Stable and unstable manifold structures in the Hénon family

1999 ◽  
Vol 19 (2) ◽  
pp. 309-338 ◽  
Author(s):  
MARCY BARGE ◽  
BEVERLY DIAMOND
Keyword(s):  
Periodic Orbit ◽  
Periodic Attractor ◽  
Quadratic Map ◽  
Attracting Set ◽  
Kneading Sequence ◽  
Unstable Manifolds ◽  
Stable And Unstable Manifold ◽  
Parameter Values ◽  
Basin Boundaries ◽  
Distinct Period

For parameter values $a$ where the quadratic map $f_a(x) = 1-ax^2$ has an attracting periodic point, and small values of $b$, the Hénon map $H_{a,b}(x, y) = (1 + y - ax^2, bx)$ has a periodic attractor and an attracting set that is homeomorphic with the inverse limit of $f_a|_{[1-a, 1]}$. This attracting set consists of the collection of unstable manifolds of a hyperbolic invariant set together with the attracting periodic orbit. In the case in which $f_a$ is also not renormalizable with smaller period and $b < 0$, we give a symbolic description of the collection of stable manifolds of this hyperbolic set and show that this collection is homeomorphic with the collection of unstable manifolds precisely when the attracting periodic orbit is accessible from the complement of the attracting set, a condition that can be characterized in terms of the kneading sequence of the quadratic map. As an application, we answer a question raised by Hubbard and Oberste-Vorth by proving that the basin boundaries corresponding to three distinct period five sinks in the Hénon family are non-homeomorphic.

ANALYTICAL PREDICTION OF THE TWO FIRST PERIOD-DOUBLINGS IN A THREE-DIMENSIONAL SYSTEM

2000 ◽  
Vol 10 (06) ◽  
pp. 1497-1508 ◽  
Author(s):  
M. BELHAQ ◽  
M. HOUSSNI ◽  
E. FREIRE ◽  
A. J. RODRÍGUEZ-LUIS
Keyword(s):  
Periodic Orbit ◽  
Critical Parameter ◽  
Order Approximation ◽  
Three Dimensional ◽  
Period Doubling ◽  
Dimensional System ◽  
Analytical Prediction ◽  
Parameter Values ◽  
Three Dimensional System ◽  
Period Doubling Bifurcations

Analytical study of the two first period-doubling bifurcations in a three-dimensional system is reported. The multiple scales method is first applied to construct a higher-order approximation of the periodic orbit following Hopf bifurcation. The stability analysis of this periodic orbit is then performed in terms of Floquet theory to derive the critical parameter values corresponding to the first and second period-doubling bifurcations. By introducing suitable subharmonic components in the first order of the multiple scale analysis the two critical parameter values are obtained simultaneously solving analytically the resulting system of two algebraic equations. Comparisons of analytic predictions to numerical simulations are also provided.

BIFURCATIONS OF 2-2-1 HETERODIMENSIONAL CYCLES UNDER TRANSVERSALITY CONDITION

2012 ◽  
Vol 22 (08) ◽  
pp. 1250191
Author(s):  
DAN LIU ◽  
MAOAN HAN ◽  
WEIPENG ZHANG
Keyword(s):  
Periodic Orbit ◽  
Sufficient Conditions ◽  
Transversality Condition ◽  
Moving Frame ◽  
Two Dimensional ◽  
Poincare Mapping ◽  
Unstable Manifolds ◽  
Local Moving Frame ◽  
Vector Formula ◽  
Homoclinic Cycle

Bifurcations of generic 2-2-1 heterodimensional cycles connecting to three saddles, in which two of them have two-dimensional unstable manifolds, are studied by setting up a local moving frame. Under a certain transversal condition, we firstly present the existence, uniqueness and noncoexistence of a 3-point heterodimensional cycle, 2-point heterodimensional or equidimensional cycle, 1-homoclinic cycle and 1-periodic orbit bifurcated from the 3-point heterodimensional cycle, and the bifurcation surfaces and bifurcation regions are located when the u-component [Formula: see text] of the vector [Formula: see text] under the Poincaré mapping [Formula: see text] is nonzero. Conversely, we obtain some sufficient conditions such that the bifurcation of a 2-fold 1-periodic orbit occurs and a 1-periodic orbit coexists with the surviving heterodimensional cycle, showing some new bifurcation behaviors different from the well-known equidimensional cycles.

scholarly journals Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo

2020 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhouchao Wei ◽  
Fanrui Wang ◽  
Huijuan Li ◽  
Wei Zhang
Keyword(s):  
Stability Analysis ◽  
Periodic Orbit ◽  
Impulsive Control ◽  
Hidden Attractor ◽  
Geometric Methods ◽  
Onset Of Chaos ◽  
Specific Parameter ◽  
The Matrix ◽  
Parameter Values ◽  
Deviation Vector

