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Searching recurrent patterns in complex systems with high-dimensional phase spaces is an important task in diverse fields. In the current work, an improved scheme is proposed to accelerate the recently designed variational approach for finding periodic orbits in systems with chaotic dynamics based on the existence of inertial manifold widely observed in various spatially extended systems, especially those with high dimensions. On the premise of keeping exponential convergence of the variational method, an effective loop evolution equation is derived to greatly reduce the storage and computing time. With repeated modification of local coordinates and evolution of the guess loop being carried out alternately, the rapid convergence and the stability of the reduction scheme are effectively achieved. The dimension of local coordinate subspaces is generally larger than the number of nonnegative Lyapunov exponents to ensure the exponential convergence. The proposed scheme is successfully demonstrated on several well-known examples and expected to supply a powerful tool in the exploration of high-dimensional nonlinear systems., 25pages

Nonlinear dynamics is a rapidly developing subject across all disciplines involving spatial or temporal evolution. The reconstruction of the equations of motion for a nonlinear system from observed time series has been a hot topic for a long time. Nevertheless, in practice only partial information is available for many systems which are very likely contaminated with noise. Here, based on the invariance of the evolution equation during time translation, a globally valid local approximation of the trajectory is determined, which could be reliably used for the reconstruction of the vector fields with unknown parameters or functional forms, even with partial information of the state evolution. The global consideration very effectively alleviates noise interference and bestows exceptional robustness to the technique, which asks only for solution of linear equations and thus is very efficient. The new scheme is nicely demonstrated in the Lorenz equation in different conditions, while an FHN neural network model is used to show its strength in high-dimensions.

In the study of network synchronization, an outstanding question of both theoretical and practical significance is how to allocate a given set of heterogenous oscillators on a complex network in order for improving the synchronization performance. Whereas methods have been proposed to address this question in literature, the methods are based on accurate models describing the system dynamics, which, however, are normally unavailable in realistic situations. Here we show that this question can be addressed by the model-free technique of feed-forward neural network (FNN) in machine learning. Specifically, we measure the synchronization performance of a number of allocation schemes and use the measured data to train a machine. It is found that the trained machine is able to not only infer the synchronization performance of any new allocation scheme, but also find from a huge amount of candidates the optimal allocation scheme for synchronization., 5 page2, 2 figures

As periodic orbit theory works badly on computing the observable averages of dynamical systems with intermittency, we propose a scheme to cooperate with cycle expansion and perturbation theory so that we can deal with intermittent systems and compute the averages more precisely. Periodic orbit theory assumes that the shortest unstable periodic orbits build the framework of the system and provides cycles expansion to compute dynamical quantities based on them, while the perturbation theory can locally analyze the structure of dynamical systems. The dynamical averages may be obtained more precisely by combining the two techniques together. Based on the integrability near the marginal orbits and the hyperbolicity in the part away from the singularities in intermittent systems, the chief idea of this paper is to revise intermittent maps and maintain the natural measure produced by the original maps. We get the natural measure near the singularity through the Taylor expansions and periodic orbit theory captures the natural measure in the other parts of the phase space. We try this method on 1-dimensional intermittent maps with single singularity, and more precise results are achieved., 15 pages, 10 figures

Symbolic dynamics is a powerful tool to describe topological features of a nonlinear system, where the required partition, however, remains a challenge for some time due to the complications involved in determining the partition boundaries. In this article, we show that it is possible to carry out interesting symbolic partitions for chaotic maps based on properly constructed eigenfunctions of the finite-dimensional approximation of the Koopman operator. The partition boundaries overlap with the extrema of these eigenfunctions, the accuracy of which is improved by including more basis functions in the numerical computation. The validity of this scheme is demonstrated in well-known 1D and 2D maps.

