Reverse Bifurcation Research Articles

Chaotic behavior and multistability of a tree trunk structure driven by self-and parametric hydrodynamic forces

Abstract This study investigates the chaotic behavior of a tree trunk under dynamic wind loads, focusing on control strategies and multistability. We consider time-varying wind speeds and analyze a specific case where hydrodynamic drag forces align with the flow velocity. The stability of the model’s equilibrium points is analyzed theoretically and numerically. Melnikov’s method is employed to identify conditions for homoclinic bifurcation. Numerical simulations employing basin of attraction confirm the analytical predictions. Our findings show a decrease in the threshold for chaos with increasing amplitudes of external excitation, damping coefficient, and parametric damping. The global dynamics are explored numerically using a fourth-order Runge-Kutta method. When solely subjected to external excitation, the system exhibits period doubling bifurcations, multiperiodic oscillations, mixed-mode oscillations, and chaos. Conversely, with self- and parametric drag forces, the system displays reverse periodic bifurcations, periodic bubbling oscillations, antimonotonicity, transient chaos, and chaos. Poincaré maps analyze the geometric structure of chaotic attractors, revealing a strong influence of dimensionless drag parameters. These parameters can be manipulated to control or eliminate chaos. Furthermore, the system exhibits multistability, with coexisting attractors. Beyond its application in protecting infrastructure from wind damage, this research can contribute to ecological balance by improving our understanding of tree wind resistance.

Self-organization of PIP3 waves is controlled by the topology and curvature of cell membranes

Phosphatidylinositol (3,4,5)-trisphosphate (PIP3) is a signaling lipid on the plasma membrane that plays a fundamental role in cell signaling with a strong impact on cell physiology and diseases. It is responsible for the protruding edge formation, cell polarization, macropinocytosis, and other membrane remodeling dynamics in cells. It has been shown that the membrane confinement and curvature affects the wave formation of PIP3 and F-actin. But, even in the absence of F-actin, a complex self-organization of the spatiotemporal PIP3 waves is observed. In recent findings, we have shown that these waves can be guided and pinned on strongly bended Dictyostelium membranes caused by molecular crowding and curvature-limited diffusion. Based on these experimental findings, we investigate the spatiotemporal PIP3 wave dynamics on realistic three-dimensional cell-like membranes to explore the effect of curvature-limited diffusion, as observed experimentally. We use an established stochastic reaction-diffusion model with enzymatic Michaelis-Menten-type reactions that mimics the dynamics of Dictyostelium cells. As these cells mimic the three-dimensional shape and size observed experimentally, we found that the PIP3 wave directionality can be explained by a Hopf-like and a reverse periodic-doubling bifurcation for uniform diffusion and curvature-limited diffusion properties. Finally, we compare the results with recent experimental findings and discuss the discrepancy between the biological and numerical results.

Energy function and complex dynamics from a jerk system

Jerk, as a mathematical concept, is used in mechanics to describe the rate of change of acceleration and plays a crucial role in the design of control systems for machines and vehicles. Therefore, it is important to master the various states and the energy released during the change of acceleration. This is why a new simple jerk function introduced afterward, energy released, is derived from a Hamilton function using the Helmholtz theorem. The condition of having a stable or unstable rate of change of acceleration is established using Hopf bifurcation theory. Some two-parameter stability charts are then computed for a suitable selection region of the study. Using some nonlinear analysis metrics, in the unstable region of the study, the occurrence of phenomena is found, such as reverse period doubling bifurcation, antimonotonicity, and hysteresis involving the coexistence of the states in the considered jerk system. An electronic circuit is built and used to implement the mathematical expression of the jerk equation and validate the result of the theoretical investigation.

Experiment of a thermal plume on an open cylinder

In this study, we performed a set of experiments of the thermal plume on an open cylinder heated from below and a simple scaling analysis. The flow structure of the plume was visualized using the shadowgraph technique, and the temperature at specific points was measured using a thermistor. Transient plumes in a developing stage, an equilibrating stage and a fully developed stage were described. The new scaling laws of the stem radius and the velocity of the rising plume were presented, which are different from those on a two-dimensional heated plate due to the cylindrical effect and agree with experimental results. In the fully developed stage, there is a transition route of the plume from a steady to chaotic state with an increase in the Rayleigh number, which involves a series of bifurcations. A reverse bifurcation from a periodic (Ra = 1.17 × 106) back to a steady (Ra = 1.30 × 106) state has been observed in the experiment, which is referred to as the period bubbling bifurcation. Thus, the present experimental results validate the previous numerical results presented by Zhang et al. (Phys. Fluids, vol. 33 (6), 2021, 064110). The bifurcation diagram, spectral analysis and attractor are used to characterize the transitional plume. In addition, the heat and mass transfer have also been quantified.

