Simplest normal forms of generalized Neimark-Sacker bifurcation

Abstract

Abstract：The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given. Keywords：generalized Neimark-Sacker bifurcation; simplest normal form; near identity nonlinear transformationsThe normal form method has been widely used in the fields of dynamical system, ordinary differential equations and nonlinear vibration. Normal form theory plays an important role in the study of dynamical behavior of nonlinear systems near the dynamic equilibrium points because it greatly simplifies the analysis and formulations. This simple form can be used conveniently in analyzing the dynamical behavior of the original system near the dynamic equilibrium points [1][2][3][4][5] . However, it is not a simple task to calculate the normal form for some given ordinary differential equations. The normal forms of Neimark-Sacker bifurcation, also known as the discrete cases of generalized Hopf bifurcation, as .well as their .applications. have .been .extensively .studied [6][7][8] . This phenomenon appears in some known biological models like the delayed logistic map [9] . It is also of interest because the bifurcation of a limit cycle of a vector field can be transformed into an invariant torus via Poincaré map [10] . Recently this bifurcation has also been detected in some economic models [11] .This paper concentrates on generalized NeimarkSacker bifurcation. This case is subtler due to the possible presence of resonant terms, which are different from the odd order terms in the conventional normal form.In this paper, five theorems are presented to show that the generalized Neimark-Sacker bifurcation can be further simplified to the simplest normal forms in which at most two nonlinear terms remain in the amplitude equations if appropriate nonlinear transformations are chosen. There are two kinds of simplest normal forms which are simpler than the expressions in Refs.[6] and [12]. Their algebraic expression formulas are given below.

Stability Analysis of a Discrete-time Limit Cycle Model Predictive Controller

“…in polar coordinates {r, θ} ∈ R and where the parameters {α, ρ, ϕ} ∈ R >0 (required for the attractor limit cycle supercritical case) stand for the scale factor, circular limit cycle radius, and phase step size correspondingly, in discrete time k ∈ Z, [4], [8], [9]. Negative radii are considered to ease the analysis but can be interpreted as phase jumps.…”

“…While Theorem 1 establishes M as a final set for Ω, looking at the subsets of M in (8), their reachability differs within Ω. Starting with M ∞ , it can only be reached by solutions at r k = ±ρ ∞ , i.e., within M ∞ ; which is unlikely to be hit exactly in real applications.…”

“…Theorem 2. Let X ⋆ = arg min J (X) be the solution of the optimization problem in (8), consisting of every consecutive state sequence x ⋆ ∈ Ω from time step k + 1 up to k + H p , stacked in order into a column vector. If J (X ⋆ ) = 0, then every consecutive state sequence x ⋆ of the optimization problem solution X ⋆ approaches M as k → ∞.…”

Harmonic Compensation by Limit Cycle Model Predictive Control

“…with parameters {µ, α, φ} ∈ R, in discrete time k ∈ Z, with sampling time τ = t k , [28]. A system transformation to analyze the phase-space dynamics is proposed as follows.…”