Accumulating evidences show that the cerebral cortex is operating near a critical state featured by power-law size distribution of neural avalanche activities, yet evidence of this critical state in artificial neural networks mimicking the cerebral cortex is lacking. Here we design an artificial neural network of coupled phase oscillators and, by the technique of reservoir computing in machine learning, train it for predicting chaos. It is found that when the machine is properly trained, oscillators in the reservoir are synchronized into clusters whose sizes follow a power-law distribution. This feature, however, is absent when the machine is poorly trained. Additionally, it is found that despite the synchronization degree of the original network, once properly trained, the reservoir network is always developed to the same critical state, exemplifying the "attractor" nature of this state in machine learning. The generality of the results is verified in different reservoir models and by different target systems, and it is found that the scaling exponent of the distribution is independent on the reservoir details and the bifurcation parameter of the target system, but is modified when the dynamics of the target system is changed to a different type. The findings shed lights on the nature of machine learning, and are helpful to the design of high-performance machine in physical systems., Comment: 8 pages, 4 figures

Finding the globally optimal network is an unsolved problem in synchrony optimization. In this paper, an efficient edge-adding optimization method based on global information is proposed. The edge-adding scheme is obtained through the eigenvector corresponding to the maximum eigenvalue of the system's Jacobian matrix. With different frequency distributions, we find that the optimized networks have similar features, concluded as three conditions: (i) The deviations of nodes' frequencies from the mean value are linear with the nodes' degrees, (ii) the oscillators form a bipartite network divided according to the frequencies of the oscillators, and (iii) oscillators are only connected to those with sufficiently large frequency differences. An optimal network can be constructed directly based on these three conditions for a given distribution of natural frequencies. We show that the critical coupling strengths of these constructed networks approach the theoretical lower bound. The constructed networks are or at least close to the globally optimal ones for synchrony.

Molecular communications rely on diffusive propagation to transport information, which is attractive for a variety of nano-scale applications. Due to the long-tail channel response, spatial-temporal coding of information may lead to severe inter-symbol interference (ISI). Classical linear signal processing in wireless communications is usually operating with high complexity and high signal-to-noise ratios, whereas signal processing in molecular communication system requires operating in opposite conditions. In this work, we propose a novel signal processing paradigm inspired by the biological principle, which enables low-complexity signal detection in extremely noisy environments. We first propose a non-linear filter inspired by stochastic resonance, which is found in a variety of biological systems, and it can significantly improve the output SNR by converting noise to useful signals. Then, we design a non-coherent detection method, one which exploits the generally transient trend of observed signals (i.e. quick-rising and slow-decaying) rather than hidden channel state information (CSI), thus excluding CSI estimation and involving only summations. Implementation issues are also discussed, including parameters configuration and adaptive threshold. Numerical results show that the proposed bio-inspired scheme can improve the performance remarkably over classical approaches. Even compared with the optimal linear methods, the required SNR of the proposed scheme can be reduced by 7 dB, which reaffirms why it can be used in noisy biological environments. As the first attempt to design bio-inspired molecular signal detectors, the proposed non-linear processing paradigm may provide the great promise to the emerging nano-machine applications.

We introduce a class of generalized Langevin-type growth equations exhibiting long-ranged temporal correlations in kinetic roughening, and investigate memory effects and dynamic scaling behavior based on scaling analysis and numerical simulations. The critical exponents in the weak- and strong-coupling regions are obtained, and the interplay among temporal correlations is discussed. Our results show that these long-range interactions affect nontrivial scaling properties of the temporally correlated growth system.

We optimize synchrony in networks of heterogeneous phase oscillators. By analyzing the eigenvalues of the Jacobian matrix at full synchronization, two structure indicators related to the critical coupling strength are constructed. When the network structure is fixed, the current scheme of optimization on the frequency picking and allocation converts the synchronization from a second-order continuous to a first-order discontinuous phase transition. When the frequency distribution is deterministic and uniform, the obtained best network structure has a critical coupling strength that approaches the theoretical minimum value. It is found that the requirement for explosive synchronization is consistent with that for optimizing synchrony, which requires that the frequency difference between connected nodes be large, and the distance from the node to the average frequency is proportional to the node degree. The connection between optimization and explosive synchronization deepens our understanding of the latter and provides an effective way of control.