Effects of periodic parametric damping and amplitude-modulated signal on vibrational resonance and torus-doubling bifurcations occurrence in an asymmetric mixed Rayleigh-Liénard oscillator

This research paper examines the effects of periodic parametric damping and amplitude-modulated signal on vibrational resonance and the occurrence of torus-doubling bifurcations in an asymmetric mixed Rayleigh-Liénard oscillator. The method of direct separation of the slow and fast motions is used to derive the approximate theoretical expression of response amplitude at the low frequency. The obtained results show that the presence of periodic parametric damping induces in the system multiple resonance peaks when the low frequency is varied. Moreover, the increase of carrier amplitude modulated increases or decreases the maximum amplitude value in certain range of the low frequency. However, when the periodic parametric damping coefficient is varied, one resonance peak occurs and the maximum amplitude value increases when the carrier amplitude modulated increases. The theoretical and direct numerical predictions have shown a fairly satisfactory agreement. On the other hand, the global dynamical changes of the system are numerically examined in context of vibrational resonance. It is found that, the system displays many torus attractors of different topologies, torus-doubling bifurcations, reverse torus-doubling bifurcations and torus-chaos. These observations are illustrated by plotting the phase portraits and their corresponding Poincaré maps.

Complex Dynamics Modes in a Simple Model of Prey-Predator Community: Bistability and Multistability

The paper proposes and studies a two-component discrete-time model of the prey-predator community considering zooplankton and fish interactions and their development features. Discrete-time systems of equations allow us to take into account naturally the rhythm of many processes occurring in marine and freshwater communities, which are subject to cyclical fluctuations due to the daily and seasonal cycle. We describe the dynamics of fish and zooplankton populations constituting the community by Ricker’s model, which is well-studied and widely used in population modeling. To consider the species interaction, we use the Holling-II type response function taking into account predator saturation. We carried out the study of the proposed model. The system is shown to have from one to three non-trivial equilibria, which gives the existence of the complete community. In addition to the saddle-node bifurcation, which generates bistability of stationary dynamics, a nontrivial equilibrium loses its stability according to the Neimark-Sacker scenario with an increase in the reproductive potential of both predator and prey species, as a result of which the community exhibits long-period oscillations similar to those observed in experiments. With the higher bifurcation parameter, the reverse Neimark-Sacker bifurcation is shown to occur followed by the closed invariant curve collapses, and dynamics of the population stabilizes, later losing stability through a cascade of period-doubling bifurcations. Multistability complicates the birth and disappearance of the invariant curve in the phase space scenario by the emergence of another irregular dynamics in the system with the single unstable nontrivial fixed point. At fixed values of the model parameters and different initial conditions, the system considered is shown to demonstrate various quasi-periodic oscillations. Despite extreme simplicity, the proposed discrete-time model of community dynamics demonstrates a wide variety and variability of dynamic modes. It shows that the influence of environmental conditions can change the type and nature of the observed dynamics.

Bifurcation cascade, self-similarity, and duality in the three-rotor problem.

The three-rotor system concerns equally massive point particles moving on a circle subject to attractive cosine potentials of strength g. The quantum theory models chains of coupled Josephson junctions. Classically, it displays order-chaos-order behavior with increasing energy E along with a seemingly globally chaotic phase for 5.33g≲E≲5.6g. It is also known to admit pendulum and isosceles breather families of periodic orbits at all energies. While pendula display a doubly infinite sequence of stability transitions accumulating at their libration to rotation threshold at E=4g, breathers undergo only one stability transition. Here, we show that these stability transitions are associated with forward and reverse fork-like isochronous and period-doubling bifurcations. The new family of periodic orbits born at each of these bifurcations is found using an efficient search algorithm starting from a transverse perturbation to the parent orbit. The graphs of stability indices of various classes of orbits born at pendulum bifurcations meet at E=4g forming "fans." The transitions in the librational and rotational phases are related by an asymptotic duality between bifurcation energies and shapes of newly born periodic orbits. The latter are captured by solutions to a Lamé equation. We also find and numerically validate values of scaling constants for self-similarity in (a) stability indices of librational and rotational pendula and (b) shapes of newly born orbits as E→4g. Finally, we argue that none of the infinitely many families of periodic orbits we have found is stable for 5.33g≲E≲5.6g, providing further evidence for global chaos in this energy band.

