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Overview of dynamical systems and chaos
Published in Marcio Eisencraft, Romis Attux, Ricardo Suyama, Chaotic Signals in Digital Communications, 2018
In 1967, Allen F. Kelley Jr. (1931-) proved that, in the phase space of the original nonlinear system, there is a unique ns-dimensional stable manifold14Ws tangent to Es in x*, and there is a unique nu-dimensional unstable manifold Wu tangent to Eu in x*. If x0 ∈ Ws (or if x0 ∈ Es), then x(t) → x* for t → ∞ (there is convergence to x* as the time passes); if x0 ∈ Wu (or if x0 ∈ Eu), then x(t) → x* for t → −∞ (there is divergence from x* as the time progresses). These results are in agreement with the Hartman-Grobman theorem. Observe that if nu > 0, x* is unstable; if nc = 0, x* is hyperbolic; if nc > 0, x* is non-hyperbolic. Kelley also showed that for nc > 0 there is a nc-dimensional center manifold Wc tangent to Ec in x*; however, Wc needs not be unique. Notice that if nu =0, then ns + nc = n and the trajectories or orbits will tend to Wc for t → ∞. Thus, the analysis can be restricted to Wc, a space with nc dimensions; that is, the analysis (near an equilibrium or fixed point) can be performed in a reduced system of equations. In 1981, Jack Carr (1948-) showed how to determine the dynamics on Wc, which discloses the Lyapunov stability of x* when the indirect method fails. In fact, solutions on Wc can converge to x*, diverge from x*, oscillate around x*, or remain constant as the time goes by. The behavior on Wc depends on nonlinear terms of f (while the dynamics on Ws and Wu are ruled by the linear approximation of f). Center manifold reduction and normal form theory simplify the characterization of local bifurcations.
Some new instability phenomena in nonlinear discrete systems
Published in W. B. Krätzig, O. T. Bruhns, H. L. Jessberger, K. Meskouris, H.-J. Niemann, G. Schmid, F. Stangenberg, A. N. Kounadis, G. I. Schuëller, Structural Dynamics, 1991
The last fifteen years there have been published numerous studies on postbuckling analyses of non-gradient structural systems [Plaut (1976), Kounadis, Giri and Simitses (1978), Kounadis and Economou (1980), Kounadis and Mahrenholtz (1989), Sotiropoulos and Kounadis (1990)]. However,even when the loss of stability occurs in the region of existence of adjacent equilibria,the static nonlinear stability analyses are found to be inadequate for establishing the loadcarrying capacity of limit point systems under follower forces [Kounadis (1989), Sotiropoulos- and Kounadis (1990)].This is mainly due to the fact that for limit point systems even when the effects of damping and masses can be neglected the dynamic buckling loads may differ substantially from the corresponding limit point loads [Kounadis (1989), Sotiropoulos and Kounadis (1990)]. Moreover, dynamic bifurcations cannot be explored by using static methods of analysis and hence a nonlinear dynamic formulation must be employed. Some interesting nonlinear dynamic analyses dealing with different types of dynamic bifurcations of perfect non-gradient systems losing their stability through either a divergence point or an oscillatory instability are presented by various investigators [Sethna and Schapiro (1977), Mandani and Huseyin (1980), Marsden and McGracken (1976), Atadan and Huseyin (1985), Huseyin and Yu (1988)]. Dynamic bifurcations have been also studied with the aid of centre manifold theory [Holmes (1981),Carr (1981), Jin and Matsuzaki (1988)]for reducing the system dimension. The above nonlinear dynamical analyses based on the study of Jacobian eigenvalues assume that the initial trivial path is stable. The case where such an assumption is not valid (imperfect limit point systems with precritical deformation) is studied recently by Kounadis (1989)using Ziegler's two-degree-of freedom system as model. For such system dynamic buckling load is defined as the smallest load for which an escaped motion may occur. In the last work chaoslike phenomena in case of three competing equilibrium points have also been detected. In this investigation a general theory for nonlinear multi-parameter dissipative or nondissipative discre te systems under partial follower loading described by a set of non-linear autonomous differential equations is presented The analysis considers both types of systems: Perfect without precritical deformation as well as imperfect with precritical deformation. Among other findings for the latter systems, the significance of the unstable complementary path on the dynamic buckling mechanism is revealed for the first time in the technical literature. Local and global bifurcations as well as all findings of this investigation are checked through numerical simulation of Ziegler's model for which a lot of numerical results are available.
Final patterns and bifurcation analysis of the odd-periodic Swift–Hohenberg equation with respect to the period
Published in Applicable Analysis, 2021
There have been many interesting results on the bifurcation and the pattern selection of the SHE depending on parameters and . For instance, see [4–14] and references therein. Particularly, the authors in [14, 15] studied the bifurcation phenomena of the trivial solution with respect to the parameter α, which means that α varies, and the interval is fixed. When α crosses over a critical number, the trivial solution bifurcates to an attractor which tells us the structure of the final patterns of solutions. On the other hand, Peletier and Williams showed in [4] that for fixed , the SHE also bifurcates from the trivial solutions to an attractor as varies. They considered the SHE (1) on the interval with the boundary condition u=0 and at . They also assumed that the initial function satisfies the following symmetric condition It is not difficult to see that this symmetry is preserved during time evolution. In this case, the linearized equation has the eigenvalues , and the corresponding eigenvectors are for . With this setting, by studying -norm behavior of solutions, they presented that the patterns are formed as a by-product of bifurcation as the parameter passes through critical numbers. They provided a detailed analysis of the bifurcation phenomena observed in numerical studies by utilizing the center manifold reduction. One interesting result is that they captured the instant at which a gap is collapsed into a point, and there appears an overlapped interval Λ of . They gave a detailed description of final patterns of solutions for each . Indeed, they proved that if we write , then there exist two numbers such that we have different long-time dynamics of solutions on each subinterval , and .