With the development of probing and computing technology, the study of complex systems has become a necessity in various science and engineering problems, which may be treated efficiently with Koopman operator theory based on observed time series. In the current paper, combined with a singular value decomposition (SVD) of the constructed Hankel matrix, Koopman analysis is applied to a system of coupled oscillators. The spectral properties of the operator and the Koopman modes of a typical orbit reveal interesting invariant structures with periodic, quasiperiodic, or chaotic motion. By checking the amplitude of the principal modes along a straight line in the phase space, cusps of different sizes on the magnitude profiles are identified whenever a qualitative change of dynamics takes place. There seems to be no obstacle to extend the current analysis to high-dimensional nonlinear systems with intricate orbit structures.

Understanding key patterns in a spatially extended system is an essential task of modern physics of complex systems. Just like in low-dimensional nonlinear systems, here we show that orbit topology plays a critical role even for the investigation of spatiotemporal dynamics. First, we design a new scheme to reduce possible continuous symmetries that are prevailing in these systems based on topological consideration. The scheme is successfully demonstrated in the well-known pattern formation systems. Interesting bifurcation routes to chaos are conveniently revealed after symmetry reduction. In particular, we find that near the onset of turbulent dynamics, with an increase of instability, local phase chaos with the same spatial topological index may merge into more complex ones, while those with different indices induce defect chaos necessarily through connections docked with defects. The topological argument is so strong that the scenario presented here should be omnipresent in diverse systems.

A proper description of spin glass remains a hard subject in theoretical physics and is considered to be closely related to the emergence of chaos in the renormalization group (RG) flow. Previous efforts concentrate on models with either complicated or nonrealistic interactions in order to achieve this chaotic behavior. Here we find that the commonly used Potts model with long-range interaction could do the job nicely in a large parameter regime as long as the competition between the ferromagnetic and antiferromagnetic interaction is maintained. With this simplicity, the appearance of chaos is observed to sensitively depend on the detailed network structure: the parity of bond number in a branch of the basic RG substituting unit; chaos only emerges for even numbers of bonds. These surprising and universal findings may shed light on the study of spin glass.

Explosive synchronization(ES), as one kind of abrupt dynamical transition in nonlinearly coupled systems, is currently a subject of great interests. Given a special frequency distribution, a mixed ES is observed in a ring of coupled phase oscillators which transit from partial synchronization to ES with the increment of coupling strength. The coupling weight is found to control the size of the hysteresis region where asynchronous and synchronized states coexist. Theoretical analysis reveals that the transition varies from the mixed ES, to the ES and then to a continuous one with increasing coupling weight. Our results are helpful to extend the understanding of the ES in homogenous networks., Comment: 21 pages, 9 figures

The observation and study of nonlinear dynamical systems has been gaining popularity over years in different fields. The intrinsic complexity of their dynamics defies many existing tools based on individual orbits, while the Koopman operator governs evolution of functions defined in phase space and is thus focused on ensembles of orbits, which provides an alternative approach to investigate global features of system dynamics prescribed by spectral properties of the operator. However, it is difficult to identify and represent the most relevant eigenfunctions in practice. Here, combined with the Koopman analysis, a neural network is designed to achieve the reconstruction and evolution of complex dynamical systems. By invoking the error minimization, a fundamental set of Koopman eigenfunctions are derived, which may reproduce the input dynamics through a nonlinear transformation provided by the neural network. The corresponding eigenvalues are also directly extracted by the specific evolutionary structure built in.

Based on the thermosize effects of a confined ideal Maxwellian gas, we established a model of micro/nano-scaled heat pump including two isothermal and two isobaric processes, where the Joule-like heating effect was analyzed in detail by using the nonequilibrium thermodynamic theory. Further, in order to improve the performance of a single thermosize device, a model of two-stage micro/nano-scaled heat pump with multiple micro/nano components is for the first time put forward. The general expressions of heating load and coefficient of performance for the single and the two-stage heat pump are derived. The influences of the internal and external irreversibilities of the system, i.e., Fourier heat flow and the Joule-like heating and the finite rate heat transfer with the reservoirs, on the performance characteristic of thermosize devices are studied in detail. Moreover, we also discuss the effect of the allocation the thermosize components on the optimal performance of the two-stage micro/nano-scaled heat pump cycle. It is found that in the presence of the internal and external irreversibilities, the two kinds of the micro/nano-scaled heat pump can achieve specific heating performance. In particular, when the total number M of the thermosize components is fixed, both the maximum of the optimal coefficient of performance Φ opt , max and the maximum of the heating load Π opt , max corresponding to the optimal coefficient of performance appear near n = M / 2 with n being the number of the thermosize elements of the top stage. The results obtained here show several main irreversibility source of the micro/nano-scaled heat pump and have potential applications in the efficient utilization and conversion of energy in the microchannel system using gas.