Design of a New Chaotic System with Sine Function: Dynamic Analysis and Offset Boosting Control

A new chaotic system is presented in this research work.The proposed system has three nonlinear terms and one sine term which improves the complexity of the system. The basic properties of new system such as Lyapunov exponent, equilibrium point and stability are analyzed in detail. The dynamic analysis is conducted using classic tools such as bifurcation diagram and Lyapunov exponent plot to verify the chaotic nature in the proposed system. The changes in the states of the system is verified using bifurcation diagram and Lyapunov exponent plot. The proposed system presents some special features such as two wing attractors, forward and reverse periodic doubling bifurcation, and dc offset boosting control. The dc offset boosting behavior can be used to diagnosis the multistability behaviour in the dynamical system and to reduce the number of components in the communication system. This special feature converts the bipolar signal in to unipolar signal which can be used in many engineering applications. The theoretical study and the simulation results show that the proposed system has wealthy chaotic behaviour itself. Furthermore, the adaptive synchronization of identical new system is achieved for the application of secure communication system.

Transitions between peace and systemic war as bifurcations in a signed network dynamical system

AbstractWe investigate structural features and processes associated with the onset of systemic conflict using an approach which integrates complex systems theory with network modeling and analysis. We present a signed network model of cooperation and conflict dynamics in the context of international relations between states. The model evolves ties between nodes under the influence of a structural balance force and a dyad-specific force. Model simulations exhibit a sharp bifurcation from peace to systemic war as structural balance pressures increase, a bistable regime in which both peace and war stable equilibria exist, and a hysteretic reverse bifurcation from war to peace. We show how the analytical expression we derive for the peace-to-war bifurcation condition implies that polarized network structure increases susceptibility to systemic war. We develop a framework for identifying patterns of relationship perturbations that are most destabilizing and apply it to the network of European great powers before World War I. We also show that the model exhibits critical slowing down, in which perturbations to the peace equilibrium take longer to decay as the system draws closer to the bifurcation. We discuss how our results relate to international relations theories on the causes and catalysts of systemic war.

The disturbance influence on vibration of a belt device driven by a crank mechanism

As well as technological forces, real-life belt and conveyor systems are loaded by disturbing effects that have a mostly random character. To analyze importance of their influence on belt and conveyor system behavior, a computational model was set up in which the disturbances were represented by the discretization and round-off errors occurring during the solving of the governing equations, which is a new and original idea. The simulations that were carried out show that the disturbances can spawn hidden and co-existing attractors and transitions between vibrations of different periods or between regular and chaotic motions and reverse bifurcations. The main contribution of the paper is a new and original way to modeling of the systems’ inaccuracies and disturbances having frequently a random character by means of numerical inaccuracies caused by round-off errors and set tolerances of the solution methods for solving the equations of motion. The computational simulations give evidence that these factors have the same effect as inaccuracies and uncertainties of real machines on their behavior. The simulation showed that these factors can lead to qualitatively different system behavior, like the transitions between the regular and chaotic regimes. The physical validity of all operating regimes was proved by the theorem of the time change of the kinetic energy. The main goal of the presented article is to point out that small deviations from the rated state can cause qualitatively different behavior of the machine systems.

Analysis and microcontroller implementation of Josephson junction driven by Wien bridge circuit

This paper recounts the dynamical investigations and microcontroller validation of Josephson junction (JJ) driven by Wien bridge circuit (WBC). Relying on the stability analysis of two equilibrium points found in JJ driven by WBC (JJDWBC), it is revealed that one equilibrium point is unconditionally stable while the other equilibrium point is unstable. The JJDWBC exhibits reverse period-doubling bifurcation, periodic attractors, period-doubling bifurcation, bistable chaotic attractors, and different presentations of monostable chaotic attractors. Partial and total amplitude controls are achieved by adding two controller parameters into the rate equations of JJDWBC. Finally, the microcontroller implementation is used to validate the dynamical behaviors found in JJDWBC.

