Triple-zero Eigenvalue Research Articles

Detecting the Shilnikov Scenario in a Hopf-Hopf Bifurcation with 1:3 Resonance

We investigate the behaviour of the primary solutions at a Hopf-Hopf interaction close to a 1:3 resonance. It turns out, that the secondary bifurcations from the primary periodic solution branches are governed by Duffing and Mathieu equations.By numerical path following a homoclinic orbit at a saddle node was detected, giving rise to the Shilnikov scenario. In order to understand the creation of homoclinic orbits, a continuation of that orbit was applied, which terminated at an equilibrium with a triple zero eigenvalue. The existence of different types of homoclinic and heteroclinic orbits in the vicinity of triple zero bifurcation points has already been established. A short discussion of the local bifurcations at the triple zero eigenvalue is given.

An exact homoclinic orbit and its connection with the Rössler system

Abstract In this Letter we consider a three parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue of an equilibrium point. Using blow-up techniques we obtain a system where an exact homoclinic connection is determined. The numerical continuation of this global connection shows that it exhibits three different kinds of codimension-two degeneracies. Finally, these same codimension-two homoclinic bifurcations are detected in the Rossler system, ensuring in this way the existence of chaotic dynamics.

Inverted Pendulum Stabilization Via a Pyragas-Type Controller: Revisiting the Triple Zero Singularity

The use of Pyragas-Type controller proved interest in the stabilization of unstable periodic orbits. The stabilization problem of a balancing inverted pendulum on an horizontally moving cart by the use of such a controller is considered. The main objective of the paper is to propose delayed control law containing only proportional gains able to stabilize the inverted pendulum by avoiding the existence of a triple zero eigenvalue at the origin. We analyze the center dynamics described by a three dimensional system of ordinary differential equations (ODEs) with a codimension-three triple zero bifurcation. Furthermore, the stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium is also proposed. This analysis is done in order to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed. Time-Delay, Stability, Delayed Feedback, Control, Center Manifold Theorem, Normal Forms, Local Bifurcation, Inverted Pendulum.

A Theorem for the Triple Zero bifurcation

Given a three-dimensional m-paremeterized vector field, which has a non-hyperbolic equilibrium point with a triple-zero eigenvalue with geometric multiplicity one, the called non-degeneracy and transversality conditions are given such that system undergoes the triple-zero bifurcation.

RESONANCES OF PERIODIC ORBITS IN RÖSSLER SYSTEM IN PRESENCE OF A TRIPLE-ZERO BIFURCATION

This paper focuses on resonance phenomena that occur in a vicinity of a linear degeneracy corresponding to a triple-zero eigenvalue of an equilibrium point in an autonomous tridimensional system. Namely, by means of blow-up techniques that relate the triple-zero bifurcation to the Kuramoto–Sivashinsky system, we characterize the resonances that appear near the triple-zero bifurcation. Using numerical tools, the results are applied to the Rössler equation, showing a number of interesting bifurcation behaviors associated to these resonance phenomena. In particular, the merging of the periodic orbits appeared in resonances, the existence of two types of Takens–Bogdanov bifurcations of periodic orbits and the presence of Feigenbaum cascades of these bifurcations, joined by invariant tori curves, are pointed out.

Equality conditions for the singular values of 3 × 3 matrices with one-point spectrum

Let Γa be an upper triangular 3 × 3 matrix with diagonal entries equal to a complex scalar a. Necessary and su.cient conditions are found for two of the singular values of Γa to be equal. These conditions are much simpler than the equality discr ϕ = 0, where the expression in the left-hand side is the discriminant of the characteristic polynomial ϕ of the matrix Ga = ΓaΓa. Understanding the behavior of singular values of Γa is important in the problem of finding a matrix with a triple zero eigenvalue that is closest to a given normal matrix A.

Extending the permissible control loop latency for the controlled inverted pendulum

A pendulum can be stabilized in its upright position by proportional-plus-derivative (PD) feedback control only if the latency in the control loop is smaller than a certain critical delay. This critical delay is determined by the presence of a fully symmetric triple-zero eigenvalue singularity, a bifurcation of codimension three. We investigate three possible modifications of the PD scheme with the aim of extending the range of permissible delays. Effectively, these modifications introduce another parameter. This additional parameter can be used to continue the triple-zero singularity in four parameters until it gains a higher-order degeneracy imposing a new limit on the permissible delay. It turns out that the most effective modification is to feed back the value of the position with a small (intentional) additional delay on top of the control loop latency.

Complex balancing motions of an inverted pendulum subject to delayed feedback control

We show that an inverted pendulum that is balanced on a cart by linear delayed feedback control may exhibit small chaotic motion about the upside-down position. In periodic windows associated with this chaotic regime we find periodic orbits of arbitrarily high period that correspond to complex balancing motion of the pendulum with bounded velocity of the cart. This result shows that complex balancing is possible in a controlled mechanical system with a geometric nonlinearity even when the control law is only linear. This is in contrast to other proposed models that require a nonlinearity of the controller, such as round-off due to digitization. We find the complex motion by studying homoclinic bifurcations of a reduced three-dimensional vector field near a triple-zero eigenvalue singularity. More generally, the dynamics we describe must be expected in any system with a triple-zero singularity and reflection symmetry.

An Algorithm for Computing Quasi-Homogeneous Formal Normal Forms under Equivalence

This paper presents a recursive algorithm which is useful for computing normal forms of vector fields using smooth orbital equivalence. The case of vector fields with a singularity corresponding to a triple-zero eigenvalue with geometric multiplicity one is considered in detail. The results obtained are applied to the study of a simple electronic device, with only one nonlinearity.

Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity

We investigate a delay differential equation that models a pendulum stabilized in the upright position by a delayed linear horizontal control force. Linear stability analysis reveals that the region of stability of the origin (the upright position of the pendulum) is bounded for positive delay. We find that a codimension-three triple-zero eigenvalue bifurcation acts as an organizing centre of the dynamics. It is studied by computing and then analysing a reduced three-dimensional vector field on the centre manifold. The validity of this analysis is checked in the full delay model with the continuation software DDE-BIFTOOL. Among other things, we find stable small-amplitude solutions outside the region of linear stability of the pendulum, which can be interpreted as a relaxed form of successful control.

A NOTE ON THE TRIPLE-ZERO LINEAR DEGENERACY: NORMAL FORMS, DYNAMICAL AND BIFURCATION BEHAVIORS OF AN UNFOLDING

This paper is devoted to the analysis of bifurcations in a three-parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue. We carry out the study of codimension-two local bifurcations of equilibria (Takens–Bogdanov and Hopf-zero) and show that they are nondegenerate. This allows to put in evidence the presence of several kinds of bifurcations of periodic orbits (secondary Hopf,…) and of global phenomena (homoclinic, heteroclinic). The results obtained are applied in the study of the Rössler equation.

Tendril Perversion in Intrinsically Curved Rods

A straight elastic rod with intrinsic curvature under varying tension can undergo an instability and bifurcate to a filament made out of two helices with opposite handedness. This inversion of handedness, known as perversion, appears in a wide range of biological and physical systems and is investigated here within the framework of thin elastic rods described by the static Kirchhoff equations. In this context, a perversion is represented by a heteroclinic orbit joining asymptotically two fixed points representing helices with opposite torsion. A center manifold reduction and a normal form transformation for a triple zero eigenvalue reduce the dynamics to a third-order reversible dynamical system. The analysis of this reduced system reveals that the heteroclinic connection representing the physical solution results from the collapse of pairs of symmetric homoclinic orbits. Results of the normal form calculation are compared with numerical solutions obtained by continuation methods. The possibility of self-contact and the elastic characteristics of the perverted rod are also studied.

The simplest normal forms associated with a triple zero eigenvalue of indices one and two

The simplest normal forms associated with a triple zero eigenvalue of indices one and two

Hypernormal form calculation for triple-zero degeneracies

A computational approach to obtain normal forms for equilibrium points of three-dimensional autonomous systems, having a linear degeneracy corresponding to a triple-zero eigenvalue, is presented. Also, we provide the explicit expressions for the normal form coecients, and analyze some additional simplications that can be achieved. The results are applied in the analysis of bifurcation behaviours in an autonomous electronic oscillator.

Computation of Normal Forms of Differential Equations Associated with Non-Semisimple Zero Eigenvalues

This paper presents a method to compute the normal forms of differential equations whose Jacobian evaluated at an equilibrium includes a double zero or a triple zero eigenvalue. The method combines normal form theory with center manifold theory to deal with a general n-dimensional system. Explicit formulas are derived and symbolic computer programs have been developed using a symbolic computation language Maple. This enables one to easily compute normal forms and nonlinear transformations up to any order for a given specific problem. The programs can be conveniently executed on a main frame, workstation or a PC machine without any interaction. Mathematical and practical examples are presented to show the applicability of the method.

On generic dynamics related to a 3-fold zero eigenvalue

In this paper, we consider the bifurcation and stability behaviour of a nonlinear autonomous system in the vicinity of a generic codimension-3 critical point characterized by a triple-zero eigenvalue. The analysis is based on simplified differential equations in normal form and the unification technique coupled with the intrinsic harmonic balancing procedure. The equilibrium solutions, Hopf bifurcations and bifurcations into two-dimensional tori are studied in detail, and the associated stability conditions are presented as well. All results are expressed in terms of system coefficients. An example drawn from nonlinear control systems is analysed to demonstrate the direct applicability of the theory

Homoclinic explosion in the vicinity of bifurcation at the triple-zero eigenvalue

Equations describing small-amplitude motion in the proximity of bifurcation at the triple-zero eigenvalue, after being reduced to a normal form by nonlinear transformations, display a double-scale structure that combines relatively fast conservative orbital motion with slow dissipative evolution of two integrals of motion. The homoclinic explosion occurs when the attractor of the averaged dissipative equations comes close to a homoclinic trajectory of the fast conservative subsystem. Applying the method of matched asymptotic expansions allows one to reduce the dynamical system near this point to a simple next return map possessing an infinite family of invariant manifolds that can turn into attractors of different character under small parametric perturbation. After the attractor of the averaged system has been expelled from the region of closed orbits, one observes large-amplitude dynamics with rare events of anomalous intermittency superimposed upon a quasiperiodic or chaotic attractor of nonuniversal character.

Dynamics of lumped chemically reacting systems near singular bifurcation points

The systematic method of locating and classifying complex dynamical regimes of lumped reactors involves analysis of dynamics near singular bifurcation points at multiple eigenvalues. Amplitude expansion in the proximity of a singular bifurcation combined with multiple time scaling are used to derive universal equations that, though valid quantitatively only in a narrow patch of the parametric space, give full qualitative information on dynamics of a system possessing the respective highest singularity. The principal general result is that an algebraic singularity of a certain order is required to obtain dynamics of a certain dimensionality. In the lowest order, this is cusp singularity for two-dimensional dynamics in the proximity of bifurcation at double zero eigenvalue, and swallowtail singularity for three-dimensional dynamics at triple zero eigenvalue. The basic equation of two-dimensional dynamics describes various transitions between stationary and periodic states as observed, for example, in a first-order exothermic reaction. The equation of three-dimensional dynamics generates more complex phenomena, in particular, chaotic regimes.