Two-parameter Space Research Articles

Hopf Bifurcation of a Type of Neuron Model with Multiple Time Delays

By applying a geometrical scheme developed to tackle the eigenvalue problem of delay differential equations with multiple time delays, Hopf bifurcation of Hopfield neuron model is analyzed in two-parameter space. By the introduction of two new angles, the calculation of imaginary roots is carried out analytically and effectively. By increasing the parameter to cross over the Hopf bifurcation lines, the stability switching direction is confirmed. The method is a useful tool to show the partition of stable and unstable regions in two-parameter space and detect double Hopf bifurcation further. The typified dynamical behaviors based on nearby double Hopf points are analyzed by applying the normal form technique and center manifold method.

On the dynamics of a simplified canonical Chua's oscillator with smooth hyperbolic sine nonlinearity: Hyperchaos, multistability and multistability control.

A simplified hyperchaotic canonical Chua's oscillator (referred as SHCCO hereafter) made of only seven terms and one nonlinear function of type hyperbolic sine is analyzed. The system is found to be self-excited, and bifurcation tools associated with the spectrum of Lyapunov exponents reveal the rich dynamical behaviors of the system including hyperchaos, torus, period-doubling route to chaos, and hysteresis when turning the system control parameters. Wide ranges of hyperchaotic dynamics are highlighted in various two-parameter spaces based on two-parameter Lyapunov diagrams. The analysis of the hysteretic window using a basin of attraction as argument reveals that the SHCCO exhibits three coexisting attractors. Laboratory measurements further confirm the performed numerical investigations and henceforth validate the mathematical model. Of most/particular interest, multistability observed in the SHCCO is further controlled based on a linear augmentation scheme. Numerical results show the effectiveness of the control strategy through annihilation of the asymmetric pair of coexisting attractors. For higher values of the coupling strength, only a unique symmetric periodic attractor survives.

Effect of hunting cooperation and fear in a predator-prey model

Dynamics of predator-prey systems under the influence of cooperative hunting among predators and the fear thus imposed on the prey population is of great importance from ecological point of view. The role of hunting cooperation and the fear effect in the predator-prey system is gaining considerable attention by the researchers recently. But the study on combined effect of hunting cooperation and fear in the predator-prey system is not yet studied. In the present paper, we investigate the impact of hunting cooperation among predators and predator induced fear in prey population by using the classical predator-prey model. We consider that predator populations cooperate during hunting. We also consider that hunting cooperation induces fear among prey, which has far richer and complex dynamics. We observe that without hunting cooperation, the unique coexistence equilibrium point is globally asymptotically stable. However, an increase in the hunting cooperation induced fear may destabilize the system and produce periodic solution via Hopf-bifurcation. The stability of the Hopf-bifurcating periodic solution is obtained by computing the Lyapunov coefficient. The limit cycles thus obtained may be supercritical or subcritical. We also observe that the system undergoes the Bogdanov-Takens bifurcation in two-parameter space. Further, we observe that the system exhibits backward bifurcation between predator-free equilibrium and coexisting equilibrium. The system also exhibits two different types of bi-stabilities due to subcritical Hopf-bifurcation (between interior equilibrium and stable limit cycle) and backward bifurcation (between predator-free and interior equilibrium points). Further, we observe strong demographic Allee phenomenon in the system. To visualize the dynamical behavior of the system, extensive numerical experiments are performed by using MATLAB and MATCONT softwares.

Effects of task constraint on action dynamics

The actualization of action possibilities (i.e., affordances) can often be accomplished in numerous ways. For instance, an individual could walk over to a rubbish bin to drop an item in or throw the piece of rubbish into the bin from some distance away. The aim of the current study was to investigate the action dynamics that emerge from such under-constrained task or action spaces using an object transportation task. Participants were instructed to transport balls between a starting location and a large wooden box located 9 m away. The temporal interval between the sequential presentation of balls was manipulated as a control parameter and was expected to influence the distance participants moved prior to throwing or dropping the ball into the target box. A two-parameter state space derived from the Cusp Catastrophe Model was employed to illustrate how behavioral variability emerged as a consequence of the under-constrained task context. Two follow-up experiments demonstrated direct correspondence between model predictions and observed action dynamics as a function of increasing task constraints. Implications for modeling, the theory of affordances, and empirical studies more generally are discussed.

Floquet scalar dynamics in global AdS

We study periodically driven scalar fields and the resulting geometries with global AdS asymptotics. These solutions describe the strongly coupled dynamics of dual finite-size quantum systems under a periodic driving which we interpret as Floquet condensates. They span a continuous two-parameter space that extends the linearized solutions on AdS. We map the regions of stability in the solution space. In a significant portion of the unstable subspace, two very different endpoints are reached depending upon the sign of the perturbation. Collapse into a black hole occurs for one sign. For the opposite sign instead one attains a regular solution with periodic modulation. We also construct quenches where the driving frequency and amplitude are continuously varied. Quasistatic quenches can interpolate between pure AdS and sourced solutions with time periodic vev. By suitably choosing the quasistatic path one can obtain boson stars dual to Floquet condensates at zero driving field. We characterize the adiabaticity of the quenching processes. Besides, we speculate on the possible connections of this framework with time crystals.

