This paper proposes a universal impulse-function-based method for extending discrete chaotic maps,enabling flexible construction of multicavity chaotic attractors.The proposed method achieves one-directional(1D)/two-d...This paper proposes a universal impulse-function-based method for extending discrete chaotic maps,enabling flexible construction of multicavity chaotic attractors.The proposed method achieves one-directional(1D)/two-directional(2D)extensions without introducing additional nonlinear terms or altering system stability.Theoretically,the cavity quantity in arbitrary directions is controlled by adjusting impulse levels N,while the amplitude regulation is implemented through modifications to the proportionality parameter r.Theoretical analyses,including Lyapunov exponents(LEs)and bifurcation diagrams,are conducted,confirming that the extended maps retain the intrinsic dynamics of five rational map classes.The field-programmable gate array(FPGA)implementation results are consistent with the numerical simulation results,verifying the correctness of the theoretical analysis.The method enables the expansion of unipolar attractors and enhances entropy metrics,offering a robust framework for applications in secure communication,encryption,and chaos-based technologies.展开更多
Existing chaotic encryption schemes primarily focus on single types of images,making the design of hybrid image encryption schemes more suitable for practical applications.In this paper,a hyperchaotic map with a spher...Existing chaotic encryption schemes primarily focus on single types of images,making the design of hybrid image encryption schemes more suitable for practical applications.In this paper,a hyperchaotic map with a spherical attractor is proposed,which is constructed using spherical coordinates.Dynamical analyses reveal that the hyperchaotic map exhibits global hyperchaos and high complexity,making it capable of generating more complex chaotic sequences suitable for image encryption.A hybrid encryption scheme based on a hyperchaotic map is proposed for two-dimensional(2D)images,three-dimensional(3D)models,and 3D point clouds.Firstly,the pixels of 2D image and the coordinate data of 3D image are fused into a plaintext cube,which is combined with Hash-512 to obtain the initial value of the hyperchaotic map.Chaotic sequences are utilized for cube space internal confusion and dynamic cross-diffusion.The encrypted images demonstrate high information entropy,and the test results show that the encryption scheme effectively protects the images.The proposed hybrid image encryption scheme provides an efficient solution for securing various types of images.展开更多
The neuron model has been widely employed in neural-morphic computing systems and chaotic circuits.This study aims to develop a novel circuit simulation of a three-neuron Hopfield neural network(HNN)with coupled hyper...The neuron model has been widely employed in neural-morphic computing systems and chaotic circuits.This study aims to develop a novel circuit simulation of a three-neuron Hopfield neural network(HNN)with coupled hyperbolic memristors through the modification of a single coupling connection weight.The bistable mode of the hyperbolic memristive HNN(mHNN),characterized by the coexistence of asymmetric chaos and periodic attractors,is effectively demonstrated through the utilization of conventional nonlinear analysis techniques.These techniques include bifurcation diagrams,two-parameter maximum Lyapunov exponent plots,local attractor basins,and phase trajectory diagrams.Moreover,an encryption technique for color images is devised by leveraging the mHNN model and asymmetric structural attractors.This method demonstrates significant benefits in correlation,information entropy,and resistance to differential attacks,providing strong evidence for its effectiveness in encryption.Additionally,an improved modular circuit design method is employed to create the analog equivalent circuit of the memristive HNN.The correctness of the circuit design is confirmed through Multisim simulations,which align with numerical simulations conducted in Matlab.展开更多
A method of generating multi-double scroll attractors is proposed based on the memristor Hopfield neural network(HNN)under pulse control.First,the original hyperbolic-type memristor is added to the neural network math...A method of generating multi-double scroll attractors is proposed based on the memristor Hopfield neural network(HNN)under pulse control.First,the original hyperbolic-type memristor is added to the neural network mathematical model,and the influence of this memristor on the dynamic behavior of the new HNN is analyzed.The numerical results show that after adding the memristor,the abundant dynamic behaviors such as chaos coexistence,period coexistence and chaos period coexistence can be observed when the initial value of the system is changed.Then the logic pulse is added to the external memristor.It is found that the equilibrium point of the HNN can multiply and generate multi-double scroll attractors after the pulse stimulation.When the number of logical pulses is changed,the number of multi-double scroll attractors will also change,so that the pulse can control the generation of multi-double scroll attractors.Finally,the HNN circuit under pulsed stimulation was realized by circuit simulation,and the results verified the correctness of the numerical results.展开更多