We propose theoretically an optomechanical system with double cavities to explore the thermophonon transport from the thermal bath of the mechanical oscillator to the coupled system. We find that the direction and magnitude of thermophonon flux in the system can be controlled flexibly by coupling an active cavity with gain to the driven cavity. In particular, the injected squeezing vacuum can reverse the nonequilibrium characteristics of the system and change the thermophonon flux from positive to negative. We also investigate in detail the influence of the driving power and the photon tunneling strength on the flux, which can widen the energy transfer channel of the system. The results obtained here have a potential application in the thermal noise energy harvesting and rectification by the optomechanical setup.

In this work, we only use data on the unstable manifold to locate the partition boundaries by checking folding points at different levels, which practically coincide with homoclinic tangencies (HTs). The method is then applied to the classic two-dimensional Henon map and a well-known three-dimensional map. Comparison with previous results is made in the Henon case and Lyapunov exponents are computed through the metric entropy based on the partition, to show the validity of the current scheme.

We theoretically investigate the generation of steady-state entanglement in a levitated optomechanical system with a cascade three-level atomic medium and a dielectric nanosphere. There are two cavity modes, driven by two external laser beams with different frequencies, which are responsible for trapping and cooling the nanosphere. Atoms prepared in coherent superposition are injected into the cavity to manipulate the optical fields and the effective optomechanical coupling. We show that the entanglement between the nanosphere and the cavity modes can be enhanced by selecting properly the initial population of the lower atomic level. In particular, when the atoms are prepared at that level, the corresponding optomechanical entanglement increases with the increase of the atom-field coupling strength. We also investigate in detail the influence of the effective cavity field detunings, the atomic detunings, the driving power, and the nanosphere's radius and found that there exist optimal values of the effective cavity field detuning and the driving power that maximize the optomechanical entanglement. By selecting the system's parameters properly, we further demonstrate the generation of genuine tripartite entanglement between three degrees of freedom in the system. The findings here may help flexible control of the steady-state entanglement in hybrid optomechanical systems.

We investigate thermo-phonon transport and its nontrivial Berry-phase-like effect in an optomechanical system with a squeezed vacuum injection. By taking the cumulant generation function approach the exact expressions of the thermo-phonon flux and optomechanical Berry phase are derived analytically. Further, the quantum master equation approach is invoked to verify the analytical results of the transport properties. It is shown that the steady-state thermo-phonon flux can be modulated by varying optically the nonequilibrium characteristics of the system via the squeezed vacuum. In particular, an adiabatic modulation of squeezing parameters induces an optomechanical Berry-phase-like effect and as a result provides an additional geometric phonon response across the macroscopic mechanical motion near the quantum regime, which can also be seen as a consequence of the asymmetric jumping probability between transition associated with phonon absorption and emission in a thermal bath. The present method and results are general and can be straightforwardly extended to any multimode oscillator systems and therefore pave the way to the thermal noise energy harvesting and rectification in coupled oscillator systems with inertial terms.

Synchronization is an important dynamical phenomenon in coupled nonlinear systems, which has been studied extensively in recent years. However, analysis focused on individual orbits seems hard to extend to complex systems, while a global statistical approach is overly cursory. Koopman operator technique seems to balance well the two approaches. In this paper we extend Koopman analysis to the study of synchronization of coupled oscillators by extracting important eigenvalues and eigenfunctions from the observed time series. A renormalization group analysis is designed to derive an analytic approximation of the eigenfunction in the case of weak coupling that dominates the oscillation. For moderate or strong couplings, numerical computation further confirms the importance of the average frequencies and the associated eigenfunctions. The synchronization transition points could be located with quite high accuracy by checking the correlation of neighboring eigenfunctions at different coupling strengths, which is readily applied to other nonlinear systems.

