Dynamics at infinity and other global dynamical aspects of Shimizu–Morioka equations (original) (raw)
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Nonlinear Dynamics, 2015
We present a global dynamical analysis of the following quadratic differential systeṁ x=a(y−x),ẏ = dy − xz,ż =−bz + f x 2 + gx y, where (x, y, z) ∈ R 3 are the state variables and a, b, d, f, g are real parameters. This system has been proposed as a new type of chaotic system, having additional complex dynamical properties to the well-known chaotic systems defined in R 3 , alike Lorenz, Rössler, Chen and other. By using the Poincaré compactification for a polynomial vector field in R 3 , we study the dynamics of this system on the Poincaré ball, showing that it undergoes interesting types of bifurcations at infinity. We also investigate the existence of first integrals and study the dynamical behavior of the system on the invariant algebraic surfaces defined by these first integrals, showing the existence of families of homoclinic and heteroclinic orbits and centers contained on these invariant surfaces.
The Lorenz System Has a Global Repeller at Infinity
Journal of Nonlinear Mathematical Physics, 2011
It is well known that the celebrated Lorenz system has an attractor such that every orbit ends inside a certain ellipsoid in forward time. We complement this result by a new phenomenon and by a new interpretation. We show that "infinity" is a global repeller for a set of parameters wider than that usually treated. We construct in a compacted space, a unit sphere that serves as the image of an ideal set at infinity. This sphere is shown to be the union of a family of periodic solutions. Each periodic solution is viewed as a limit cycle, or an isolated periodic orbit when restricted to a certain plane. The unconventional compactification y = x 1−x † x is used.
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Communications in Mathematical Physics, 1978
A 14-dimensional generalized Lorenz system of ordinary differential equations is constructed and its bifurcation sequence is then studied numerically. Several fundamental differences are found which serve to distinguish this model from Lorenz's original one, the most unexpected of which is a family of invariant two-tori whose ultimate bifurcation leads to a strange attractor. The strange attractor seems to have many of the gross features observed in Lorenz's model and therefore is an excellent candidate for a higher dimensional analogue.
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Physica D: Nonlinear Phenomena, 2000
A simplified, one-parameter version of the Lorenz model is obtained in the limit of high Rayleigh-and Prandtl-numbers, physically corresponding to diffusionless convection. It is argued that the bifurcation structure of this simplified Lorenz model essentially involves only Shil'nikov bifurcations. An exact solution to this simplified dynamical system is given which serves as the limit for strong forcing and appears to be a new integrable case of the Lorenz equations. For small values of the bifurcation parameter, an approximate, analytical and multipeaked map is obtained which gives successive periods of the pulse-like motion. This map leads to self-similar behaviour in parameter-space.
A Family of Periodic Motions to Chaos with Infinite Homoclinic Orbits in the Lorenz System
Lobachevskii Journal of Mathematics 42(14):3382-3437, 2022
In this paper, the bifurcation dynamics of a family of (n 1 , 1, n 2)-periodic motions to chaos with infinite homoclinic orbits (min {n 1 , n 2 } = 1, max {n 1 , n 2 } = 1, 2,. . .,) in the Lorenz system is studied through the discrete mapping method. The bifurcation trees of (n 1 , 1, n 2)periodic motion to chaos are presented through discrete nodes and harmonic amplitudes. The stability and bifurcations of periodic motions are determined through eigenvalue analysis. The bifurcation scenarios of (n 1 , 1, n 2)-period-1 motions to chaos are similar each other. The critical values for existence of (n 1 , 1, n 2)-related periodic motions are determined for saddle-node and period-doubling bifurcations. The homoclinic orbits are associated with unstable periodic motions on the bifurcation trees of the (n 1 , 1, n 2)-periodic motions to chaos. The homoclinic obits and periodic motions are illustrated from the bifurcation trees of the (n 1 , 1, n 2)-periodic motions to chaos. The numerical and analytical trajectories of unstable periodic motions were presented for comparison. If the numerical simulations did not have any computational errors, the numerical and analytical solutions of unstable periodic motions in the Lorenz system should be identical. Thus, one observed so-called strange attractors in the Lorenz system through numerical simulations, which are not real strange attractors. This paper is specially dedicated to the good friend and colleague in memory of Gennady A. Leonov for his contributions on nonlinear dynamics.
Limit cycle bifurcations in polynomial Liénard and Lorenz dynamical systems
Developing a bifurcational geometric approach to the qualitative analysis of polynomial dynamical systems, we first generalize the results on the classical two-dimensional Liénard system and solve the limit cycle problem for the general Liénard system with arbitrary polynomial restoring and damping functions. Then, using a similar approach and some numerical results, we present a new scenario of chaos transition for the classical three-dimensional Lorenz system. The general Liénard polynomial system We consider first Liénard equations x + f (x) ˙ x + g(x) = 0 (1) and the corresponding dynamical systems in the form ˙ x = y, ˙ y = −g(x) − f (x)y. (2) There are many examples in the natural sciences and technology in which these and related systems are applied; see [4, 7, 8, 10, 11, 13]. Such systems are often used to model either mechanical or electrical, or biomedical systems, and in the literature, many systems are transformed into Liénard type to aid in the investigations. They can ...
Nonlinear Processes in Geophysics Discussions, 2016
In this study, we discuss the role of the linear heating term and nonlinear terms associated with a nonlinear feedback loop in the energy cycle of the three-dimensional (X-Y-Z) nondissipative Lorenz model (3D-NLM), where (X, Y, Z) represent the solutions in the phase space. Using trigonometric functions, we first present the closed-form solution of the nonlinear equation d 2 X/dτ 2 + (X 2 /2)X = 0 without the heating term (i.e., rX), (where τ is a non-dimensional time 5 and r is the normalized Rayleigh number), a solution that has not been previously documented. Since the solution of the simplfied 3D-NLM is oscillatory (wave-like) and since the nonlinear term (X 3) is associated with the nonlinear feedback loop, here, we suggest that the nonlinear feedback loop may act as a restoring force. When the heating term is considered, the system yields three critical points. A linear analysis suggests that the origin (i.e., the trivial critical point) is a saddle point 10 and that the other two non-trivial critical points are stable. Here, we provide an analysis for three types of solutions that are associated with these critical points. Two of the solutions represent closed curves that travel around one non-trivial critical point or all three critical points. The third type of solution, appearing to connect the stable and unstable manifolds of the saddle point, is called the homoclinic orbit. Using the solution that contains one non-trivial critical point, here, we show that 15 the competing impact of the nonlinear restoring force and the linear (heating) force determines the partitions of the averaged available potential energy from the Y and Z modes. Based on the energy analysis, an energy cycle with four different regimes is identified. The cycle is only half of a "large" cycle, displaying the wing pattern of a glasswinged butterfly. The other half cycle is antisymmetric with respect to the origin. The two types of oscillatory solutions with either a small cycle or a 20 large cycle are orbitally stable. As compared to the oscillatory solutions, the homoclinic orbit is not periodic because it "takes forever" to reach the origin. Two trajectories with starting points near the homoclinic orbit may be diverged because one moves with a small cycle and the other moves with 1