A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynam- ical system. It is shown that the compact equi-att...A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynam- ical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.展开更多
In this paper,we investigate the existence of strange nonchaotic attractors(SNAs)in a slender rigid rocking block under quasi-periodic forcing with two frequencies.We find that an SNA can exist between a quasi-periodi...In this paper,we investigate the existence of strange nonchaotic attractors(SNAs)in a slender rigid rocking block under quasi-periodic forcing with two frequencies.We find that an SNA can exist between a quasi-periodic attractor and a chaotic attractor,or between two chaotic attractors.In particular,we demonstrate that a torus doubling bifurcation of a quasi-periodic attractor can result in SNAs via the fractal route before transforming into chaotic attractors.This phenomenon is rarely reported in quasiperiodically forced discontinuous differential equations and vibro-impact systems.The properties of SNAs are verified by the Lyapunov exponent,rational approximation,phase sensitivity,power spectrum,and separation of nearby trajectories.展开更多
This article proposes a non-ideal flux-controlled memristor with a bisymmetric sawtooth piecewise function, and a new multi-wing memristive chaotic system(MMCS) based on the memristor is generated. Compared with other...This article proposes a non-ideal flux-controlled memristor with a bisymmetric sawtooth piecewise function, and a new multi-wing memristive chaotic system(MMCS) based on the memristor is generated. Compared with other existing MMCSs, the most eye-catching point of the proposed MMCS is that the amplitude of the wing will enlarge towards the poles as the number of wings increases. Diverse coexisting attractors are numerically found in the MMCS, including chaos,quasi-period, and stable point. The circuits of the proposed memristor and MMCS are designed and the obtained results demonstrate their validity and reliability.展开更多
To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets <em>B<sub>k</sub></em>. Then we prove that the solution semigroup h...To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets <em>B<sub>k</sub></em>. Then we prove that the solution semigroup has Lipschitz property and discrete squeezing property. Finally, we obtain a family of exponential attractors and its estimation of dimension by combining them with previous theories. Next, we obtain Kirchhoff-type random equation by adding product white noise to the right-hand side of the equation. To study the existence of random attractors, firstly we transform the equation by using Ornstein-Uhlenbeck process. Then we obtain a family of bounded random absorbing sets via estimating the solution of the random differential equation. Finally, we prove the asymptotic compactness of semigroup of the stochastic dynamic system;thereby we obtain a family of random attractors.展开更多
Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are...Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.展开更多
This paper presents the problem of generating four-wing （eight-wing） chaotic attractors. The adopted method consists in suitably coupling two （three） identical Lorenz systems. In analogy with the original Lorenz s...This paper presents the problem of generating four-wing （eight-wing） chaotic attractors. The adopted method consists in suitably coupling two （three） identical Lorenz systems. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings （eight wings） of these novel attractors axe located around the four （eight） equilibria with two （three） pairs of unstable complex-conjugate eigenvalues.展开更多
We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability...We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.展开更多
In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state...In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.展开更多
In this paper,we introduce a new two-dimensional nonlinear oscillator with an infinite number of coexisting limit cycles.These limit cycles form a layer-by-layer structure which is very unusual.Forty percent of these ...In this paper,we introduce a new two-dimensional nonlinear oscillator with an infinite number of coexisting limit cycles.These limit cycles form a layer-by-layer structure which is very unusual.Forty percent of these limit cycles are self-excited attractors while sixty percent of them are hidden attractors.Changing this new system to its forced version,we introduce a new chaotic system with an infinite number of coexisting strange attractors.We implement this system through field programmable gate arrays.展开更多