Spatiotemporal Patterns in a Predator–Prey Model with Cross-Diffusion Effect

The present research deals with the emergence of spatiotemporal patterns of a two-dimensional (2D) continuous predator–prey system with cross-diffusion effect. First, we work out the critical lines of Hopf and Turing bifurcations of the current model system in a 2D spatial domain by means of bifurcation theory. More specifically, the exact Turing region is specified in a two-parameter space. In effect, by choosing the cross-diffusion coefficient as one of the momentous parameter, we demonstrate that the model system undergoes a sequence of spatiotemporal patterns in a homogeneous environment through diffusion-driven instability. Our results via numerical simulation authenticate that cross-diffusion be able to create stationary patterns which enrich the findings of pattern formation in an ecosystem.

Cluster synchronization of dry friction oscillators

Synchronization is a well known phenomenon in non-linear dynamics and is treated as correlation in time of at least two different processes. In scope of this article, we focus on complete and cluster synchronization in the systems of coupled dry friction oscillators, coupled by linear springs. The building block of the system is the classic stick-slip oscillator, which consists of mass, spring and belt-mass friction interface. The Stribeck friction itself is modelled using Stribeck friction model with exponential non-linearity. The oscillators in the systems are connected in nearest neighbour fashion, both in open and closed ring topology. We perform a numerical study of the properties of the dynamics of the systems in question, in two-parameter space (coupling coefficient vs. angular excitation frequency) and explore the possible configurations of cluster synchronization.

Multistability Analysis and Function Projective Synchronization in Relay Coupled Oscillators

Regions of stability phases discovered in a general class of Genesio−Tesi chaotic oscillators are proposed. In a relatively large region of two-parameter space, the system has coexisting point attractors and limit cycles. The variation of two parameters is used to characterize the multistability by plotting the isospike diagrams for two nonsymmetric initial conditions. The parameters window in which the jerk system exhibits the unusual and striking feature of multiple attractors (e.g., coexistence of six disconnected periodic chaotic attractors and three-point attraction) is investigated. The second aspect of this study presents the synchronization of systems that act as mediators between two dynamical units that, in turn, show function projective synchronization (FPS) with each other. These are the so-called relay systems. In a wide range of operating parameters; this setup leads to synchronization between the outer circuits, while the relaying element remains unsynchronized. The results show that the coupled systems can achieve function projective synchronization in a determined time despite the unpredictability of the scaling function. In the coupling path, the outer dynamical systems show finite-time synchronization of their outputs, that is, displaying the same dynamics at exactly the same moment. Further, this effect is rather general and it has a wide range of applications where sustained oscillations should be retained for proper functioning of the systems.

On the First Hyperchaotic Hyperjerk System With No Equilibria: A Simple Circuit for Hidden Attractors

A fourth-order hyperchaotic hyperjerk system with no equilibria has never been reported, whereas a search for its existence has been an open research problem. A new system in a rare type of 4D seven-term no-equilibrium hyperchaotic systems is proposed and also compared with two existing systems of such a rare type. The proposed system offers 10 concurrent advantages, six of which appear to be superior to both existing systems, that is; 1) the first report of a fourth-order hyperchaotic hyperjerk system with no equilibria; 2) a much simpler circuit based on only 21 electronic components; 3) a higher value of the Lyapunov dimension at 3.2280; 4) a higher value of the largest Lyapunov exponent at 0.2525; 5) a large two-parameter space of hyperchaos; and 6) a boostable variable for offset control. The other four advantages offer either equal or better features, that is; 7) the number of nonlinear terms is two; 8) no potential dangers of multistability; 9) hyperchaos, chaos, and periodic behavior are possible; and 10) attractors are readily hidden attractors. The system is unique in the sense that there are no equilibria in both dynamical and hyperjerk forms.

Quantum versus classical dynamics in the optical centrifuge

The interplay between classical and quantum-mechanical evolution in the optical centrifuge (OC) is discussed. The analysis is based on the quantum-mechanical formalism starting from either the ground state or a thermal ensemble. Two resonant mechanisms are identified, i.e., the classical autoresonance and the quantum-mechanical ladder climbing, yielding different dynamics and rotational excitation efficiencies. The rotating-wave approximation is used to analyze the two resonant regimes in the associated dimensionless two-parameter space and calculate the OC excitation efficiency. The results show good agreement between numerical simulations and theory and are relevant to existing experimental setups.