Dynamics and bifurcation characteristics of actively Q-switched Erbium-doped fiber ring laser based on EOM with large modulation index

Dynamic and bifurcation characteristics of actively Q-switched Erbium-doped fiber ring laser (EDFRL) with large modulation index are investigated theoretically and experimentally. Using pump power and modulation index as control parameters, the laser exhibits a pulse splitting dynamic response process. More pulse splitting states such as double pulse, triple pulse, and quadruple pulse can be achieved in the process of increasing pump power and modulation index, while chaotic outputs are present during the pulse splitting evolution. The dynamic response of the output due to modulation index and modulation frequency change is also discussed. At large modulation index, the formed multiple periodic route exhibits a reverse multiple periodic bifurcation after the maximum multiple period. The dynamic response of Q-switched EDFRL with large modulation index contributes to a more comprehensive understanding of the dynamic characteristics of EDFRL and its applications.

Dynamics of full-coupled chains of a great number of oscillators with a large delay in couplings

The subject of this work is the study of local dynamics of full-coupled chains of a great number of oscillators with a large delay in couplings. From a discrete model describing the dynamics of a great number of coupled oscillators, a transition has been made to a nonlinear integro-differential equation, continuously depending on time and space variable. A class of full-coupled systems has been considered. The main assumption is that the amount of delay in the couplings is large enough. This assumption opens the way to the use of special asymptotic methods of study. The parameters under which the critical case is realized in the problem of the equilibrium state stability have been distinguished. It is shown that they have infinite dimension. The analogues of normal forms — nonlinear boundary value problems of Ginzburg–Landau type have been constructed. In some cases, these boundary value problems contain integral components too. Their nonlocal dynamics describes the behavior of all solutions of the original equations in the balance state neighbourhood. Methods. As applied to the considered problems, methods of constructing quasinormal forms on central manifolds are developed. An algorithm for constructing the asymptotics of solutions based on the use of quasinormal forms for determining slowly varying amplitudes has been created. Results. Quasinormal forms that determine the dynamics of the original boundary value problem have been constructed. The dominant terms of asymptotic approximations for solutions of the considered chains have been obtained. On the basis of the given statements, a number of interesting dynamical effects have been revealed. For example, an infinite alternation of direct and reverse bifurcations when the delay coefficient increases. Their distinguishing feature is that they have two or three spatial variables.

A Memristor-Based Colpitts Oscillator Circuit

This paper investigates a simple memristor emulator consisting of a diode bridge and a capacitor. It exhibits pinched hysteresis loops, and what is more striking is the higher frequency, as it operates up to greater than 5 MHz. Based on the proposed memristor, a higher-frequency Colpitts circuit was established. According to the mathematical model of the system, the system only possesses one unstable equilibrium point. Period doubling bifurcation, reverse periodic doubling bifurcation, different types of periodic and chaotic orbits, transient chaos, coexisting bifurcations and offset boosting are depicted. More interestingly, it has coexisting multiple attractors with different topologies, such as a chaotic attractor accompanied with periodic orbits, period-1 orbits with bicuspid structure and periodic-2 orbits with tridentate structure. Moreover, a hardware circuit using discrete components was fabricated and experimental measurements were consistent with the MATLAB numerical results, further confirming the real feasibility of the proposed circuit.

New 4D and 3D models of chaotic systems developed from the dynamic behavior of nuclear reactors.

The complex, highly nonlinear dynamic behavior of nuclear reactors can be captured qualitatively by novel four-dimensional (that is, fourth order) and three-dimensional (that is, third order) models of chaotic systems and analyzed with Lyapunov spectra, bifurcation diagrams, and phase diagrams. The chaotic systems exhibit a rich variety of bifurcation phenomena, including the periodic-doubling route to chaos, reverse bifurcations, anti-monotonicity, and merging chaos. The offset boosting method, which relocates the attractor's basin of attraction in any direction, is demonstrated in these chaotic systems. Both constant parameters and periodic functions are seen in offset boosting phenomena, yielding chaotic attractors with controlled mean values and coexisting attractors.