This paper introduces a four-dimensional （4D） segmented disc dynamo which possesses coexisting hidden attractors with one stable equilibrium or a line equilibrium when parameters vary. In addition, by choosing an ap...This paper introduces a four-dimensional （4D） segmented disc dynamo which possesses coexisting hidden attractors with one stable equilibrium or a line equilibrium when parameters vary. In addition, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcation and pitchfork bifurcation occur in the system. The ultimate bound is also estimated. Some numerical investigations are also exploited to demonstrate and visualize the corresponding theoretical results.展开更多
In this paper, we investigate the existence of random attractor for the random dynamical system generated by the Kirchhoff-type suspension bridge equations with strong damping and white noises. We first prove the exis...In this paper, we investigate the existence of random attractor for the random dynamical system generated by the Kirchhoff-type suspension bridge equations with strong damping and white noises. We first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the existence of the global attractors of the equation.展开更多
We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div（o（x）↓△u） and multiplicative noises. Under some mild conditions on the diffusion variable o（x） an...We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div（o（x）↓△u） and multiplicative noises. Under some mild conditions on the diffusion variable o（x） and without any restriction on the upper growth p of nonlinearity, except that p 〉 2, we show the existences of random attractor in D0^1,2（DN, σ） space, where DN is an arbitrary （bounded or unbounded） domain in R^N N 〉 2. For this purpose, some abstract results based on the omega-limit compactness are established.展开更多
In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a si...In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.展开更多
A novel 5-dimensional（5D） memristive chaotic system is proposed, in which multi-scroll hidden attractors and multiwing hidden attractors can be observed on different phase planes. The dynamical system has multiple l...A novel 5-dimensional（5D） memristive chaotic system is proposed, in which multi-scroll hidden attractors and multiwing hidden attractors can be observed on different phase planes. The dynamical system has multiple lines of equilibria or no equilibrium when the system parameters are appropriately selected, and the multi-scroll hidden attractors and multi-wing hidden attractors have nothing to do with the system equilibria. Particularly, the numbers of multi-scroll hidden attractors and multi-wing hidden attractors are sensitive to the transient simulation time and the initial values. Dynamical properties of the system, such as phase plane, time series, frequency spectra, Lyapunov exponent, and Poincar′e map, are studied in detail. In addition, a state feedback controller is designed to select multiple hidden attractors within a long enough simulation time. Finally, an electronic circuit is realized in Pspice, and the experimental results are in agreement with the numerical ones.展开更多
The aim of this paper is to provide explicitly a sequence of m-dimensional approximate inertial manifolds M(m,j,)j = 1,2,, for each positive integer m, for the Kuramoto-Sivashinsky equations. A very thin neighborhood ...The aim of this paper is to provide explicitly a sequence of m-dimensional approximate inertial manifolds M(m,j,)j = 1,2,, for each positive integer m, for the Kuramoto-Sivashinsky equations. A very thin neighborhood into which the orbits enter with an exponential speed and in a finite time is associated with each manifold. The thickness of these neighborhoods decreases with increasing m for a fixed order j. Besides, the neighborhoods localize the global attractor and aid in the approximate computation of large-time solutions of the Kuramoto-Sivashinsky equations.展开更多
As an important research branch,memristor has attracted a range of scholars to study the property of memristive chaotic systems.Additionally,time⁃delayed systems are considered a significant and newly⁃developing field...As an important research branch,memristor has attracted a range of scholars to study the property of memristive chaotic systems.Additionally,time⁃delayed systems are considered a significant and newly⁃developing field in modern research.By combining memristor and time⁃delay,a delayed memristive differential system with fractional order is proposed in this paper,which can generate hidden attractors.First,we discussed the dynamics of the proposed system where the parameter was set as the bifurcation parameter,and showed that with the increase of the parameter,the system generated rich chaotic phenomena such as bifurcation,chaos,and hypherchaos.Then we derived adequate and appropriate stability criteria to guarantee the system to achieve synchronization.Lastly,examples were provided to analyze and confirm the influence of parameter a,fractional order q,and time delayτon chaos synchronization.The simulation results confirm that the chaotic synchronization is affected by a,q andτ.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.62001391)the Guangdong Basic and Applied Basic Research Foundation(Grant No.2024A1515010308)+1 种基金the Natural Science Basic Research Program of Shaanxi(Grant No.2024JC-YBQN-0464)the Scientific Research Program Funded by Education Department of Shaanxi Provincial Government(Grant No.24JK0559).