Amplitude death in intrinsic time-delayed chaotic oscillators with direct–indirect coupling: the existence of death islands

We investigate the transition between oscillatory and amplitude death (AD) states and the existence of death islands in intrinsic time-delayed chaotic oscillators under the simultaneous presence of diffusive (direct) and environmental (indirect) coupling. Studies in two-parameter space reveal that depending upon parameters and intrinsic time delays the coupling can bring the oscillators to the AD state and again can revive the system to oscillatory states, thus creating death islands in parameter space; this observation is in sharp contrast to the death scenario of non-delayed oscillators under the same coupling scheme where no death islands are formed. Using a linear stability analysis, we derive the explicit conditions for different transition scenarios. We use a continuation package for the time-delay systems to precisely identify the zone of AD and its islands and their origin. We also extend our study to the network of oscillators and show that the observed results are general for a large number of oscillators, too. Finally, we demonstrate our results experimentally to verify the analytical and numerical findings.

Floquet–Weyl and Floquet-topological-insulator phases in a stacked two-dimensional ring-network lattice

We show the presence of Floquet–Weyl and Floquet-topological-insulator phases in a stacked two-dimensional ring-network lattice. The Weyl points in the three-dimensional Brillouin zone and Fermi-arc surface states are clearly demonstrated in the quasienergy spectrum of the system in the Floquet–Weyl phase. In addition, chiral surface states coexist in this phase. The Floquet-topological-insulator phase is characterized by the winding number of two in the reflection matrices of the semi-infinite system and resulting two gapless surface states in the quasienergy gap of the bulk. The phase diagram of the system is derived in the two-parameter space of hopping S-matrices among the rings. We also discuss a possible optical realization of the system together with the introduction of synthetic gauge fields.

Existence of spatiotemporal patterns in the reaction–diffusion predator–prey model incorporating prey refuge

The pattern formation in reaction–diffusion system has long been the subject of interest to the researchers in the domain of mathematical ecology because of its universal existence and importance. The present investigation deals with a spatial dynamics of the Beddington–DeAngelis predator–prey model in the presence of a constant proportion of prey refuge. The model system representing boundary value problem under study is subjected to homogeneous Neumann boundary conditions. The asymptotic stability including the local and the global stability and the bifurcation as well of the unique positive homogeneous steady state of the corresponding temporal model has been analyzed. The Turing instability region in two-parameter space and the condition of diffusion-driven instability of the spatiotemporal model are investigated. Based on the appropriate numerical simulations, the present model dynamics in Turing space appears to get influenced by prey refuge while it exhibits diffusion-controlled pattern formation growth to spots, stripe-spot mixtures, labyrinthine, stripe-hole mixtures and holes replication. The results obtained appear to enrich the findings of the model system under consideration.

Ordered and isomorphic mapping of periodic structures in the parametrically forced logistic map

We investigate the periodic domains found in the parametrically forced logistic map, the classical logistic map when its control parameter changes dynamically. Phase diagrams in two-parameter spaces reveal intricate periodic structures composed of patterns of intersecting superstable orbits curves, defining the cell of a periodic window. Cells appear multifoliated and ordered, and they are isomorphically mapped when one changes the map parameters. Also, we identify the characteristics of simplest cell and apply them to other more complex, discussing how the topography on parameter space is affected. By use of the winding number as defined in periodically forced oscillators, we show that the hierarchical organization of the periodic domains is manifested in global and local scales.

Spatiotemporal Dynamics of a Network of Coupled Time-Delay Digital Tanlock Loops

The time-delay digital tanlock loop (TDTLs) is an important class of phase-locked loop that is widely used in electronic communication systems. Although nonlinear dynamics of an isolated TDTL has been studied in the past but the collective behavior of TDTLs in a network is an important topic of research and deserves special attention as in practical communication systems separate entities are rarely isolated. In this paper, we carry out the detailed analysis and numerical simulations to explore the spatiotemporal dynamics of a network of a one-dimensional ring of coupled TDTLs with nearest neighbor coupling. The equation representing the network is derived and we carry out analytical calculations using the circulant matrix formalism to obtain the stability criteria. An extensive numerical simulation reveals that with the variation of gain parameter and coupling strength the network shows a variety of spatiotemporal dynamics such as frozen random pattern, pattern selection, spatiotemporal intermittency and fully developed spatiotemporal chaos. We map the distinct dynamical regions of the system in two-parameter space. Finally, we quantify the spatiotemporal dynamics by using quantitative measures like Lyapunov exponent and the average quadratic deviation of the full network.

