63 research outputs found
Reduction of dimension for nonlinear dynamical systems
We consider reduction of dimension for nonlinear dynamical systems. We
demonstrate that in some cases, one can reduce a nonlinear system of equations
into a single equation for one of the state variables, and this can be useful
for computing the solution when using a variety of analytical approaches. In
the case where this reduction is possible, we employ differential elimination
to obtain the reduced system. While analytical, the approach is algorithmic,
and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}.
In other cases, the reduction cannot be performed strictly in terms of
differential operators, and one obtains integro-differential operators, which
may still be useful. In either case, one can use the reduced equation to both
approximate solutions for the state variables and perform chaos diagnostics
more efficiently than could be done for the original higher-dimensional system,
as well as to construct Lyapunov functions which help in the large-time study
of the state variables. A number of chaotic and hyperchaotic dynamical systems
are used as examples in order to motivate the approach.Comment: 16 pages, no figure
Coexistence of generalized synchronization and inverse generalized synchronization between chaotic and hyperchaotic systems
In this paper, we present new schemes to synchronize different dimensional chaotic and hyperchaotic systems. Based on coexistence of generalized synchronization (GS) and inverse generalized synchronization (IGS), a new type of hybrid chaos synchronization is constructed. Using Lyapunov stability theory and stability theory of linear continuous-time systems, some sufficient conditions are derived to prove the coexistence of generalized synchronization and inverse generalized synchronization between 3D master chaotic system and 4D slave hyperchaotic system. Finally, two numerical examples are illustrated with the aim to show the effectiveness of the approaches developed herein
Emergence of a common generalized synchronization manifold in network motifs of structurally different time-delay systems
We point out the existence of a transition from partial to global generalized
synchronization (GS) in symmetrically coupled structurally different time-delay
systems of different orders using the auxiliary system approach and the mutual
false nearest neighbor method. The present authors have recently reported that
there exists a common GS manifold even in an ensemble of structurally
nonidentical scalar time-delay systems with different fractal dimensions and
shown that GS occurs simultaneously with phase synchronization (PS). In this
paper we confirm that the above result is not confined just to scalar
one-dimensional time-delay systems alone but there exists a similar type of
transition even in the case of time-delay systems with different orders. We
calculate the maximal transverse Lyapunov exponent to evaluate the asymptotic
stability of the complete synchronization manifold of each of the main and the
corresponding auxiliary systems, which in turn ensures the stability of the GS
manifold between the main systems. Further we estimate the correlation
coefficient and the correlation of probability of recurrence to establish the
relation between GS and PS. We also calculate the mutual false nearest neighbor
parameter which doubly confirms the occurrence of the global GS manifold.Comment: 17 pages, 11 figures, Accepted for publication in Chaos Solitons
Fractal
A proposed lightweight image encryption using ChaCha with hyperchaotic maps
Image encryption plays a pivotal rule in enhancing telecommunications media. Since Privacy is necessary in our daily life in many areas, the personal image will be encrypted when it sent it over the Internet to the recipient to maintain privacy issue. In this paper, the image is encrypted using ChaCha symmetric stream cipher with Hyperchaotic Map. Due to the sensitivity characteristics of initial conditions, pseudo randomness chaotic maps and control parameters in chaotic, Hyperchaotic maps is use, higher security is obtained via using initial seed number, variance of parameters, and unpredictable direction of chaotic. The suggested lightweight image encryption has confirmed robustness contra brute force attacks by providing a massive key space. Furthermore, the suggested lightweight image encryption is eligible to defense from statistical cracking, insecurity of image based on criteria's histogram correlation and entropy
A Novel Four-Wing Hyperchaotic Complex System and Its Complex Modified Hybrid Projective Synchronization with Different Dimensions
We introduce a new Dadras system with complex variables
which can exhibit both four-wing hyperchaotic and chaotic attractors. Some dynamic properties of the system have been described including Lyapunov exponents, fractal dimensions, and Poincaré maps. More importantly,
we focus on a new type of synchronization method of modified hybrid project
synchronization with complex transformation matrix (CMHPS) for different
dimensional hyperchaotic and chaotic complex systems with complex parameters,
where the drive and response systems can be asymptotically synchronized
up to a desired complex transformation matrix, not a diagonal matrix. Furthermore,
CMHPS between the novel hyperchaotic Dadras complex system
and other two different dimensional complex chaotic systems is provided as
an example to discuss increased order synchronization and reduced order
synchronization, respectively. Numerical results verify the feasibility and effectiveness
of the presented schemes
Dynamical Systems
Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...
Symmetry in Chaotic Systems and Circuits
Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue
- …