| 词汇 | example_english_bifurcation |
| 释义 | Examples of bifurcationThese examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. Connected simple systems, transition matrices and heteroclinic bifurcations. On some global bifurcations connected with disappearance of a saddle-node fixed point. General bifurcations from periodic solutions with spatiotemporal symmetry, including mode interactions and resonances, in preparation. Further increase of the wave amplitude leads to further bifurcations and eventually to a chaotic surface agitation. However, qualitative analysis of equilibria, bifurcations or asymptotic behaviour is more difficult than with differential equation models. We discuss the dependence of stationary measures on an auxiliary parameter, thus describing bifurcations of families of random diffeomorphisms. The following local bifurcations can be found by straightforward calculations. He gave properties of the non-wandering set, traced the genealogies of the periodic points of period 5, and described their stability types and bifurcations. Much of the original motivation for developing the theory of bifurcations with symmetry came from fluid mechanical experiments in the laboratory. The rather short length of the vortex is characteristic of the disturbance for symmetrybreaking pitchfork bifurcations. Morphological bifurcations involving reaction-diffusion processes during microtubule formation. Also, complicated price dynamics can only be generated through saddle-nodetype bifurcations. We can reduce the problem of detecting bifurcations one step further, by noting the following. Typically, transition to chaos takes place through bifurcations. The density of saddle-node bifurcations of periodic orbits implies the following corollary. Discontinuous changes of the control set, interpreted this way, are among the bifurcations identified in this paper. We observe the following sequence of instabilities leading to turbulence via subharmonic bifurcations. We define the transition matrix pair for discrete semidynamical systems, which detects bifurcations of connecting orbits. I n the meanwhile, it is noteworthy that such flows are capable of period bifurcations which should be discernible by experiment. As the parameter is increased, the family goes through an infinite sequence of periodic doubling bifurcations. The phylogram's distances were linearised to enable comparison of the relative order of bifurcations. A succession of bifurcations can lead to a strange attractor and aperiodic or chaotic dynamical behavior. Both bifurcations are "supercritical," whereby an attracting orbit or cycle emerges as the subsidy rate passes a critical value. From these computations we will show that at the critical values the integral manifolds undergo bifurcations in their topology. We once again consider whether the bifurcations are super-critical or sub-critical, to determine the stability of the solutions. Homoclinic bifurcations may lead to the creation (or annihilation) of chaotic attractors. The author s put forward theories of zonal flow generation and their suppression effects and of bifurcations in plasmas. The computation times for the network elements, both for bifurcations and microcirculation calculations, do not show linear speed-up. An analysis of bifurcations in the presence of this symmetry provides a list of array configurations which one would expect to (locally) optimise performance. We study its bifurcations at and show that the singularities of codimension two unfold versally in a neighborhood. Such bifurcations occur in relatively low-viscosity liquids as the amplitude of the acoustic field exceeds a threshold dependent on the relevant parameters of the problem. The initially cylindrical vortex tube is folded and deformed without any holes or bifurcations. Theorem 8.7 below moreover describes the possible codimension-one bifurcations. One of the most explored of these routes is via period-doubling bifurcations as a controlling parameter takes on increasing values. Our case is more involved, but one can expect that generalized homoclinic bifurcations can appear in suitable three-dimensional differential equations depending periodically on time. We continue with more specific descriptions of dynamics and bifurcations near the homoclinic tangles. We consider intermittency in one-parameter families of unimodal maps, induced by saddle node and boundary crisis bifurcations. They appear in horseshoes, 'strange attractors', at various global bifurcations and suchlike. Comparing connection matrices for two parameter values, we can also detect global bifurcations of connecting orbits in one-parameter families of flows. We investigate bifurcations in the chain recurrent set for a particular class of one-parameter families of diffeomorphisms in the plane. Here we address this problem for critical saddle-node bifurcations. The successive bifurcations are studied in detail and their physical mechanism is elucidated. All dynamic properties such as stability or bifurcations are linked to this map. Furthermore, this paper also reports on the stability of the system through associated equilibrium points and bifurcations analytically, using the simplified model. Figure 6 shows a sequence of solutions showing the most relevant steps in the hierarchy of bifurcations. In the case of transcritical bifurcations, the dimension of (39) is one. The existence of an important class of codimension-two bifurcations was also confirmed in that paper. In certain cases, global bifurcations may open a route to chaos. We investigate whether these features affect the emergence of endogenous (sunspot-driven) fluctuations, by analyzing the occurrence of local indeterminacy and local bifurcations. More precisely, we analyze the effect of a wage inequality on local stability of