文摘This paper proposes a universal impulse-function-based method for extending discrete chaotic maps,enabling flexible construction of multicavity chaotic attractors.The proposed method achieves one-directional(1D)/two-directional(2D)extensions without introducing additional nonlinear terms or altering system stability.Theoretically,the cavity quantity in arbitrary directions is controlled by adjusting impulse levels N,while the amplitude regulation is implemented through modifications to the proportionality parameter r.Theoretical analyses,including Lyapunov exponents(LEs)and bifurcation diagrams,are conducted,confirming that the extended maps retain the intrinsic dynamics of five rational map classes.The field-programmable gate array(FPGA)implementation results are consistent with the numerical simulation results,verifying the correctness of the theoretical analysis.The method enables the expansion of unipolar attractors and enhances entropy metrics,offering a robust framework for applications in secure communication,encryption,and chaos-based technologies.
基金Project supported by the Basic Scientific Research Projects of Department of Education of Liaoning Province,China(Grant No.LJ212410152049)the Technological Innovation Projects in the field of artificial intelligence of Liaoning Province,China(Grant No.2023JH26/10300011)。
文摘Existing chaotic encryption schemes primarily focus on single types of images,making the design of hybrid image encryption schemes more suitable for practical applications.In this paper,a hyperchaotic map with a spherical attractor is proposed,which is constructed using spherical coordinates.Dynamical analyses reveal that the hyperchaotic map exhibits global hyperchaos and high complexity,making it capable of generating more complex chaotic sequences suitable for image encryption.A hybrid encryption scheme based on a hyperchaotic map is proposed for two-dimensional(2D)images,three-dimensional(3D)models,and 3D point clouds.Firstly,the pixels of 2D image and the coordinate data of 3D image are fused into a plaintext cube,which is combined with Hash-512 to obtain the initial value of the hyperchaotic map.Chaotic sequences are utilized for cube space internal confusion and dynamic cross-diffusion.The encrypted images demonstrate high information entropy,and the test results show that the encryption scheme effectively protects the images.The proposed hybrid image encryption scheme provides an efficient solution for securing various types of images.
基金Project supported by the National Nature Science Foundation of China(Grant Nos.51737003 and 51977060)the Natural Science Foundation of Hebei Province(Grant No.E2011202051).
文摘The neuron model has been widely employed in neural-morphic computing systems and chaotic circuits.This study aims to develop a novel circuit simulation of a three-neuron Hopfield neural network(HNN)with coupled hyperbolic memristors through the modification of a single coupling connection weight.The bistable mode of the hyperbolic memristive HNN(mHNN),characterized by the coexistence of asymmetric chaos and periodic attractors,is effectively demonstrated through the utilization of conventional nonlinear analysis techniques.These techniques include bifurcation diagrams,two-parameter maximum Lyapunov exponent plots,local attractor basins,and phase trajectory diagrams.Moreover,an encryption technique for color images is devised by leveraging the mHNN model and asymmetric structural attractors.This method demonstrates significant benefits in correlation,information entropy,and resistance to differential attacks,providing strong evidence for its effectiveness in encryption.Additionally,an improved modular circuit design method is employed to create the analog equivalent circuit of the memristive HNN.The correctness of the circuit design is confirmed through Multisim simulations,which align with numerical simulations conducted in Matlab.
