Reduction to invariant cones for non-smooth systems
Introduction
The center manifold approach provides a powerful tool to reduce high-dimensional parameter dependent dynamical systems to lower dimensional systems carrying the essential dynamics responsible for example for bifurcation processes. In this paper we will continue to investigate if a similar approach is available for non-smooth systems. First we briefly recall the key facts for smooth systems.
Letdenote a smooth dynamical system with stationary solution .
Using linearization and transformations according to the structure of the eigenvalues of the linearization of f, Eq. (1) can be stated in the following form with ξ = (x, y, z)T and an accordingly arranged matrix A
where the submatrices A−, A0 and A+ correspond to the eigenvalues λi in the spectrum σ(A) = σ(A−) ∪ σ(A0) ∪ σ(A+) of A with negative, vanishing and positive real part, respectively:
here g−, g0 and g+ collect terms of higher order in x, y and z.
Since the stationary solution is already unstable if σ(A+) ≠ ∅, we assume for simplicity that σ(A+) = ∅, hence Eq. (1) is equivalent to
The center manifold approach performs a locally equivalent reduction to a system defined in the center space, i.e. there exists a function h, defined in a neighborhood of in the center space mapping into the stable space satisfying h(0) = 0, ( ∂ h/∂ y)(0) = 0 such that the reduced equation:is locally equivalent to (2).
The advantage of this approach relies on the fact that usually in relevant applications n0 ≔ dim y ≪ n, typically n0 = 1 or n0 = 2.
Once (3) has been established the dynamics, stability and bifurcation behavior of (1) can be obtained by studying (3).
The underlying center manifold approach essentially depends on smoothness properties of the original problem using the properties of the linearized problem.
Hopf bifurcation serves as a typical example which can be studied via the center manifold approach. The bifurcation of periodic orbits from a stationary solution is triggered by a crossing of exactly one pair of eigenvalues of the linearization A through the imaginary axis.
For non-smooth systems linearization is not at hand due to a lack of smoothness. A review of recent results concerning nonsmooth problems is given by [1], [2], [3], [4], [5], [7], [11], [12]. The growing interest to investigate non-smooth problems of high dimensions involving many parameters with regard to stability and bifurcation has stimulated the question if destabilization and bifurcation for non-smooth systems arises as well through a change in low dimensional terms and if suitable reduction techniques can be developed.
Since there is no linearized equation defined for non-smooth systems criteria by use of eigenvalues are not at hand. In previous studies [9], [10], [13], [14], [15], [16] restricted to planar systems it has been shown that the analytical criterium based on the eigenvalues crossing the imaginary axis can be substituted by an equivalent interpretation: Hopf bifurcation is due to a change in phase space from a stable focus to an unstable focus via a center. This is considered as the key observation that this situation can be mimicked for piecewise linear systems, and for that reason it has been suggested in [8], [14], [16] to replace the linearized problem by a basic piecewise linear problem. In various papers it has indeed been shown that the occurrence of periodic orbits in terms of “generalized Hopf” bifurcation can be achieved. For planar systems there is of course no need for any reduction of the system.
Based on a first approach in [8] we now continue to set up methods to reduce a high dimensional non smooth system to a low dimensional one.
Section snippets
N-dimensional piecewise systems
To describe the reduction we consider the simple situation of a non-smooth problem given by two smooth problems defined in half-spaces separated by a hyperspace M with appropriate transition rules motivated by examples described in [8]. In we take as separation manifold and assume that the dynamical system is given bywhere are smooth functions and .
Since we are particularly interested in trajectories crossing M we have to define
The nonlinear problem
We now assume that the PWLS (5) has been derived from (4) by considering linearized problems in each half-space; i.e.:where satisfying g±(0) = 0 and ( ∂ g±/∂ ξ)(0) = 0. We assume that the corresponding PWLS possesses an attractive invariant cone generated by and with the eigenvalues of satisfying (6). We further assume that (7) is already given in a truncated version; i.e for same r > 0The Poincaré map associated to (6)
3D-Example
To illustrate the previous results we consider a parameter dependent example in of the form:Since we want to study systems switching between the half-spaces eTξ > 0 and eTξ < 0, we assume that there are complex eigenvalues with non-vanishing imaginary part; i.e. λ± + iω± and μ±.
Further we assume that the eigenvectors corresponding to the complex eigenvalues are not both contained in the hyperplane M = {ξ | eTξ = 0}. Without restriction we can assume that one of the linear
Conclusions
In a previous paper the existence of invariant cones for nonsmooth piecewise linear systems has been established. This approach can be extended to include nonlinear perturbation of the basic piecewise linear systems. With regard to further bifurcation analysis the notion of a generalized “center manifold” can be formulated.
Using a class of three-dimensional examples already developed in [8] we investigate various ways to generate invariant cones.
As a typical situation we consider the a
Acknowledgement
This work was supported by Department of Mathematics, Faculty of Science, Al-Azhar University of Assiut, Egypt.
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