The Exact Nonlinear Dynamics. New Bifurcation Groups with Chaos and Rare Attractors

New nonlinear models which allows to obtain exact periodic, quasi-periodic and chaotic attractors and transients processes, are proposed. The idea of “exact” models has its corner-stone in description of restoring, damping and excitation forces only by constant forces on each its linear sub-system. Examples with a system with one potential well, two potential wells and pendulum systems are discussed. We suppose that it will be possible finding the exact simple analytical formulae for periodic, quasi-periodic and chaotic dynamics in different nonlinear models

Clusters of submerged subharmonic isles with rare attractors in nonlinear machine dynamics

This work is devoted to new nonlinear effects in the machine dynamics. This article presents a new type of subharmonic solutions which in the bifurcation diagrams have the form of the isles, all solutions of which are unstable except for small ranges of the bifurcation parameter at the edges of the isles. It is shown that such isles can form clusters, which we assume are typical topological formation for a wide class of system

Periodic Skeletons of Nonlinear Dynamical Systems in the Problems of Global Bifurcation Analysis

The construction of the periodic skeleton is a search for stable and unstable periodic regimes for a given parameter space of nonlinear dynamical periodicals systems. This stage of the main non-linear bifurcation theory, which is designed for a global analysis of nonlinear dynamical systems and the state of the parameter space that allows for complete bifurcation analysis, build complex bifurcation group and discover new previously unknown solutions

Bifurcation analysis and rare attractors in driven damped pendulum systems

The paper reports the complete bifurcation analysis of the driven damped pendulum systems by the new method of complete bifurcation groups (MCBG). Construction of complete bifurcation groups is based on the method of stable and unstable periodic regimes continuation on a parameter. Global bifurcation analysis of the driven damped pendulum systems allows determination of new bifurcation groups with rare attractors and chaotic regime

Bifurcation analysis and rare attractors in driven damped pendulum systems

The paper reports the complete bifurcation analysis of the driven damped pendulum systems by the new method of complete bifurcation groups (MCBG). Construction of complete bifurcation groups is based on the method of stable and unstable periodic regimes continuation on a parameter. Global bifurcation analysis of the driven damped pendulum systems allows determination of new bifurcation groups with rare attractors and chaotic regime

Bifurcation analysis by method of complete bifurcation groups of the driven system with two degrees of freedom with three equilibrium positions

This paper devoted to application of the new method of complete bifurcation groups (MCBG), which shows very good results in single-degree-of-freedom tasks, for global bifurcation analysis of systems with two degrees-offreedom on example of two-mass chain system with symmetrical elastic characteristic with two potential wells between masses. It is shown, that using of the MCBG allows to implement global bifurcation analysis of nonlinear oscillators with 2 DOF, and to find new nonlinear effects, bifurcation groups, and unknown before periodic and chaotic regime

Bifurcation analysis by method of complete bifurcation groups of the driven system with two degrees of freedom with three equilibrium positions

This paper devoted to application of the new method of complete bifurcation groups (MCBG), which shows very good results in single-degree-of-freedom tasks, for global bifurcation analysis of systems with two degrees-offreedom on example of two-mass chain system with symmetrical elastic characteristic with two potential wells between masses. It is shown, that using of the MCBG allows to implement global bifurcation analysis of nonlinear oscillators with 2 DOF, and to find new nonlinear effects, bifurcation groups, and unknown before periodic and chaotic regime

Bifurcation analysis by method of complete bifurcation groups of the driven system with two degrees of freedom with three equilibrium positions

This paper devoted to application of the new method of complete bifurcation groups (MCBG), which shows very good results in single-degree-of-freedom tasks, for global bifurcation analysis of systems with two degrees-offreedom on example of two-mass chain system with symmetrical elastic characteristic with two potential wells between masses. It is shown, that using of the MCBG allows to implement global bifurcation analysis of nonlinear oscillators with 2 DOF, and to find new nonlinear effects, bifurcation groups, and unknown before periodic and chaotic regime

Bifurcation analysis by method of complete bifurcation groups of the driven system with two degrees of freedom with three equilibrium positions

This paper devoted to application of the new method of complete bifurcation groups (MCBG), which shows very good results in single-degree-of-freedom tasks, for global bifurcation analysis of systems with two degrees-offreedom on example of two-mass chain system with symmetrical elastic characteristic with two potential wells between masses. It is shown, that using of the MCBG allows to implement global bifurcation analysis of nonlinear oscillators with 2 DOF, and to find new nonlinear effects, bifurcation groups, and unknown before periodic and chaotic regime

Paradoxes of increasing linear damping in the nonlinear driven oscillators

The work is devoted to the systematic study of periodic and chaotic forced oscillations. Recently, on the basis of method of complete bifurcation groups new nonlinear effects were found in driven damped systems with various nonlinearities of elastic restoring forces. Construction of complete bifurcation groups is based on the method of stable and unstable periodic regimes continuation on parameter. Aim of the work – to study new nonlinear effects induced by varying linear dissipation in following dynamical systems with typical nonlinear restoring forces: symmetric trilinear and quadratic, bilinear, cubic with asymmetry, Duffing, pendulum. The work presents new qualitative and quantitative results of nonlinear dynamics in the systems with increasing linear dissipatio