<p style='text-indent:20px;'>In this paper, we make a thorough inquiry about the Jacobi stability of 5D self-exciting homopolar disc dynamo system on the basis of differential geometric methods namely Kosambi-Cartan-Chern theory. The Jacobi stability of the equilibria under specific parameter values are discussed through the characteristic value of the matrix of second KCC invariants. Periodic orbit is proved to be Jacobi unstable. Then we make use of the deviation vector to analyze the trajectories behaviors in the neighborhood of the equilibria. Instability exponent is applicable for predicting the onset of chaos quantitatively. In addition, we also consider impulsive control problem and suppress hidden attractor effectively in the 5D self-exciting homopolar disc dynamo.</p>

Chaos in a model of forced quasi-geostrophic flow over topography: an application of Melnikov's method

1991 ◽  
Vol 226 ◽  
pp. 511-547 ◽  
Author(s):  
J. S. Allen ◽  
R. M. Samelson ◽  
P. A. Newberger
Keyword(s):  
Periodic Orbit ◽  
Saddle Point ◽  
Wind Stress ◽  
Homoclinic Orbits ◽  
Periodic Forcing ◽  
Melnikov's Method ◽  
Geostrophic Flow ◽  
Melnikov’S Method ◽  
Parameter Values ◽  
Time Periodic

We demonstrate the existence of a chaotic invariant set of solutions of an idealized model for wind-forced quasi-geostrophic flow over a continental margin with variable topography. The model (originally formulated to investigate mean flow generation by topographic wave drag) has bottom topography that slopes linearly offshore and varies sinusoidally alongshore. The alongshore topographic scales are taken to be short compared to the cross-shelf scale, allowing Hart's (1979) quasi-two-dimensional approximation, and the governing equations reduce to a non-autonomous system of three coupled nonlinear ordinary differential equations. For weak (constant plus time-periodic) forcing and weak friction, we apply a recent extension (Wiggins & Holmes 1987) of the method of Melnikov (1963) to test for the existence of transverse homoclinic orbits in the model. The inviscid unforced equations have two constants of motion, corresponding to energy E and enstrophy M, and reduce to a one-degree-of-freedom Hamiltonian system which, for a range of values of the constant G = E − M, has a pair of homoclinic orbits to a hyperbolic saddle point. Weak forcing and friction cause slow variations in G, but for a range of parameter values one saddle point is shown to persist as a hyperbolic periodic orbit and Melnikov's method may be applied to study the perturbations of the associated homoclinic orbits. In the absence of time-periodic forcing, the hyperbolic periodic orbit reduces to the unstable fixed point that occurs with steady forcing and friction. The method yields analytical expressions for the parameter values for which sets of chaotic solutions exist for sufficiently weak time-dependent forcing and friction. The predictions of the perturbation analysis are verified numerically with computations of Poincaré sections for solutions in the stable and unstable manifolds of the hyperbolic periodic orbit and with computations of solutions for general initial-value problems. In the presence of constant positive wind stress τ0 (equatorward on eastern ocean boundaries), chaotic solutions exist when the ratio of the oscillatory wind stress τ1 to the bottom friction parameter r is above a critical value that depends on τ0/r and the bottom topographic height. The analysis complements a previous study of this model (Samelson & Allen 1987), in which chaotic solutions were observed numerically for weak near-resonant forcing and weak friction.

FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS

1994 ◽  
Vol 04 (04) ◽  
pp. 933-941 ◽  
Author(s):  
ANDREW L. KATZ ◽  
EARL H. DOWELL
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Stable And Unstable Manifolds ◽  
Detailed Explanation ◽  
Single Well ◽  
Unstable Manifolds ◽  
Parameter Values

The study of stable and unstable manifolds, and their intersections with each other, is a powerful technique for interpreting complex bifurcations of nonlinear systems. The escape phenomenon in the twin-well Duffing oscillator is one such bifurcation that is elucidated through the analysis of manifold intersections. In this paper, two escape scenarios in the twin-well Duffing oscillator are presented. In each scenario, the relevant manifold structures are examined for parameter values on either side of the escape bifurcation. Included is a description of the role of the hilltop saddle stable manifolds, which are known to separate the single well basins (should single well attractors exist). In each of the two bifurcation scenarios, it is shown through a detailed analysis of Poincaré maps that a homoclinic intersection of the manifolds of a specific period-3 saddle implies the destruction of the single well chaotic attractor. Although the Duffing oscillator is used to illustrate the ideas advanced here, it is thought that the approach will be useful for a variety of dynamical systems.

