Bifurcation and invariant manifolds of the logistic competition model

In this paper, we study a new logistic competition model. We will investigate stability and bifurcation of the model. In particular, we compute the invariant manifolds, including the important centre manifolds, and study their bifurcation. Saddle-node and period-doubling bifurcation route to chaos are exhibited via numerical simulations.     

Invariant manifolds of dynamical systems close to a rotation: Transverse to the rotation axis

We consider one parameter families of vector fields depending on a parameter ? such that for ?=0 the system becomes a rotation of R²×Rⁿ around {0}×Rⁿ and such that for ?>0 the origin is a hyperbolic singular point of saddle type with, say, attraction in the rotation plane and expansion in the complementary space. We look for a local subcenter invariant manifold extending the stable manifolds to ?=0. Afterwards the analogous case for maps is considered. In contrast with the previous case the arithmetic properties of the angle of rotation play an important role.     

Families of stable manifolds for one-dimensional parabolic equations

We prove that to any invariant subset of the dynamical system generated by a one-dimensional quasilinear parabolic equation there corresponds an invariant family of stable manifolds of finite codimension. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 11–23, July, 1996.     

Invariant manifolds of periodic orbits for piecewise linear three-dimensional systems

This paper analyses the existence of invariant manifolds ofperiodic orbits for a specific piecewise linear three-dimensionalsystem with two zones, whose linear parts share a pair of imaginaryeigenvalues. This degenerate situation is obtained from thelack of controllability. The analysis proceeds by its reductionto a periodic one-dimensional equation for which some resultsof the Ambrosetti–Prodi type are given.     

The manifold structure of maps between open manifolds

We establish in a canonical manner a manifold structure for the completed space of bounded maps between open manifoldsM andN, assuming thatM andN are endowed with Riemannian metrics of bounded geometry up to a certain order. The identity component of the corresponding diffeomorphisms is a Banach manifold and metrizable topological group.     

An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary

We establish an index theorem for Toeplitz operators on odd-dimensional spin manifolds with boundary. It may be thought of as an odd-dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of η type associated to K¹ representatives on even-dimensional manifolds, which should be of independent interests. For example, it gives an intrinsic interpretation of the so called Wess-Zumino term in the WZW theory in physics.     

Two-dimensional invariant manifolds and global bifurcations: some approximation and visualization studies

We illustrate and discuss the computer-assisted study (approximation and visualization) of two-dimensional (un)stable manifolds of steady states and saddle-type limit cycles for flows in R ⁿ . Our investigation highlights a number of computational issues arising in this task, along with our solutions and “quick-fixes” for some of these problems. Two examples illustrative of both successes and shortcomings of our current approach are presented. Representative “snapshots” demonstrate the dependence of two-dimensional invariant manifolds on a bifurcation parameter as well as their interactions. Such approximation and visualization studies are a necessary component of the computer-assisted study and understanding of global bifurcations. This revised version was published online in June 2006 with corrections to the Cover Date.     

求解混沌映射不变分布的符号计算法

根据Frobenius-Perron方程,可以对混沌映射的不变分布从理论上加以分析,从而对混沌映射不变分布作出大致的估计.由此可以利用符号计算的方法求解得到迭代函数系统不变分布的密度近似函数,从而逼近理论解.用几个计算实例和常见数值解法作了比较,试验结果表明符号计算方法具有一定的优势.     

Distributions of orientations on Stiefel manifolds

The Riemann space whose elements are m × k (m k) matrices X, i.e., orientations, such that X′X = I_k is called the Stiefel manifold V_k,m. The matrix Langevin (or von Mises-Fisher) and matrix Bingham distributions have been suggested as distributions on V_k,m. In this paper, we present some distributional results on V_k,m. Two kinds of decomposition are given of the differential form for the invariant measure on V_k,m, and they are utilized to derive distributions on the component Stiefel manifolds and subspaces of V_k,m for the above-mentioned two distributions. The singular value decomposition of the sum of a random sample from the matrix Langevin distribution gives the maximum likelihood estimators of the population orientations and modal orientation. We derive sampling distributions of matrix statistics including these sample estimators. Furthermore, representations in terms of the Hankel transform and multi-sample distribution theory are briefly discussed.     