Germinal center (GC) is a particular biological structure produced for affinity maturation in the lymphoid follicle during the T-dependent immune response and is an important component of the humoral immune system. However, the impact of morphological features of the GC on antibody production is not clear. According to the latest biological experiments, we establish a spatiotemporal stochastic model to simulate the whole self-organization process of the GC including the appearance of two specific zones: the dark zone(DZ) and the light zone (LZ). We find that the development of light and dark zones in GC serves to maintain an effective competition among different cells and promote affinity maturation. On the other hand, by varying the GC size, a phase transition is discovered, which determines a critical GC volume for best performance in both the stochastic and the deterministic model. This critical volume is determined by the distance between the activated B Cell Receptor(BCR) and the target epitope of the antigen. The conclusion is confirmed in both the 2D and the 3D simulations and explains partly the variability in the GC size.Author summaryGerminal center (GC) is an important component of the humoral immune system, which supports antibody affinity maturation and the generation of immunity memory. However, the impact of special morphological features of the GC on antibody production is not clear. According to the latest biological experiments, we establish a spatiotemporal stochastic model to simulate the whole self-organization process of the GC. We use the mixing index of different B cells to quantitatively describe the polarization in GC. With the increase of the mixing index, the affinity of plasma cells decreases gradually, even GC might collapse. Therefore, the development of light and dark zones in GC serves to maintain effective competition among different cells and promote affinity maturation. On the other hand, by varying the GC volume, a phase transition is discovered, which determines a critical GC volume for best performance in both the stochastic and the deterministic model. This critical volume is determined by the distance between the activated B Cell Receptor (BCR) and the target epitope of antigen. The conclusion is confirmed in both the 2D and the 3D simulations and explains partly the variability in the GC size.

In this paper, we construct a stochastic model of the 2019-nCoV transmission in a confined space, which gives a detailed account of the interaction between the spreading virus and mobile individuals. Different aspects of the interaction at mesoscopic level, such as the human motion, the shedding and spreading of the virus, its contamination and invasion of the human body and the response of the human immune system, are touched upon in the model, thier relative importance during the course of infection being evaluated. The model provides a bridge linking the epidemic statistics to the physiological parameters of individuals and may serve a theoretical guidance for epidemic prevention and control.

The pandemic of coronavirus disease 2019(COVID-19) has threatened the social and economic structure all around the world. Generally, COVID-19 has three possible transmission routes, including pre-symptomatic, symptomatic, and asymptomatic transmission, among which the last one has brought a severe challenge for the containment of the disease. One core scientific question is to understand the influence of asymptomatic individuals and of the strength of control measures on the evolution of the disease, particularly on a second outbreak of the disease. To explore these issues, we proposed a novel compartmental model that takes the infection of asymptomatic individuals into account. We get the relationship between asymptomatic individuals and critical strength of control measures theoretically. Furthermore, we verify the reliability of our model and the accuracy of the theoretical analysis by using the real confirmed cases of COVID-19 contamination. Our results, showing the importance of the asymptomatic population on the control measures, would provide useful theoretical reference to the policymakers and fuel future studies of COVID-19.

Amplitude chimera states, representing a spontaneous symmetry breaking of a population of coupled identical oscillators into two distinct clusters with one oscillating in spatial coherent amplitude, while the other displaying oscillations in a spatially incoherent manner, have been observed as a kind of transient dynamics in the process of transition to the in-phase synchronization in coupled limit-cycle oscillators. Here, we obtain a kind of stable amplitude chimera state in the chaotic regime of a system of repulsively coupled Lorenz oscillators. With the increment of the coupling strength, the coupled oscillators transit from spatiotemporal chaos to amplitude chimera states then to coherent oscillation death or chimera death states. Moreover, the number of clusters in amplitude chimera patterns has a power-law dependence on the number of coupled neighbors. The amplitude chimera and the chimera death states coexist at certain coupling strength. Moreover, the amplitude chimera and the amplitude death patterns are related to the initial condition for given coupling strength. Our findings of amplitude chimera states and chimera death states in coupled chaotic system may enrich the knowledge of the symmetry-breaking-induced pattern formation.