the steady state and occurrence of bifurcations. In this way, we study the local indeterminacy of the steady state and the occurrence of local bifurcations. A rigorous justification of such non-local bifurcations is an open problem. They gave a quite thorough treatment of the local bifurcations which can occur from the normal state [11, 12]. We explain how random saddle node bifurcations occur in both examples and random homoclinic bifurcations in random logistic maps. We observe that these solutions exhibit bifurcations at points where the adiabatic profile leaves the trivial profile. Travelling water-waves, as a paradigm for bifurcations in reversible infinitedimensional 'dynamical' systems. We first study the sort of bifurcations associated to a homoclinic class in terms of its dominated splitting. Previous attempts to study stochastic bifurcations fall into two categories. We distinguish bifurcations where the density function of a stationary measure varies discontinuously or where the support of a stationary measure varies discontinuously. Local bifurcations are by definition the bifurcations that occur in the neighborhood of a non-hyperbolic equilibrium or a nonhyperbolic periodic solution. They can also occur due to saddle-node type bifurcations, in which the saddle and node could each be part of a larger chain component. Techniques in the theory of local bifurcations: blow up, normal form, nilpotent bifurcations, singular perturbations. The following discussion is made on the assumption that such additional bifurcations are not the first to occur. We note that the complex solutions occur as secondary bifurcations from the real ones. Section 3 analyzes and discusses the local dynamic properties of the model and the occurrence of local bifurcations. In this section, we investigate the control of bifurcations using fiscal feedback laws. Consequently, in order to find all versal unfoldings of (1) we need to find all surfaces in the bifurcations set of (2). When the control parameter is further decreased, the periodic window is succeed by a sequence of period-doubling bifurcations. The microrobot will have a tridimensional contortion with a minimum curve radius, to explore 3-dimensional networks having curves and bifurcations. The end points of these curves, other than (0, 0), are bifurcations from a normal state. Spatial dynamics has also been utilized [37, 38] to investigate bifurcations of modulated fronts with large amplitude. The solutions form a series of pitch-fork bifurcations. As c is increased, further period-doubling ## bifurcations occur until a period-2 attractor coexists with a chaotic attractor. Another route to turbulence involves successive subharmonic (or period doubling) bifurcations of a periodic flow. Stationary bifurcations from large squares: note that the last four solutions generically bifurcate at the same point. We will derive results which together with the general observations of [10] give rise to a complete picture of the bifurcations of the invertibility picture of the map. Therefore, bifurcations occur at nonhyperbolic equilibria only. There are finitely many such bifurcations. Hard bifurcations in dynamical systems with bounded random perturbations. The starting point of the present paper will be the family of non-autonomous systems after rescaling, although our results are concerned with bifurcations of limit cycles in the autonomous truncation. We show also how considerations of symmetry and associated group theory can be used to explain the nature of these transitions and the sequence in which the relevant bifurcations occur. However, the prevalence of relatively simple phenomena (period doubling bifurcations, quasi-periodic motion, and phase locking) does offer some hope for the possibility of achieving qualitative understanding of the basic processes. The advantage of the group-theoretic approach is to enumerate all possibilities systematically, so that we know that the above five types are the only possible simple stationary bifurcations. Thus, these bifurcations are sources of intermittency. A more general division strategy is needed that does not depend on splitting at bifurcations. Estimating the speed of decay of correlations as a function of a parameter gives further details of changes through bifurcations. As a second issue, take a one-parameter family of maps that exhibits bifurcations and add small bounded noise to it. In these bifurcations either a periodic orbit or a periodic interval disappears to give rise to chaotic bursts. We then describe how the above analytic result is applied to bifurcations of limit cycles. A number of socalled fold bifurcations were thus detected, one of which would seem to be of particular importance for the present discussion. Again, it can be shown that the occurrence of bifurcations is necessary to explain transitions to more powerful structures. Sub-critical bifurcations may be analogous to flexible transitions in neural activity. One would not expect to observe the more degenerate (higher codimension) singularities like transcritical bifurcations. None of the two new bifurcations can be absorbed by a change in the frame of reference. Similarly, solutions lacking all symmetry are only attainable after several bifurcations. We describe the different kinds of dynamic behavior observed, and we characterize the bifurcations that mark the transitions between qualitatively different time evolutions. In particular, those bifurcations can be related to the fact that certain eigenvalues go through zero (continuous time) or through unity in modulus (discrete time). The main reason for this choice of model is the hope of solutions which may include local bifurcations. These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. |
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