Birkhoff signature change: a criterion for the instability of chaotic resonance

1992 ◽  
Vol 338 (1651) ◽  
pp. 557-568 ◽  
Keyword(s):  
Homoclinic Orbit ◽  
Invariant Manifolds ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Global Dynamics ◽  
Significant Information ◽  
Signature Change ◽  
Numerical Effort ◽  
Initial Change

For periodically forced nonlinear oscillators permitting escape from a potential well a relation is observed between two well-known phenomena, the period-doubling cascade leading to the chaotic escape of the resonant attractor and the complex dynamics associated with the creation of a structurally unstable homoclinic orbit. The particular homoclinic orbit is identified as that created at the initial change of the period one Birkhoff signature of the invariant manifolds of the hilltop saddle. The primary resonant attractor may thus be viewed as the period one simple Newhouse orbit. Significant subharmonic and superharmonic escape events may likewise be associated with nearby Birkhoff signature changes. Significant information about the global dynamics may thus be obtained with little numerical effort by inspection of the signatures of the invariant manifolds of the hilltop saddle.

Topology of the invariant manifolds of period-doubling attractors for some forced nonlinear oscillators

1983 ◽  
Vol 96 (6) ◽  
pp. 269-272 ◽  
Author(s):  
Peter Beiersdorfer ◽  
Jean-Marie Wersinger ◽  
Yvain Treve
Keyword(s):  
Invariant Manifolds ◽  
Period Doubling ◽  
Nonlinear Oscillators

Complex dynamics resulting from repeated stimulation of nonlinear oscillators at a fixed phase

1987 ◽  
Vol 125 (2-3) ◽  
pp. 119-122 ◽  
Author(s):  
John Lewis ◽  
Manjit Bachoo ◽  
Leon Glass ◽  
Canio Polosa
Keyword(s):  
Complex Dynamics ◽  
Nonlinear Oscillators ◽  
Repeated Stimulation ◽  
Fixed Phase ◽  
Stimulation Of

LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

2011 ◽  
Vol 21 (06) ◽  
pp. 1617-1636 ◽  
Author(s):  
SOMA DE ◽  
PARTHA SHARATHI DUTTA ◽  
SOUMITRO BANERJEE ◽  
AKHIL RANJAN ROY
Keyword(s):  
Three Dimensional ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Numerical Continuation ◽  
Homoclinic Tangency ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Border Collision Bifurcation ◽  
Smooth Map

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

COMPLEX DYNAMICS AND FAST-SLOW SCALE INSTABILITY IN CURRENT-MODE CONTROLLED BUCK CONVERTER WITH CONSTANT CURRENT LOAD

2013 ◽  
Vol 23 (04) ◽  
pp. 1350062 ◽  
Author(s):  
GUOHUA ZHOU ◽  
BOCHENG BAO ◽  
JIANPING XU
Keyword(s):  
Complex Dynamics ◽  
Period Doubling ◽  
Current Mode ◽  
Input Voltage ◽  
Second Order ◽  
Buck Converter ◽  
Constant Current ◽  
Current Load ◽  
Time Model ◽  
Conduction Mode

The complex dynamics and coexisting fast-slow scale instability in current-mode controlled buck converter with constant current load (CCL), operating in both continuous conduction mode (CCM) and discontinuous conduction mode (DCM), are investigated in this paper. Via cycle-by-cycle computer simulation and experimental measurement of current-mode controlled buck converter with CCL, it is found that a unique fast-slow scale instability exists in the second-order switching converter. It is also found that a unique period-doubling accompanied by Neimark–Sacker bifurcation exists in this simple second-order converter, which is different from period-doubling or Neimark–Sacker bifurcations reported previously. Based on a nonlinear discrete-time model and the corresponding Jacobian, the effects of CCL and input voltage on the dynamics of current-mode controlled buck converter are investigated and verified theoretically. Fixed point analysis for slow-scale low-frequency oscillation is also given to verify the dynamics and the coexisting fast-slow scale instability.

Bifurcations and chaos control in a discrete-time biological model

2020 ◽  
Vol 13 (04) ◽  
pp. 2050022 ◽  
Author(s):  
A. Q. Khan ◽  
T. Khalique
Keyword(s):  
Discrete Time ◽  
Chaos Control ◽  
Control Method ◽  
Complex Dynamics ◽  
Small Neighborhood ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Flip Bifurcation ◽  
Time Model ◽  
Predator Prey Model

In this paper, bifurcations and chaos control in a discrete-time Lotka–Volterra predator–prey model have been studied in quadrant-[Formula: see text]. It is shown that for all parametric values, model has boundary equilibria: [Formula: see text], and the unique positive equilibrium point: [Formula: see text] if [Formula: see text]. By Linearization method, we explored the local dynamics along with different topological classifications about equilibria. We also explored the boundedness of positive solution, global dynamics, and existence of prime-period and periodic points of the model. It is explored that flip bifurcation occurs about boundary equilibria: [Formula: see text], and also there exists a flip bifurcation when parameters of the discrete-time model vary in a small neighborhood of [Formula: see text]. Further, it is also explored that about [Formula: see text] the model undergoes a N–S bifurcation, and meanwhile a stable close invariant curves appears. From the perspective of biology, these curves imply that between predator and prey populations, there exist periodic or quasi-periodic oscillations. Some simulations are presented to illustrate not only main results but also reveals the complex dynamics such as the orbits of period-2,3,13,15,17 and 23. The Maximum Lyapunov exponents as well as fractal dimension are computed numerically to justify the chaotic behaviors in the model. Finally, feedback control method is applied to stabilize chaos existing in the model.

scholarly journals Complex Dynamics of an Impulsive Chemostat Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950101 ◽  
Author(s):  
Jin Yang ◽  
Yuanshun Tan ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Global Stability ◽  
Substrate Concentration ◽  
Complex Dynamics ◽  
Control Strategies ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Chemostat Model ◽  
Existence And Stability ◽  
Theoretical Results

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincaré map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-[Formula: see text] [Formula: see text] periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed.

