Narrow Results

The dynamics of a system defined by an endomorphism is essentially different from that of a system defined by a diffeomorphism due to interaction of invariant objects with the so-called critical locus. A planar endomorphism typically folds the phase space along curves J₀ where the Jacobian of the map is singular. The critical locus, denoted J₁, is the image of J₀. It is often only piecewise smooth due to the presence of isolated cusp points that are persistent under perturbation. We investigate what happens when the stable set W_s of a fixed point or periodic orbit interacts with J₁ near such a cusp point C₁. Our approach is in the spirit of bifurcation theory, and we classify the different unfoldings of the codimension-two singularity where the curve W_s is tangent to J₁ exactly at C₁. The analysis uses a local normal-form setup that identifies the possible local phase portraits. These local phase portraits give rise to different global manifestations of the behavior as organized by five different global bifurcation diagrams.

The objective of this paper is to study the sliding bifurcation in a planar piecewise smooth system with an elliptic switching curve. Some new phenomena are observed, such as a crossing limit cycle containing four intersections with the switching curve, sliding cycles having four sliding segments, and sliding cycles consisting of the entire switching curve. Firstly, we investigate the bifurcation of sliding cycle from a sliding heteroclinic connection to two cusps and show the appearance of one sliding cycle with two folds. To plot the bifurcation diagram, a planar piecewise linear system with two zones separated by an ellipse are considered. Moreover, we study in more detail the unfolding of a sliding cycle connecting four cusps by exhibiting its complete bifurcation diagram. More precisely, we explore the necessary and sufficient conditions for the existence of limit cycles and derive the concrete bifurcation curves. Additionally, a simple piecewise smooth system with nonlinear subsystems is studied, which shows the possibility of the existence of two nested limit cycles. Finally, numerical simulations are given to confirm the theoretical analysis.

Studies on nonlinear analysis of system dynamics have increased in recent years. Since most systems that exist in nature have complex dynamics and therefore exhibit nonlinear behavior; there are various methods and theories developed in this context. Self-similar functions are mathematical functions exhibiting self-similar and scale-invariant behaviors performing fractal structures. Singularities are the basis for producing the self-similar behavior of these functions. Singularity analysis is mainly carried out by using wavelet transform (WT). The cusp singularities are non-oscillating singularities which are characterized by their singularity strength. However, the representation and behavior of this type of singularity differs depending on the sign of the exponent, known as the singularity strength. The cusp singularity functions with negative exponent show irregular behavior progressively different than positively valued functions since the value of the function is undefined at that particular singular point. It is commonly accepted that the singularity strength is studied as Hölder exponent of the cusp function, but by definition, the value of this exponent cannot take negative values. We present a new method to estimate the singularity strength of cusp singularities with negative exponent. The developed method is based on analyzing and redistributing the amplitudes of a cusp function with negative exponent by taking the WT. The redistribution of amplitudes over time is achieved by applying curve fitting process to frequency values of the analyzing function.

In this paper, we are concerned with the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation with typical discontinuous initial data (u(0, x), ∂_tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x₁,…,x_n), n ≥ 2, and f(t, x, u) is C^∞ smooth on its arguments. When the initial data φ(x) is homogeneous of degree zero or piecewise smooth along the hyperplane {t = x₁ = 0}, it is shown that the local solution u(t, x) ∈ L^∞([0, T] × ℝⁿ) exists and is C^∞ away from the forward cuspidal conic surface or the cuspidal wedge-shaped surfaces respectively. On the other hand, for n = 2 and piecewise smooth initial data φ(x) along the two straight lines {t = x₁ = 0} and {t = x₂ = 0}, we establish the local existence of a solution and further show that in general due to the degenerate character of the equation under study, where . This is an essential difference to the well-known result for solution to the two-dimensional semilinear wave equation with (v(0, x), ∂_tv(0, x)) = (0, φ(x)), where Σ₀ = {t = |x|}, and .