基金supported by the Guizhou Province Natural Science Foundation(Qiankehe Fundamentals-ZK[2023]General-055)Guizhou Province Science and Technology Support Plan Project(Qiankehe Fundamentals[2023]General-465)。
文摘A method of generating multi-double scroll attractors is proposed based on the memristor Hopfield neural network(HNN)under pulse control.First,the original hyperbolic-type memristor is added to the neural network mathematical model,and the influence of this memristor on the dynamic behavior of the new HNN is analyzed.The numerical results show that after adding the memristor,the abundant dynamic behaviors such as chaos coexistence,period coexistence and chaos period coexistence can be observed when the initial value of the system is changed.Then the logic pulse is added to the external memristor.It is found that the equilibrium point of the HNN can multiply and generate multi-double scroll attractors after the pulse stimulation.When the number of logical pulses is changed,the number of multi-double scroll attractors will also change,so that the pulse can control the generation of multi-double scroll attractors.Finally,the HNN circuit under pulsed stimulation was realized by circuit simulation,and the results verified the correctness of the numerical results.
基金supported by the National Natural Science Foundation of China(11571283)supported by Natural Science Foundation of Guizhou Province
文摘A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynam- ical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.
基金supported by the National Natural Science Foundation of China under grant number 11971019.
文摘In this paper,we investigate the existence of strange nonchaotic attractors(SNAs)in a slender rigid rocking block under quasi-periodic forcing with two frequencies.We find that an SNA can exist between a quasi-periodic attractor and a chaotic attractor,or between two chaotic attractors.In particular,we demonstrate that a torus doubling bifurcation of a quasi-periodic attractor can result in SNAs via the fractal route before transforming into chaotic attractors.This phenomenon is rarely reported in quasiperiodically forced discontinuous differential equations and vibro-impact systems.The properties of SNAs are verified by the Lyapunov exponent,rational approximation,phase sensitivity,power spectrum,and separation of nearby trajectories.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 62366014 and 61961019)the Natural Science Foundation of Jiangxi Province, China (Grant No. 20232BAB202008)。
文摘This article proposes a non-ideal flux-controlled memristor with a bisymmetric sawtooth piecewise function, and a new multi-wing memristive chaotic system(MMCS) based on the memristor is generated. Compared with other existing MMCSs, the most eye-catching point of the proposed MMCS is that the amplitude of the wing will enlarge towards the poles as the number of wings increases. Diverse coexisting attractors are numerically found in the MMCS, including chaos,quasi-period, and stable point. The circuits of the proposed memristor and MMCS are designed and the obtained results demonstrate their validity and reliability.
文摘To prove the existence of the family of exponential attractors, we first define a family of compact, invariant absorbing sets <em>B<sub>k</sub></em>. Then we prove that the solution semigroup has Lipschitz property and discrete squeezing property. Finally, we obtain a family of exponential attractors and its estimation of dimension by combining them with previous theories. Next, we obtain Kirchhoff-type random equation by adding product white noise to the right-hand side of the equation. To study the existence of random attractors, firstly we transform the equation by using Ornstein-Uhlenbeck process. Then we obtain a family of bounded random absorbing sets via estimating the solution of the random differential equation. Finally, we prove the asymptotic compactness of semigroup of the stochastic dynamic system;thereby we obtain a family of random attractors.
基金Project supported by the National Natural Science Foundation of China (Grant No 10275053)
文摘Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.
文摘This paper presents the problem of generating four-wing （eight-wing） chaotic attractors. The adopted method consists in suitably coupling two （three） identical Lorenz systems. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings （eight wings） of these novel attractors axe located around the four （eight） equilibria with two （three） pairs of unstable complex-conjugate eigenvalues.
基金National Natural Science Foundation of China(Grant Nos.11672257,11632008,11772306,and 11972173)the Natural Science Foundation of Jiangsu Province of China(Grant No.BK20161314)+1 种基金the 5th 333 High-level Personnel Training Project of Jiangsu Province of China(Grant No.BRA2018324)the Excellent Scientific and Technological Innovation Team of Jiangsu University.
文摘We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.
文摘In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.