Efficient control of qubits plays a key role in quantum information processing. In the current work, an alternative set of differential equations are derived for an optimal quantum control of single or multiple qubits with or without interaction. The new formulation enables a great reduction of the computation load by eliminating possible redundant complexity involved in previous algorithms. A relaxation technique is designed for numerically detecting optimal paths involving entanglement. Interesting continuous symmetries are identified in the Lagrangian, which indicates the existence of physically equivalent classes of paths and may be utilized to remove neutral directions in the Jacobian of the evolution. In the ‘ground state’ solution among the set of optimal paths, the time-reversal symmetry of the system shows up, which is expected to be universal for the symmetry-related initial and final state.

The dynamics of a two-layered rotating liquid at low angular velocity in a cylindrical container is explored, both experimentally and theoretically, using an oil-water system. In the two-layered liquid, a transient concave down paraboloid interface, containing undulations, is observed before the final, stable concave up paraboloid interface develops. A simple yet effective model is developed to explain the observed profile dynamics, in which the predicted maximal height of the paraboloid agrees well with experimental measurements over a fairly large range of rotation speeds.

Noise and fluctuations play vital roles in signal transduction in cells. Various numerical techniques for its simulation have been proposed, most of which are not efficient in cellular networks with a wide spectrum of timescales. In this paper, based on a recently developed variational technique, low-dimensional structures embedded in complex stochastic reaction dynamics are unfolded which sheds light on new design principles of efficient simulation algorithm for treating noise in the mesoscopic world. This idea is effectively demonstrated in several popular regulation models with an empirical selection of test functions according to their reaction geometry, which not only captures complex distribution profiles of different molecular species but also considerably speeds up the computation.

This study highlights the theoretical investigation ofquantum coherence in mechanical oscillators and its transfer between the cavityand mechanical modes of an optomechanical system comprising an optical cavity and twomechanical oscillators that, in this study, were simultaneously coupled tothe optical cavity at different optomechanical coupling strengths. The quantumcoherence transfer between the optical and mechanical modes is found to dependstrongly on the relative magnitude of the two optomechanical couplings.The laser power, decay rates of the cavity and mechanical oscillators,environmental temperature, and frequency of the mechanical oscillator areobserved to significantly influence the investigated quantum coherences.Moreover, quantum coherence generation in the optomechanical system is restrictedby the systems stability condition, which helps sustain high and stablequantum coherence in the optomechanical system.

We propose theoretically a four-mode coupled optomechanical system to explore the optical-response properties of an optical trimer system consisting of a passive cavity, a no-loss-gain cavity, and an active cavity coupled with a mechanical oscillator. In the study, the passive cavity is driven by an external laser so that the stability of the coupled system depends strongly on the gain-to-loss ratio and the photon tunneling between the adjacent cavities. We find that in the regime near the stable-unstable critical point, the resonance absorption of the optical trimer system changes quickly with the increasing gain of the active cavity. In contrast, when the mechanical oscillator is coupled to the passive cavity, the center absorption peak is split and a phenomenon of optomechanically induced transparency appears. Consequently, two additional resonance peaks in the absorption profile are induced by the optomechanical coupling, which hinges on the tunneling between the optical cavities and also on the gain of the active cavity. The dependence of the width of the transparency window on the optomechanical coupling and the driving strength is also discussed in detail. The results obtained here indicate that the optical properties of the trimer system can be manipulated by coupling a mechanical oscillator to the system and therefore have a potential application in quantum optics and quantum information processing.

Several schemes are proposed to accelerate the recently designed variational approach for finding periodic orbits in a nonlinear system with chaotic dynamics, especially those close to a singularity or with long period. An effective equation is derived to implement an automatic allocation of lattice points to capture local fine orbit structures while keeping the exponential convergence of the variational approach. Utilizing the special structure of the matrix involved in the homotopy evolution, the lower--upper (LU) decomposition could be done with much greater efficiency and much less memory, which considerably facilitates the search for long cycles. Three examples are used to demonstrate the validity of the accelerated algorithm. For Hamiltonian systems, an interesting scheme is designed to remove the neutral direction associated with the conservation law while keeping the dynamics on the desired energy surface.