To the question about the maximum principle for manifolds over local algebras

We consider manifolds over a local algebra A. We study basis functions of the canonical foliation which represent the real parts of A-differentiable functions. We prove that these are constant functions. We find the form of A-differentiable functions on some manifolds over local algebras, in particular, on compact manifolds. We obtain an estimate for the dimension of some spaces of 1-forms and analogs of the above results for the projective mappings of foliations.     

3-Quasi-Sasakian manifolds

In the present article we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular, we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.      

Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds

In this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a Banach space. The situation considered is such that the perturbed dynamical system is non-autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points. Starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non-autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium. Then we prove that the local unstable and stable manifolds associated to them are given as graphs (roughness of dichotomy plays a fundamental role here). Moreover, we prove the continuity of this local unstable and stable manifolds with respect to the perturbation. With that result we conclude the lower semicontinuity of pullback attractors.     

C Invariant Manifolds for Maps of Banach Spaces

In this work we give a unified proof for the existence, under appropriate gap conditions, of the standard invariant manifolds for a C^k map of a Banach space near a fixed point where k ≥ 1 is an integer.     

Computation of local radius of information in SM-IBC identification of nonlinear systems

System identification consists in finding a model of an unknown system starting from a finite set of noise-corrupted data. A fundamental problem in this context is to asses the accuracy of the identified model. In this paper, the problem is investigated for the case of nonlinear systems within the Set Membership—Information Based Complexity framework of [M. Milanese, C. Novara, Set membership identification of nonlinear systems, Automatica 40(6) (2004) 957–975]. In that paper, a (locally) optimal algorithm has been derived, giving (locally) optimal models in nonlinear regression form. The corresponding (local) radius of information, providing the worst-case identification error, can be consequently used to measure the quality of the identified model. In the present paper, two algorithms are proposed for the computation of the local radius of information: The first provides the exact value but requires a computational complexity exponential in the dimension of the regressor space. The second is approximate but involves a polynomial (quadratic) complexity.     

Invariant manifolds of admissible classes for semi-linear evolution equations

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and a semi-linear integral equation . We prove the existence of invariant manifolds of this equation. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory. The existence of such manifolds is obtained in the case that (U(t,s))_t?s?0 has an exponential dichotomy and the nonlinear forcing term f(t,x) satisfies the non-uniform Lipschitz conditions: ‖f(t,x₁)−f(t,x₂)‖?φ(t)‖x₁−x₂‖ for φ being a real and positive function which belongs to certain classes of admissible function spaces.     

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

Pitchfork bifurcations of invariant manifolds

A pitchfork bifurcation of an (m−1)-dimensional invariant submanifold of a dynamical system in Rm is defined analogous to that in R. Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is proved under the stated hypotheses. For discrete dynamical systems, the existence of locally attracting manifolds M₊ and M₋, after the bifurcation has taken place is proved by constructing a diffeomorphism of the unstable manifold M. Techniques used for proving the theorem involve differential topology and analysis. The theorem is illustrated by means of a canonical example.     

Unstable manifolds for endomorphisms

A detailed presentation of an unstable manifold theorem for non-invertible differentiable maps of finite-dimensional manifolds is given.     

Numerical calculation of invariant manifolds for maps

Many processes in the sciences and in engineering are modelled by dynamical systems and—in discretized version—by nonlinear maps. To understand the often complicated dynamical behaviour it is a well established tool to use the concept of invariant manifolds of the system. In this way it is often possible to reduce the dimension of the system considerably. In this paper we propose a new method to calculate numerically invariant manifolds near fixed points of maps. We prove convergence of our procedure and provide an error estimation. Finally, the application of the method is illustrated by examples.     

A notion of robustness and stability of manifolds

Starting from the notion of thickness of Parks we define a notion of robustness for arbitrary subsets of Rk and we investigate its relationship with the notion of positive reach of Federer. We prove that if a set M is robust, then its boundary ∂M is of positive reach and conversely (under very mild restrictions) if ∂M is of positive reach, then M is robust. We then prove that a closed non-empty robust set in Rk (different from Rk) is a codimension zero submanifold of class C¹ with boundary. As a partial converse we show that any compact codimension zero submanifold with boundary of class C² is robust. Using the notion of robustness we prove a kind of stability theorem for codimension zero compact submanifolds with boundary: two such submanifolds, whose boundaries are close enough (in the sense of Hausdorff distance), are diffeomorphic.