文摘In this paper,we introduce a new two-dimensional nonlinear oscillator with an infinite number of coexisting limit cycles.These limit cycles form a layer-by-layer structure which is very unusual.Forty percent of these limit cycles are self-excited attractors while sixty percent of them are hidden attractors.Changing this new system to its forced version,we introduce a new chaotic system with an infinite number of coexisting strange attractors.We implement this system through field programmable gate arrays.
基金supported by the National Natural Science Foundation of China(Grant No.11671149)
文摘This paper introduces a four-dimensional （4D） segmented disc dynamo which possesses coexisting hidden attractors with one stable equilibrium or a line equilibrium when parameters vary. In addition, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcation and pitchfork bifurcation occur in the system. The ultimate bound is also estimated. Some numerical investigations are also exploited to demonstrate and visualize the corresponding theoretical results.
文摘In this paper, we investigate the existence of random attractor for the random dynamical system generated by the Kirchhoff-type suspension bridge equations with strong damping and white noises. We first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the existence of the global attractors of the equation.
基金supported by China NSF(11271388)Scientific and Technological Research Program of Chongqing Municipal Education Commission(KJ1400430)Basis and Frontier Research Project of Chongqing(cstc2014jcyj A00035)
文摘We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div（o（x）↓△u） and multiplicative noises. Under some mild conditions on the diffusion variable o（x） and without any restriction on the upper growth p of nonlinearity, except that p 〉 2, we show the existences of random attractor in D0^1,2（DN, σ） space, where DN is an arbitrary （bounded or unbounded） domain in R^N N 〉 2. For this purpose, some abstract results based on the omega-limit compactness are established.
文摘In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.51177117 and 51307130)
文摘A novel 5-dimensional（5D） memristive chaotic system is proposed, in which multi-scroll hidden attractors and multiwing hidden attractors can be observed on different phase planes. The dynamical system has multiple lines of equilibria or no equilibrium when the system parameters are appropriately selected, and the multi-scroll hidden attractors and multi-wing hidden attractors have nothing to do with the system equilibria. Particularly, the numbers of multi-scroll hidden attractors and multi-wing hidden attractors are sensitive to the transient simulation time and the initial values. Dynamical properties of the system, such as phase plane, time series, frequency spectra, Lyapunov exponent, and Poincar′e map, are studied in detail. In addition, a state feedback controller is designed to select multiple hidden attractors within a long enough simulation time. Finally, an electronic circuit is realized in Pspice, and the experimental results are in agreement with the numerical ones.
文摘The aim of this paper is to provide explicitly a sequence of m-dimensional approximate inertial manifolds M(m,j,)j = 1,2,, for each positive integer m, for the Kuramoto-Sivashinsky equations. A very thin neighborhood into which the orbits enter with an exponential speed and in a finite time is associated with each manifold. The thickness of these neighborhoods decreases with increasing m for a fixed order j. Besides, the neighborhoods localize the global attractor and aid in the approximate computation of large-time solutions of the Kuramoto-Sivashinsky equations.
基金Sponsored by the National Natural Science Foundation of China(Grant No.61201227)the Funding of China Scholarship Council,the Natural Science the Foundation of Anhui Province(Grant No.1208085M F93)the 211 Innovation Team of Anhui University(Grant Nos.KJTD007A and KJTD001B)
文摘As an important research branch,memristor has attracted a range of scholars to study the property of memristive chaotic systems.Additionally,time⁃delayed systems are considered a significant and newly⁃developing field in modern research.By combining memristor and time⁃delay,a delayed memristive differential system with fractional order is proposed in this paper,which can generate hidden attractors.First,we discussed the dynamics of the proposed system where the parameter was set as the bifurcation parameter,and showed that with the increase of the parameter,the system generated rich chaotic phenomena such as bifurcation,chaos,and hypherchaos.Then we derived adequate and appropriate stability criteria to guarantee the system to achieve synchronization.Lastly,examples were provided to analyze and confirm the influence of parameter a,fractional order q,and time delayτon chaos synchronization.The simulation results confirm that the chaotic synchronization is affected by a,q andτ.