Dynamic Effects of a MCNN-CS Under Electromagnetic Induction and Its Application

Cellular neural network (CNN) is one of the best artificial neural network models due to its hardware implementability and many applications for image processing. The memristor-based cellular neural network (MCNN) combines the distinct benefits of memristors and neural networks, and it excels in mimicking chaotic systems under the effect of electromagnetic induction. In this work, a novel CNN chaotic system based on a flux-controlled memristor (MCNN-CS) with electromagnetic induction effects is constructed. Dynamical behaviors are examined by controlling the magnetic flux leakage strength and the intensity of the electromagnetic induction related to the memristor. Based on stability analyses and numerical simulation, forward and reverse period-doubling bifurcations, various coexisting attractors, multiperiods, chaos, several periodic windows, and chaotic crisis are observed. Besides, a number of complicated phenomena, including transient chaos, intermittent chaos, sustained chaos and bistability are also observed, demonstrating that this chaotic system has a wide range of dynamic properties. Finally, we generate pseudo-random number sequences based on this new system and apply them to image encryption. The obtained results show that this chaotic system offers superior randomness and is hence appropriate for image encryption applications.

On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps

We consider reversible nonconservative perturbations of the conservative cubic Hénon maps \(H_{3}^{\pm}:\bar{x}=y,\bar{y}=-x+M_{1}+M_{2}y\pm y^{3}\) and study their influence on the 1:3 resonance, i. e., bifurcations of fixed points with eigenvalues \(e^{\pm i2\pi/3}\). It follows from [1] that this resonance is degenerate for \(M_{1}=0,M_{2}=-1\) when the corresponding fixed point is elliptic. We show that bifurcations of this point under reversible perturbations give rise to four 3-periodic orbits, two of them are symmetric and conservative (saddles in the case of map \(H_{3}^{+}\) and elliptic orbits in the case of map \(H_{3}^{-}\)), the other two orbits are nonsymmetric and they compose symmetric couples of dissipative orbits (attracting and repelling orbits in the case of map \(H_{3}^{+}\) and saddles with the Jacobians less than 1 and greater than 1 in the case of map \(H_{3}^{-}\)). We show that these local symmetry-breaking bifurcations can lead to mixed dynamics due to accompanying global reversible bifurcations of symmetric nontransversal homo- and heteroclinic cycles. We also generalize the results of [1] to the case of the \(p:q\) resonances with odd \(q\) and show that all of them are also degenerate for the maps \(H_{3}^{\pm}\) with \(M_{1}=0\).

Solitary waves in a Whitham equation with small surface tension

AbstractUsing a nonlocal version of the center‐manifold theorem and a normal form reduction, we prove the existence of small‐amplitude generalized solitary‐wave solutions and modulated solitary‐wave solutions to the steady gravity‐capillary Whitham equation with weak surface tension. Through the application of the center‐manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four‐dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity‐capillary water wave problem, the associated linear operator is seen to undergo either a reversible bifurcation or a reversible bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second‐order or third‐order terms. These truncated systems relate directly to systems obtained in the study of the full gravity‐capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity‐capillary Whitham equation due to reversibility. Consequently, this work illuminates further connections between the gravity‐capillary Whitham equation and the full two‐dimensional gravity‐capillary water wave problem.

Analytic and Algebraic Conditions for Bifurcations of Homoclinic Orbits II: Reversible Systems

Following Part I, we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. First, we modify and extend arguments of Part I to show in a form applicable to general systems discussed there that if such bifurcations occur in four-dimensional systems, then variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under some conditions. We next extend the Melnikov method of Part I to reversible systems and obtain some theorems on saddle-node, transcritical and pitchfork bifurcations of symmetric homoclinic orbits. We illustrate our theory for a four-dimensional system, and demonstrate the theoretical results by numerical ones.

From chaos to stability in soliton mode-locked fibre laser system

Intracavity energy rate in a soliton mode-locked fibre laser is derived by solving the Haus master equation. The influence of net gain, absorber response, saturation energy, nonlinearity and absorption are investigated on stable/unstable states. Intracavity modes include the zeroth, first and higher order solitons. Accordingly, chaotic regime as well as breather modes is recognized as a conventional intracavity state. However, tuning the control parameters also results in a reverse bifurcation and thus returning to a stable state. Accordingly, a chaos-based encryption/decryption system is proposed taking the advantage of using a single-side control process; both the encryption and decryption procedures can be achieved by one of the actions of increasing/decreasing the control parameters.