Pattern Selection in Network of Coupled Multi-Scroll Attractors

Multi-scroll chaotic attractor makes the oscillator become more complex in dynamic behaviors. The collective behaviors of coupled oscillators with multi-scroll attractors are investigated in the regular network in two-dimensional array, which the local kinetics is described by an improved Chua circuit. A feasible scheme of negative feedback with diversity is imposed on the network to stabilize the spatial patterns. Firstly, the Chua circuit is improved by replacing the nonlinear term with Sine function to generate infinite aquariums so that multi-scroll chaotic attractors could be generated under appropriate parameters, which could be detected by calculating the Lyapunov exponent in the parameter region. Furthermore, negative feedback with different gains (D1, D2) is imposed on the local square center area A2 and outer area A1 of the network, it is found that spiral wave, target wave could be developed in the network under appropriate feedback gain with diversity and size of controlled area. Particularly, homogeneous state could be reached after synchronization by selecting appropriate feedback gain and controlled size in the network. Finally, the distribution for statistical factors of synchronization is calculated in the two-parameter space to understand the transition of pattern region. It is found that developed spiral waves, target waves often are associated with smaller factor of synchronization. These results show that emergence of sustained spiral wave and continuous target wave could be effective for further suppression of spatiotemporal chaos in network by generating stable pacemaker completely.

The role of three-body interactions in two-dimensional polymer collapse**Dedicated to Professor Tony Guttmann on the occasion of his 70th birthday.

Various interacting lattice path models of polymer collapse in two dimensions demonstrate different critical behaviours. This difference has been without a clear explanation. The collapse transition has been variously seen to be in the Duplantier–Saleur θ-point university class (specific heat cusp), the interacting trail class (specific heat divergence) or even first-order. Here we study via Monte Carlo simulation a generalisation of the Duplantier–Saleur model on the honeycomb lattice and also a generalisation of the so-called vertex-interacting self-avoiding walk model (configurations are actually restricted trails known as grooves) on the triangular lattice. Crucially for both models we have three- and two-body interactions explicitly and differentially weighted. We show that both models have similar phase diagrams when considered in these larger two-parameter spaces. They demonstrate regions for which the collapse transition is first-order for high three-body interactions and regions where the collapse is second order. We conjecture a higher order multiple critical point separating these two types of collapse. It remains to be tested whether the second order lines in both models are in the Duplantier–Saleur θ-point university class for all values of the parameters.

Constraints onCP-violating gauge-Higgs operators

We consider the most general set of $SU(2) \times U(1)$ invariant CP-violating operators of dimension six, which contribute to $VVh$ interactions ($V = W, Z, \gamma$). Our aim is to constrain any CP-violating new physics above the electroweak scale via the effective couplings that arise when such physics is integrated out. For this purpose, we use, in turn, electroweak precision data, global fits of Higgs data at the Large Hadron Collider and the electric dipole moments of the neutron and the electron. We thus impose constraints mainly on two-parameter and three-parameter spaces. We find that the constraints from the electroweak precision data are the weakest. Among the existing Higgs search channels, considerable constraints come from the diphoton signal strength. We note that potential contribution to $h \rightarrow \gamma Z$ may in principle be a useful constraining factor, but it can be utilized only in the high energy run. The contributions to electric dipole moments mostly lead to the strongest constraints, though somewhat fine-tuned combinations of more than one parameter with large magnitudes are allowed. We also discuss constraints on gauge boson trilinear couplings which depend on the parameters of the CP-violating operators .

Transition Mechanisms of Bursting in a Two-Cell Network Model of the Pre-Bötzinger Complex

Persistent sodium and calcium activated nonspecific cationic currents play important roles in the respiratory rhythm generation of the pre-Bötzinger complex. In this paper, we study the bursting patterns and their transition mechanisms in the two-parameter space of a two-cell network model of the pre-Bötzinger complex with synaptic coupling. Using the methods of fast/slow decomposition and two-parameter bifurcation analysis, we divide the two-parameter space into four different regions according to the multiphase oscillations, and reveal the possible transition mechanisms of bursting between these different regions. We also study the dynamics of the system with varying synaptic coupling strength. This work provides insights of how currents and synaptic coupling work on the respiratory rhythm generation.

Structural control design and defective systems

The intersection between the two concepts of structural control and defectiveness is discussed. Two simple oscillators differently connected by serial spring-dashpot arrangement are used to simply simulate technically relevant cases: dissipatively coupled adjacent free-standing structures, structures equipped by TMD and base-isolated structures. Eigensolution loci of the two classes of systems are tracked against one or more significant parameters to determine the potential benefits realized by different combinations of stiffness and viscosity. In both studied cases, codimension-two manifolds in the four-parameter space corresponding to coalescing eigenvalues are determined by analytical expressions. Conditions to discern semi-simple eigenvalues from defective ones confirm that the latter is the generic case laying in a two-parameter space while the former span a one-parameter subspace. The knowledge of the location of the defective systems in the parameter space permits to determine regions with specific dynamical properties useful for control design purpose.