Previous research has suggested that well-connected nodes in a network (commonly referred to as hubs) are better at spreading information than those with fewer connections (ordinary users). Here we investigate the roles of nodes with different numbers of connections by studying how people share news online. Quantitative analysis shows that users without many connections can sometimes spread news more effectively than well-connected users when the diffusion pattern has dendrite-like paths that reach far into the network, leading to a non-Gaussian distance distribution. When the hubs dominate, however, the distribution is Gaussian. Enhanced interactions among ordinary users are the key to the emergence of non-Gaussian characteristics. Finally, we introduce a message-passing model that reproduces the observed diffusion features. This model shows that patterns dominated by either hubs or ordinary users can be clearly demarcated by measuring the average number of forwards.

In this paper, we study an optomechanical device consisting of a Fabry-Perot cavity with two dielectric nanospheres trapped near the cavity mirrors by an external driving laser. In the condition where the distances between the nanospheres and cavity mirrors are small enough, the Casimir force helps the optomechanical coupling to induce a steady-state optomechanical entanglement of the mechanical and optical modes in a certain regime of parameters. We investigate in detail the dependence of the steady-state optomechanical entanglement on external control parameters of the system, i.e., the effective detuning, the pump powers of the cavity, the cavity decay rate and the wavelength of the driving field. It is found that the large steady-state optomechanical entanglement, i.e. E N = 5.76, can be generated with experimentally feasible parameters, i.e. the pump power P = 18.2 μW, the cavity decay rate κ = 0.5 MHz and the wavelength of the laser λL=1064 nm, which should be checked by optical measurement.

A systematic study of spatially periodic steady solutions of the Kuramoto–Sivashinsky equation (KSe) is undertaken from a dynamical systems’ point of view. A recently devised variational method is employed and one new variant is developed. At fixed system size L = 43.5 , important equilibria are identified and shown to organize the dynamics. The first integral of the steady KSe leads to a 3D dynamical system with an integration constant c. At a typical value of c = 0.40194 , four simplest cycles are identified and used as basic building blocks to construct longer cycles. The symbolic dynamics based on trajectory topology are very effective in classifying all short periodic orbits. The probation of the return map on a chosen Poincare section shows the complexity of the dynamics and the bifurcation of building blocks provides a chart to look for possible cycles at given periods. The current study may be conveniently adapted to the identification and classification of cycles in other nonlinear systems.

We propose a variational method for determining homoclinic and heteroclinic orbits including spiral-shaped ones in nonlinear dynamical systems. Starting from a suitable initial curve, a homotopy evolution equation is used to approach a true connecting orbit. The procedure is an extension of a variational method that has been used previously for locating cycles, and avoids the need for linearization in search of simple connecting orbits. Examples of homoclinic and heteroclinic orbits for typical dynamical systems are presented. In particular, several heteroclinic orbits of the steady-state Kuramoto–Sivashinsky equation are found, which display interesting topological structures, closely related to those of the corresponding periodic orbits.

In this article we address the problem of the depinning transition for driven interfaces in random media. We introduce a fractional Kardar–Parisi–Zhang equation with quenched noise, in which the normal diffusion term is replaced by a fractional Laplacian accounting for long-range interactions through quenched disorder. The critical values of the external driving force and nonlinear term coefficient evidently depend on the system size at the depinning transition. For a fixed value of the external driving force, the fractional order much determines the value of the nonlinear term coefficient that leads to a depinned interface. Near the depinning threshold, the critical exponent obtained numerically is nonuniversal, and weakly depends on the fractional order.

To analyze long-range spatial correlations in surface growth, we study numerically a class of generalized Kardar-Parisi-Zhang equation with a fractional Laplacian and driven by long-range spatially correlated noise, and investigate interplay of the fractional Laplacian and correlated noise. We find that the growth system with long-range correlation exhibits nontrivial scaling properties, such as strong dependence on the noise correlation and weak dependence on the fractional order. The growth instability is also discussed in various parameter regimes.