Dynamics of Convective Thermal Explosion in Porous Media

2020 ◽  
Vol 30 (06) ◽  
pp. 2050081 ◽  
Author(s):  
Karam Allali ◽  
Youssef Joundy ◽  
Ahmed Taik ◽  
Vitaly Volpert
Keyword(s):  
Porous Media ◽  
Thermal Explosion ◽  
Complex Dynamics ◽  
Saturated Porous Medium ◽  
Period Doubling ◽  
Boussinesq Approximation ◽  
Rectangular Domain ◽  
Chaotic Regime ◽  
Nonlinear Heat Equation ◽  
Convective Thermal

In this paper, we study complex dynamics of the interaction between natural convection and thermal explosion in porous media. This process is modeled with the nonlinear heat equation coupled with the nonstationary Darcy equation under the Boussinesq approximation for a fluid-saturated porous medium in a rectangular domain. Numerical simulations with the Radial Basis Functions Method (RBFM) reveal complex dynamics of solutions and transitions to chaos after a sequence of period doubling bifurcations. Several periodic windows alternate with chaotic regimes due to intermittence or crisis. After the last chaotic regime, a final periodic solution precedes transition to thermal explosion.

scholarly journals Local and Global Dynamics of a Constraint Profit Maximization for Bischi–Naimzada Competition Duopoly Game

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1458 ◽  
Author(s):  
Sameh S Askar ◽  
Abdulrahman Al-Khedhairi
Keyword(s):  
Complex Dynamics ◽  
Market Competition ◽  
Equilibrium Points ◽  
Profit Maximization ◽  
Stable Complex ◽  
Global Dynamics ◽  
Limited Information ◽  
Critical Curves ◽  
Local And Global Dynamics ◽  
Realistic Situation

The Bischi–Naimzada game is a market competition between two firms with the objective of maximizing profits under limited information. In this article, we study a more generalized and realistic situation that takes into account the sales constraints. we generalize the economic model suggested by Bischi–Naimzada by introducing and studying the maximization of profits based on sales constraints. Our motivation in this paper is the studying of profit and sales constraints maximization and their influences on the game’s dynamics. The local stability of the equilibrium points of the proposed model is discussed. It examines how the dynamics of the proposed two-dimensional competition game model focusing on changes in both the speed of the adjustment and the sales constraint parameters. The map describing the game is proven to be noninvertible and yields many multi-stable, complex dynamics and the coexistence chaotic attractors may arise. The global behavior of the map is achieved by studying the critical curves. The numerical simulations demonstrate the coexistence of two attractors and complex structures of the attraction basins. Several examples are discussed in order to confirm all the analytical results obtained.

Bifurcation Dynamics and Pattern Formation of a Three-Species Food Chain System with Discrete Spatiotemporal Variables

2019 ◽  
Vol 74 (11) ◽  
pp. 945-959
Author(s):  
Huayong Zhang ◽  
Ge Pan ◽  
Tousheng Huang ◽  
Tianxiang Meng ◽  
Jieru Wang ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Food Chain ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Turing Instability ◽  
Flip Bifurcation ◽  
Chain System ◽  
Self Organized ◽  
New Perspective ◽  
Centre Manifold Theorem

AbstractThe bifurcation dynamics and pattern formation of a discrete-time three-species food chain system with Beddington–DeAngelis functional response are investigated. Via applying the centre manifold theorem and bifurcation theorems, the occurrence conditions for flip bifurcation and Neimark–Sacker bifurcation as well as Turing instability are determined. Numerical simulations verify the theoretical results and reveal many interesting dynamic behaviours. The flip bifurcation and the Neimark–Sacker bifurcation both induce routes to chaos, on which we find period-doubling cascades, invariant curves, chaotic attractors, sub–Neimark–Sacker bifurcation, sub–flip bifurcation, chaotic interior crisis, sub–period-doubling cascade, periodic windows, sub–periodic windows, and various periodic behaviours. Moreover, the food chain system exhibits various self-organized patterns, including regular and irregular patterns of stripes, labyrinth, and spiral waves, suggesting the populations can coexist in space as many spatiotemporal structures. These analysis and results provide a new perspective into the complex dynamics of discrete food chain systems.

BIFURCATION STRUCTURE OF SCALAR DIFFERENTIAL DELAYED EQUATIONS

1991 ◽  
Vol 01 (03) ◽  
pp. 657-665 ◽  
Author(s):  
C. P. MALTA ◽  
C. GROTTA RAGAZZO
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Negative Feedback ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Feedback Condition ◽  
Numerical Results ◽  
The Other ◽  
Bifurcation Structure ◽  
Dominant Frequencies

We study periodic solutions of the equation [Formula: see text], with f(X) given by f1(X) = AX(1 − X) or f2(X) = πµ (1 − sin X), grouped in some sets characterized by different dominant frequencies. Numerical results with f(X) = f1(X) are given. One of these sets is shown to exhibit period-doubling cascade in the direction of both parameters A and τ. The other sets are shown to exhibit many other period-doubling cascades as τ is varied establishing a relation between the bifurcation structure within the sets. Furthermore we obtain a lower bound on A and µ for the existence of more complex dynamics. We conjecture that this fact is related to the violation of the so-called "negative-feedback condition."

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