FRACTAL AND WADA EXIT BASIN BOUNDARIES IN TOKAMAKS

2007 ◽  
Vol 17 (11) ◽  
pp. 4067-4079 ◽  
Author(s):  
JEFFERSON S. E. PORTELA ◽  
IBERÊ L. CALDAS ◽  
RICARDO L. VIANA ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Fractal Structure ◽  
Dynamical Structure ◽  
Stable And Unstable Manifolds ◽  
Chaotic Region ◽  
Chaotic Field ◽  
Unstable Manifolds ◽  
Field Lines ◽  
Basin Boundaries ◽  
Wada Property ◽  
Control Plasma

The creation of an outer layer of chaotic magnetic field lines in a tokamak is useful to control plasma-wall interactions. Chaotic field lines (in the Lagrangian sense) in this region eventually hit the tokamak wall and are considered lost. Due to the underlying dynamical structure of this chaotic region, namely a chaotic saddle formed by intersections of invariant stable and unstable manifolds, the exit patterns are far from being uniform, rather presenting an involved fractal structure. If three or more exit basins are considered, the respective basins exhibit an even stronger Wada property, for which a boundary point is arbitrarily close to points belonging to all exit basins. We describe such a structure for a tokamak with an ergodic limiter by means of an analytical Poincaré field line mapping.

scholarly journals A model for aperiodicity in earthquakes

2008 ◽  
Vol 15 (1) ◽  
pp. 1-12 ◽  
Author(s):  
B. Erickson ◽  
B. Birnir ◽  
D. Lavallée
Keyword(s):  
Periodic Orbit ◽  
Numerical Solutions ◽  
Period Doubling ◽  
Stationary Solutions ◽  
Broadband Noise ◽  
Single Oscillator Model ◽  
State Dependent ◽  
Single Oscillator ◽  
Block Models ◽  
Parameter Values

Abstract. Conditions under which a single oscillator model coupled with Dieterich-Ruina's rate and state dependent friction exhibits chaotic dynamics is studied. Properties of spring-block models are discussed. The parameter values of the system are explored and the corresponding numerical solutions presented. Bifurcation analysis is performed to determine the bifurcations and stability of stationary solutions and we find that the system undergoes a Hopf bifurcation to a periodic orbit. This periodic orbit then undergoes a period doubling cascade into a strange attractor, recognized as broadband noise in the power spectrum. The implications for earthquakes are discussed.

Dynamic Stabilization of Unstable Periodic Orbits in a CO2 Laser by Slow Modulation of Cavity Detuning

1998 ◽  
Vol 08 (09) ◽  
pp. 1783-1789 ◽  
Author(s):  
A. N. Pisarchik ◽  
R. Corbalán ◽  
V. N. Chizhevsky ◽  
R. Vilaseca ◽  
B. F. Kuntsevich
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Transient Response ◽  
Co2 Laser ◽  
Dynamic Stabilization ◽  
Dynamical Regime ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Periodic Attractor ◽  
Slow Modulation

We demonstrate numerically and experimentally that a slow modulation of cavity detuning in a loss-modulated CO 2 laser can stabilize unstable periodic orbits even when the system remains in a particular dynamical regime for adiabatic changes of the detuning. When the parameter changes faster than the transient response of deformation of the original periodic attractor, the system can evolve toward an unstable periodic orbit.

The Dynamics of Symmetric Bimodal Maps

1997 ◽  
Vol 07 (12) ◽  
pp. 2735-2744 ◽  
Author(s):  
Thomas Lofaro
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Period Doubling ◽  
General Setting ◽  
The Family ◽  
Pitchfork Bifurcations ◽  
Parameter Values ◽  
Kneading Theory ◽  
Period Doubling Bifurcations

The dynamics and bifurcations of a family of odd, symmetric, bimodal maps, fα are discussed. We show that for a large class of parameter values the dynamics of fα can be described via an identification with a unimodal map uα. In this parameter regime, a periodic orbit of period 2n + 1 of uα corresponds to a periodic orbit of period 4n + 2 for fα. A periodic orbit of period 2n of uα corresponds to a pair of distinct periodic orbits also of period 2n for fα. In a more general setting we describe the genealogy of periodic orbits in the family fα using symbolic dynamics and kneading theory. We identify which periodic orbits of even periods are born in period-doubling bifurcations and which are born in pitchfork bifurcations and provide a method of describing the "ancestors" and "descendants" of these orbits. We also show that certain periodic orbits of odd periods are born in saddle-node bifurcations.

ANALYTICS OF HOMOCLINIC BIFURCATIONS IN THREE-DIMENSIONAL SYSTEMS

2002 ◽  
Vol 12 (11) ◽  
pp. 2479-2486 ◽  
Author(s):  
MOHAMED BELHAQ ◽  
FAOUZI LAKRAD
Keyword(s):  
Periodic Orbit ◽  
Analytical Approach ◽  
Critical Parameter ◽  
Multiple Scales ◽  
Three Dimensional ◽  
Specific System ◽  
Homoclinic Bifurcations ◽  
Hyperbolic Fixed Point ◽  
Parameter Values ◽  
Multiple Scales Perturbation Method

An analytical approach to determine critical parameter values of homoclinic bifurcations in three-dimensional systems is reported. The homoclinic orbit is supposed to be a limit of a unique periodic orbit. Hence, the multiple scales perturbation method is performed to construct an approximation of the periodic solution and its frequency. Then, two simple criteria are used. The first criterion is based on the collision between the periodic and the hyperbolic fixed point involved in the bifurcation. The second uses the infinity condition of the period of the periodic orbit. For illustration a specific system is investigated.

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