Lyapunov exponents. The parameter plane is scanned and the types of modes are identified at each point. The method is effective in that it allows one to study all types of possible regimes and fine details of the parameter space arrangement. Results: For the analysis of the dynamics of non-identical Josephson junctions, the method of Lyapunov exponent charts is effective. With its help, the regions of periodic regimes, regimes of two-frequency and three-frequency quasiperiodicity, chaos have been revealed. As a rule, the regions of two-frequency quasiperiodicity have the form of bands of different widths immersed in the region of three-frequency quasiperiodicity, which form the structure of the Arnold’s resonant web. Conclusion: The boundaries of the regions of two-frequency quasiperiodicity are the lines of saddle-node bifurcations of invariant tori. As the area of chaos increases, the Arnold’s resonant web can collapse. Changing the type of external circuit coupling the junctions does not fundamentally affect the dynamics of the system.

Background and Objectives: Quasi-periodic oscillations are widespread in nature and technology. In the phase space, quasi-periodic oscillations with a different number of incommensurable frequencies correspond to invariant tori of different dimensions. Multiparametric analysis of high-dimensional systems is quite difficult. The way out of this situation can be the transition from systems with continuous time to discrete maps. Materials and Methods: In the paper we use a discretization method for the transition from a continuous-time system (two coupled quasi-periodic generators) to a new high-dimensional map. The time derivatives are replaced by finite differences. There is a new additional parameter corresponding to the discretization step, with the variation of which the system can demonstrate new interesting properties. Results: A new high-dimensional map has been obtained by the discretization method of differential equation system of coupled quasi-periodic generators. For this map, charts of Lyapunov exponents have been constructed in the plane of the frequency detuning of generators and the coupling magnitude. The existence of invariant tori of different dimensions has been demonstrated. Graphs of Lyapunov exponents and Fourier spectra have been presented. The evolution of maps with an increase of the discretization parameter has been investigated, the destruction of high-dimensional tori has been demonstrated. The influence of noise of different intensity has been studied. Conclusion: The two-parameter Lyapunov analysis of the new high-dimensional map made it possible to identify regions of invariant tori of different dimensions, up to fivefrequency ones. The map demonstrates Fourier spectra characteristic of quasi-periodicity of increasing dimension. Quasi-periodic bifurcations of invariant tori and an Arnold resonance web based on invariant tori of different dimensions have been observed. With the growth of the discretization parameter, the destruction of high-dimensional tori occurs. An increase in the discretization parameter leads to the destruction of high-dimensional tori with an increase in the noise intensity.

Background and Objectives: The basic model of study is the simplest three - dimensional map with two-frequency and three-frequency quasiperiodicity at adding of noise. The main objective is to examine the effect of noise on the quasiperiodic Hopf bifurcation of the 3-torus birth. Materials and Methods: To study the torus map in the presence of noise we use such numerical methods as computing of Lyapunov exponents, calculation of Fourier spectra, drawing of attractor portraits. Results: Quasi-periodic bifurcations under the influence of noise occupy certain intervals in the parameter, but their main classification features (equality or not of the corresponding Lyapunov exponents) are preserved at the qualitative level. Conclusion: We considered the effect of noise on the simplest system with two- and three-frequency quasiperiodicity. The three-frequency quasiperiodicity is preserved at certain noise amplitudes, but then turns into a two-frequency one. In the Fourier spectra, this process develops according to the scenario of "blurring" the noise components of the corresponding spectral components.

Background and Objectives: Chaotic behavior is one of the fundamental properties of nonlinear dynamical systems, including maps. Chaos can be most easily and reliably diagnosed using the largest Lyapunov exponent, which will be positive for the chaotic mode. Unlike flow dynamical systems, the presence of zero Lyapunov exponent in the spectrum is not an obligatory condition for maps. The zero exponent in the spectrum of a map will indicate the possibility of embedding such a map in a flow. In the framework the present paper, using as an example a three-dimensional discrete Lorenz-84 model, it is shown that there can appear chaotic attractors whose Lyapunov exponent spectrum contains one positive, one zero, and one negative exponents. Such a specific attractor represents the production of a two-dimensional torus and the Hénon attractor and was called the Quasi-periodic Hénon attractor. A scenario of development of such kind behavior is an open problem. Materials and Methods: The discrete Lorenz-84 oscillator obtained by discretizing the three-dimensional flow Lorenz-84 model is considered as an object of the present research. The dynamics of the map is studied numerically. The main analysis is carried out using the charts of dynamical regimes based on the calculation of Lyapunov exponents. Lyapunov exponents were calculated by the Benettin method with Gramm-Schmidt orthogonalization. Results: The scenario of occurrence of the Quasi-periodic Hénon attractor via a cascade of invariant curve doubling bifurcations is described. Conclusion: We study the discrete Lorenz-84 oscillator, which is a three-dimensional map (three-dimensional diffeomorphism). In the map the possibility of implementing a steady state of equilibrium, an invariant curve, a torus-attractor, and chaos with zero Lyapunov exponent in the spectrum was shown. It is also demonstrated that the chaotic mode with zero Lyapunov exponent, the Quasi-periodic Hénon attractor, can appear as a result of a cascade of doubling bifurcations of the invariant curve. Typical structures on the parameter plane, such as CrossRoad-Area, Spring-Area, whose base mode is an invariant curve, are illustrated. These structures and chaotic oscillations can arise on the basis of various invariant curves.